In this paper, we derive variational inference upper-bounds on the log-partition function of a pairwize Markov random fields on the Boolean hypercube, based on quantum relaxations of the Kullback-Leibler divergence. We then propose an efficient algorithm to compute these bounds based on primal-dual optimization. An improvement of these bounds through the use of "hierarchies", similar to sum-of-squares (SoS) hierarchies is proposed, and we present a greedy algorithm to select among these relaxations. We carry extensive numerical experiments and compare with state-of-the-art methods for this inference problem.

Adversarial training can be used to learn models that are robust against perturbations. For linear models, it can be formulated as a convex optimization problem. Compared to methods proposed in the context of deep learning, leveraging the optimization structure allows significantly faster convergence rates. Still, the use of generic convex solvers can be inefficient for large-scale problems. Here, we propose tailored optimization algorithms for the adversarial training of linear models, which render large-scale regression and classification problems more tractable. For regression problems, we propose a family of solvers based on iterative ridge regression and, for classification, a family of solvers based on projected gradient descent. The methods are based on extended variable reformulations of the original problem. We illustrate their efficiency in numerical examples.

We consider linear recurrent neural networks, which have become a key building block of sequence modeling due to their ability for stable and effective long-range modeling. In this paper, we aim at characterizing this ability on the simple but core copy task, whose goal is to build a linear filter of order $S$ that approximates the filter that looks $K$ time steps in the past (which we refer to as the shift-$K$ filter), where $K$ is larger than $S$. Using classical signal models and quadratic cost, we fully characterize the problem by providing lower bounds of approximation, as well as explicit filters that achieve this lower bound up to constants. The optimal performance highlights an uncertainty principle for this task: the optimal filter has to average values around the $K$-th time step in the past with a range (width) that is proportional to $K/S$.

Gaussian smoothing combined with a probabilistic framework for denoising via the empirical Bayes formalism, i.e., the Tweedie-Miyasawa formula (TMF), are the two key ingredients in the success of score-based generative models in Euclidean spaces. Smoothing holds the key for easing the problem of learning and sampling in high dimensions, denoising is needed for recovering the original signal, and TMF ties these together via the score function of noisy data. In this work, we extend this paradigm to the problem of learning and sampling the distribution of binary data on the Boolean hypercube by adopting Bernoulli noise, instead of Gaussian noise, as a smoothing device. We first derive a TMF-like expression for the optimal denoiser for the Hamming loss, where a score function naturally appears. Sampling noisy binary data is then achieved using a Langevin-like sampler which we theoretically analyze for different noise levels. At high Bernoulli noise levels sampling becomes easy, akin to log-concave sampling in Euclidean spaces. In addition, we extend the sequential multi-measurement sampling of Saremi et al. (2024) to the binary setting where we can bring the "effective noise" down by sampling multiple noisy measurements at a fixed noise level, without the need for continuous-time stochastic processes. We validate our formalism and theoretical findings by experiments on synthetic data and binarized images.

As platforms increasingly rely on learning algorithms, collectives may form and seek ways to influence these platforms to align with their own interests. This can be achieved by coordinated submission of altered data. To evaluate the potential impact of such behavior, it is essential to understand the computations that collectives must perform to impact platforms in this way. In particular, collectives need to make a priori assessments of the effect of the collective before taking action, as they may face potential risks when modifying their data. Moreover they need to develop implementable coordination algorithms based on quantities that can be inferred from observed data. We develop a framework that provides a theoretical and algorithmic treatment of these issues and present experimental results in a product evaluation domain.

We show that learning-rate schedules for large model training behave surprisingly similar to a performance bound from non-smooth convex optimization theory. We provide a bound for the constant schedule with linear cooldown; in particular, the practical benefit of cooldown is reflected in the bound due to the absence of logarithmic terms. Further, we show that this surprisingly close match between optimization theory and practice can be exploited for learning-rate tuning: we achieve noticeable improvements for training 124M and 210M Llama-type models by (i) extending the schedule for continued training with optimal learning-rate, and (ii) transferring the optimal learning-rate across schedules.

We analyse the convergence of one-hidden-layer ReLU networks trained by gradient flow on $n$ data points. Our main contribution leverages the high dimensionality of the ambient space, which implies low correlation of the input samples, to demonstrate that a network with width of order $\log(n)$ neurons suffices for global convergence with high probability. Our analysis uses a Polyak–Łojasiewicz viewpoint along the gradient-flow trajectory, which provides an exponential rate of convergence of $\frac{1}{n}$. When the data are exactly orthogonal, we give further refined characterizations of the convergence speed, proving its asymptotic behavior lies between the orders $\frac{1}{n}$ and $\frac{1}{\sqrt{n}}$, and exhibiting a phase-transition phenomenon in the convergence rate, during which it evolves from the lower bound to the upper, and in a relative time of order $\frac{1}{\log(n)}$.

We introduce *Backward Conformal Prediction*, a method that guarantees conformal coverage while providing flexible control over the size of prediction sets. Unlike standard conformal prediction, which fixes the coverage level and allows the conformal set size to vary, our approach defines a rule that constrains how prediction set sizes behave based on the observed data, and adapts the coverage level accordingly. Our method builds on two key foundations: (i) recent results by Gauthier et al. [2025] on post-hoc validity using e-values, which ensure marginal coverage of the form $\mathbb{P}(Y_{\rm test} \in \hat C_n^{\tilde{\alpha}}(X_{\rm test})) \ge 1 - \mathbb{E}[\tilde{\alpha}]$ for any data-dependent miscoverage $\tilde{\alpha}$, and (ii) a novel leave-one-out estimator $\hat{\alpha}^{\rm LOO}$ of the marginal miscoverage $\mathbb{E}[\tilde{\alpha}]$ based on the calibration set, ensuring that the theoretical guarantees remain computable in practice. This approach is particularly useful in applications where large prediction sets are impractical such as medical diagnosis. We provide theoretical results and empirical evidence supporting the validity of our method, demonstrating that it maintains computable coverage guarantees while ensuring interpretable, well-controlled prediction set sizes.

Recent works have highlighted the optimization difficulties encountered by gradient descent in training the first and last layer of transformer-based language models, which are overcome by optimizers such as Adam. The problem appears linked to the heavy-tailed distribution of words in text data, where the frequency of the $k$th most frequent word $\pi_k$ is proportional to $1/k$, following Zipf's law. To better understand the impact of the data distribution on training performance, we study a linear bigram model for next-token prediction when the tokens follow a power-law $\pi_k \propto 1/k^\alpha$ parameterized by the exponent $\alpha$. We derive optimization scaling laws for deterministic gradient descent and sign descent as a proxy for Adam as a function of the power $\alpha \geq 0$. This setting differs from existing theoretical investigations in scaling laws which assume that the eigenvalues of the data decay as a power with power $\alpha > 1$. This assumption effectively makes the problem "finite dimensional" as most of the loss comes from a few of the largest eigencomponents. In comparison, we show that the problem is more difficult when the data have heavier tails. The case $\alpha = 1$ as found in text is ``worst-case'' for gradient descent, in that the number of iterations required to reach a small relative error scales almost linearly with dimension. While the performance of sign descent also depends on the dimension, for Zipf-distributed data the number of iterations scales only with the square-root of the dimension, leading to a large improvement over gradient descent for large vocabularies.

Adversarial training has emerged as a key technique to enhance model robustness against adversarial input perturbations. Many of the existing methods rely on computationally expensive min-max problems that limit their application in practice. We propose a novel formulation of adversarial training in reproducing kernel Hilbert spaces, shifting from input to feature-space perturbations. This reformulation enables the exact solution of inner maximization and efficient optimization. It also provides a regularized estimator that naturally adapts to the noise level and the smoothness of the underlying function. We establish conditions under which the feature-perturbed formulation is a relaxation of the original problem and propose an efficient optimization algorithm based on iterative kernel ridge regression. We provide generalization bounds that help to understand the properties of the method. We also extend the formulation to multiple kernel learning. Empirical evaluation shows good performance in both clean and adversarial settings.

Score-based generative models achieve state-of-the-art sampling performance by denoising a distribution perturbed by Gaussian noise. In this paper, we focus on a single deterministic denoising step, and compare the optimal denoiser for the quadratic loss, we name "full-denoising", to the alternative "half-denoising" introduced by Hyvärinen (2025). We show that looking at the performance in terms of distance between distributions tells a more nuanced story, with different assumptions on the data leading to very different conclusions. We prove that half-denoising is better than full-denoising for regular enough densities, while full-denoising is better for singular densities such as mixtures of Dirac measures or densities supported on a low-dimensional subspace. In the latter case, we prove that full-denoising can alleviate the curse of dimensionality under a linear manifold hypothesis.

Physics-informed machine learning typically integrates physical priors into the learning process by minimizing a loss function that includes both a data-driven term and a partial differential equation (PDE) regularization. Building on the formulation of the problem as a kernel regression task, we use Fourier methods to approximate the associated kernel, and propose a tractable estimator that minimizes the physics-informed risk function. We refer to this approach as physics-informed kernel learning (PIKL). This framework provides theoretical guarantees, enabling the quantification of the physical prior’s impact on convergence speed. We demonstrate the numerical performance of the PIKL estimator through simulations, both in the context of hybrid modeling and in solving PDEs. In particular, we show that PIKL can outperform physics-informed neural networks in terms of both accuracy and computation time. Additionally, we identify cases where PIKL surpasses traditional PDE solvers, particularly in scenarios with noisy boundary conditions.

We propose a new method for feature learning and function estimation in supervised learning via regularised empirical risk minimisation. Our approach considers functions as expectations of Sobolev functions over all possible one-dimensional projections of the data. This framework is similar to kernel ridge regression, where the kernel is E_w(k(B)(wx, wx')), with k(B)(a, b) := min(|a|, |b|)1_{ab>0} the Brownian kernel, and the distribution of the projections w is learnt. This can also be viewed as an infinite-width one-hidden layer neural network, optimising the first layer’s weights through gradient descent and explicitly adjusting the non-linearity and weights of the second layer. We introduce a gradient-based computational method for the estimator, called Brownian Kernel Neural Network (BKerNN), using particles to approximate the expectation, where the positive homogeneity of the Brownian kernel leads to improved robustness to local minima. Using Rademacher complexity, we show that BKerNN’s expected risk converges to the minimal risk with explicit high-probability rates of O(min((d/n)^1/2, n^−1/6)) (up to logarithmic factors). Numerical experiments confirm our optimisation intuitions, and BKerNN outperforms kernel ridge regression, and favourably compares to a one-hidden layer neural network with ReLU activations in various settings and real datasets.

Sampling from Gibbs distributions and computing their log-partition function are fundamental tasks in statistics, machine learning, and statistical physics. While efficient algorithms are known for log-concave densities, the worst-case non-log-concave setting necessarily suffers from the curse of dimensionality. For many numerical problems, the curse of dimensionality can be alleviated when the target function is smooth, allowing the exponent in the rate to improve linearly with the number of available derivatives. Recently, it has been shown that similarly fast convergence rates can be achieved by efficient optimization algorithms. Since optimization can be seen as the low-temperature limit of sampling from Gibbs distributions, we pose the question of whether similarly fast convergence rates can be achieved for non-log-concave sampling. We first study the information-based complexity of the sampling and log-partition estimation problems and show that the optimal rates for sampling and log-partition computation are sometimes equal and sometimes faster than for optimization. We then analyze various polynomial-time sampling algorithms, including an extension of a recent promising optimization approach, and find that they sometimes exhibit interesting behavior but no near-optimal rates. Our results also give further insights into the relation between sampling, log-partition, and optimization problems.

Greedy first-order methods, such as coordinate descent with Gauss-Southwell rule or matching pursuit, have become popular in optimization due to their natural tendency to propose sparse solutions and their refined convergence guarantees. In this work, we propose a principled approach to generating (regularized) matching pursuit algorithms adapted to the geometry of the problem at hand, as well as their convergence guarantees. Building on these results, we derive approximate convergence guarantees and describe a transition phenomenon in the convergence of (regularized) matching pursuit from underparametrized to overparametrized models.

Calibration of machine learning classifiers is necessary to obtain reliable and interpretable predictions, bridging the gap between model outputs and actual probabilities. One prominent technique, isotonic regression (IR), aims at calibrating binary classifiers by minimizing the cross entropy with respect to monotone transformations. IR acts as an adaptive binning procedure that is able to achieve a calibration error of zero but leaves open the issue of the effect on performance. We first prove that IR preserves the convex hull of the ROC curve—an essential performance metric for binary classifiers. This ensures that a classifier is calibrated while controlling for over-fitting of the calibration set. We then present a novel generalization of isotonic regression to accommodate classifiers with $K$-classes. Our method constructs a multidimensional adaptive binning scheme on the probability simplex, again achieving a multi-class calibration error equal to zero. We regularize this algorithm by imposing a form of monotony that preserves the $K$-dimensional ROC surface of the classifier. We show empirically that this general monotony criterion is effective in striking a balance between reducing cross entropy loss and avoiding over-fitting of the calibration set.

We study the training dynamics of a shallow neural network with quadratic activation functions and quadratic cost in a teacher-student setup. In line with previous works on the same neural architecture, the optimization is performed following the gradient flow on the population risk, where the average over data points is replaced by the expectation over their distribution, assumed to be Gaussian. We first derive convergence properties for the gradient flow and quantify the overparameterization that is necessary to achieve a strong signal recovery. Then, assuming that the teachers and the students at initialization form independent orthonormal families, we derive a high-dimensional limit for the flow and show that the minimal overparameterization is sufficient for strong recovery. We verify by numerical experiments that these results hold for more general initializations.

Physics-informed machine learning combines the expressiveness of data-based approaches with the interpretability of physical models. In this context, we consider a general regression problem where the empirical risk is regularized by a partial differential equation that quantifies the physical inconsistency. We prove that for linear differential priors, the problem can be formulated as a kernel regression task. Taking advantage of kernel theory, we derive convergence rates for the minimizer $\hat f_n$ of the regularized risk and show that $\hat f_n$ converges at least at the Sobolev minimax rate. However, faster rates can be achieved, depending on the physical error. This principle is illustrated with a one-dimensional example, supporting the claim that regularizing the empirical risk with physical information can be beneficial to the statistical performance of estimators.

We introduce a theoretical framework for sampling from unnormalized densities based on a smoothing scheme that uses an isotropic Gaussian kernel with a single fixed noise scale. We prove one can decompose sampling from a density (minimal assumptions made on the density) into a sequence of sampling from log-concave conditional densities via accumulation of noisy measurements with equal noise levels. Our construction is unique in that it keeps track of a history of samples, making it non-Markovian as a whole, but it is lightweight algorithmically as the history only shows up in the form of a running empirical mean of samples. Our sampling algorithm generalizes walk-jump sampling (Saremi & Hyvärinen, 2019). The "walk" phase becomes a (non-Markovian) chain of (log-concave) Markov chains. The "jump" from the accumulated measurements is obtained by empirical Bayes. We study our sampling algorithm quantitatively using the 2-Wasserstein metric and compare it with various Langevin MCMC algorithms. We also report a remarkable capacity of our algorithm to "tunnel" between modes of a distribution.

In this paper, we study the statistical and geometrical properties of the Kullback-Leibler divergence with kernel covariance operators (KKL) introduced by [Bach, 2022, Information Theory with Kernel Methods]. Unlike the classical Kullback-Leibler (KL) divergence that involves density ratios, the KKL compares probability distributions through covariance operators (embeddings) in a reproducible kernel Hilbert space (RKHS), and compute the Kullback-Leibler quantum divergence. This novel divergence hence shares parallel but different aspects with both the standard Kullback-Leibler between probability distributions and kernel embeddings metrics such as the maximum mean discrepancy. A limitation faced with the original KKL divergence is its inability to be defined for distributions with disjoint supports. To solve this problem, we propose in this paper a regularised variant that guarantees that divergence is well defined for all distributions. We derive bounds that quantify the deviation of the regularised KKL to the original one, as well as concentration bounds. In addition, we provide a closed-form expression for the regularised KKL, specifically applicable when the distributions consist of finite sets of points, which makes it implementable. Furthermore, we derive a Wasserstein gradient descent scheme of the KKL divergence in the case of discrete distributions, and study empirically its properties to transport a set of points to a target distribution.

Injecting noise within gradient descent has several desirable features, such as smoothing and regularizing properties. In this paper, we investigate the effects of injecting noise before computing a gradient step. We demonstrate that small perturbations can induce explicit regularization for simple models based on the L1-norm, group L1-norms, or nuclear norms. However, when applied to overparametrized neural networks with large widths, we show that the same perturbations can cause variance explosion. To overcome this, we propose using independent layer-wise perturbations, which provably allow for explicit regularization without variance explosion. Our empirical results show that these small perturbations lead to improved generalization performance compared to vanilla gradient descent.

Neural networks can be trained to solve regression problems by using gradient-based methods to minimize the square loss. However, practitioners often prefer to reformulate regression as a classification problem, observing that training on the cross entropy loss results in better performance. By focusing on two-layer ReLU networks, which can be fully characterized by measures over their feature space, we explore how the implicit bias induced by gradient-based optimization could partly explain the above phenomenon. We provide theoretical evidence that the regression formulation yields a measure whose support can differ greatly from that for classification, in the case of one-dimensional data. Our proposed optimal supports correspond directly to the features learned by the input layer of the network. The different nature of these supports sheds light on possible optimization difficulties the square loss could encounter during training, and we present empirical results illustrating this phenomenon.

Spectral clustering and diffusion maps are celebrated dimensionality reduction algorithms built on eigen-elements related to the diffusive structure of the data. The core of these procedures is the approximation of a Laplacian through a graph kernel approach, however this local average construction is known to be cursed by the high-dimension $d$. In this article, we build a different estimator of the Laplacian, via a reproducing kernel Hilbert spaces method, which adapts naturally to the regularity of the problem. We provide non-asymptotic statistical rates proving that the kernel estimator we build can circumvent the curse of dimensionality. Finally we discuss techniques (Nyström subsampling, Fourier features) that enable to reduce the computational cost of the estimator while not degrading its overall performance.

Mean-field theory is widely used in theoretical studies of neural networks. In this paper, we analyze the role of depth in the concentration of mean-field predictions for Gram matrices of hidden representations in deep multilayer perceptron (MLP) with batch normalization (BN) at initialization. It is postulated that the mean-field predictions suffer from layer-wise errors that amplify with depth. We demonstrate that BN avoids this error amplification with depth. When the chain of hidden representations is rapidly mixing, we establish a concentration bound for a mean-field model of Gram matrices. To our knowledge, this is the first concentration bound that does not become vacuous with depth for standard MLPs with a finite width.

We present an algorithm for minimizing an objective with hard-to-compute gradients by using a related, easier-to-access function as a proxy. Our algorithm is based on approximate proximal-point iterations on the proxy combined with relatively few stochastic gradients from the objective. When the difference between the objective and the proxy is $\delta$-smooth, our algorithm guarantees convergence at a rate matching stochastic gradient descent on a $\delta$-smooth objective, which can lead to substantially better sample efficiency. Our algorithm has many potential applications in machine learning, and provides a principled means of leveraging synthetic data, physics simulators, mixed public and private data, and more.

In this paper, we explore the structure of the penultimate Gram matrix in deep neural networks, which contains the pairwise inner products of outputs corresponding to a batch of inputs. In several architectures it has been observed that this Gram matrix becomes degenerate with depth at initialization, which dramatically slows training. Normalization layers, such as batch or layer normalization, play a pivotal role in preventing the rank collapse issue. Despite promising advances, the existing theoretical results do not extend to layer normalization, which is widely used in transformers, and can not quantitatively characterize the role of non-linear activations. To bridge this gap, we prove that layer normalization, in conjunction with activation layers, biases the Gram matrix of a multilayer perceptron towards the identity matrix at an exponential rate with depth at initialization. We quantify this rate using the Hermite expansion of the activation function.

State-of-the-art machine learning models can be vulnerable to very small input perturbations that are adversarially constructed. Adversarial training is an effective approach to defend against it. Formulated as a min-max problem, it searches for the best solution when the training data were corrupted by the worst-case attacks. Linear models are among the simple models where vulnerabilities can be observed and are the focus of our study. In this case, adversarial training leads to a convex optimization problem which can be formulated as the minimization of a finite sum. We provide a comparative analysis between the solution of adversarial training in linear regression and other regularization methods. Our main findings are that: (A) Adversarial training yields the minimum-norm interpolating solution in the overparameterized regime (more parameters than data), as long as the maximum disturbance radius is smaller than a threshold. And, conversely, the minimum-norm interpolator is the solution to adversarial training with a given radius. (B) Adversarial training can be equivalent to parameter shrinking methods (ridge regression and Lasso). This happens in the underparametrized region, for an appropriate choice of adversarial radius and zero-mean symmetrically distributed covariates. (C) For $\ell_\infty$-adversarial training---as in square-root Lasso---the choice of adversarial radius for optimal bounds does not depend on the additive noise variance. We confirm our theoretical findings with numerical examples.

We introduce a differentiable clustering method based on stochastic perturbations of minimum-weight spanning forests. This allows us to include clustering in end-to-end trainable pipelines, with efficient gradients. We show that our method performs well even in difficult settings, such as data sets with high noise and challenging geometries. We also formulate an ad hoc loss to efficiently learn from partial clustering data using this operation. We demonstrate its performance on several data sets for supervised and semi-supervised tasks.

We consider the analysis of probability distributions through their associated covariance operators from reproducing kernel Hilbert spaces. We show that the von Neumann entropy and relative entropy of these operators are intimately related to the usual notions of Shannon entropy and relative entropy, and share many of their properties. They come together with efficient estimation algorithms from various oracles on the probability distributions. We also consider product spaces and show that for tensor product kernels, we can define notions of mutual information and joint entropies, which can then characterize independence perfectly, but only partially conditional independence. We finally show how these new notions of relative entropy lead to new upper-bounds on log partition functions, that can be used together with convex optimization within variational inference methods, providing a new family of probabilistic inference methods.

In many areas of applied statistics and machine learning, generating an arbitrary number of inde- pendent and identically distributed (i.i.d.) samples from a given distribution is a key task. When the distribution is known only through evaluations of the density, current methods either scale badly with the dimension or require very involved implemen- tations. Instead, we take a two-step approach by first modeling the probability distribution and then sampling from that model. We use the recently introduced class of positive semi-definite (PSD) models which have been shown to be e

The foundational concept of Max-Margin in machine learning is ill-posed for output spaces with more than two labels such as in structured prediction. In this paper, we show that the Max-Margin loss can only be consistent to the classification task under highly restrictive assumptions on the discrete loss measuring the error between outputs. These conditions are satisfied by distances defined in tree graphs, for which we prove consistency, thus being the first losses shown to be consistent for Max-Margin beyond the binary setting. We finally address these limitations by correcting the concept of Max-Margin and introducing the Restricted-Max-Margin, where the maximization of the loss-augmented scores is maintained, but performed over a subset of the original domain. The resulting loss is also a generalization of the binary support vector machine and it is consistent under milder conditions on the discrete loss.

We consider potentially non-convex optimization problems, for which optimal rates of approximation depend on the dimension of the parameter space and the smoothness of the function to be optimized. In this paper, we propose an algorithm that achieves close to optimal a priori computational guarantees, while also providing a posteriori certificates of optimality. Our general formulation builds on infinite-dimensional sums-of-squares and Fourier analysis, and is instantiated on the minimization of periodic functions.

Injecting artificial noise into gradient descent (GD) is commonly employed to improve the performance of machine learning models. Usually, uncorrelated noise is used in such perturbed gradient descent (PGD) methods. It is, however, not known if this is optimal or whether other types of noise could provide better generalization performance. In this paper, we zoom in on the problem of correlating the perturbations of consecutive PGD steps. We consider a variety of objective functions for which we find that GD with anticorrelated perturbations ("Anti-PGD") generalizes significantly better than GD and standard (uncorrelated) PGD. To support these experimental findings, we also derive a theoretical analysis that demonstrates that Anti-PGD moves to wider minima, while GD and PGD remain stuck in suboptimal regions or even diverge. This new connection between anticorrelated noise and generalization opens the field to novel ways to exploit noise for training machine learning models.

Uncertainty sampling in active learning is heavily used in practice to reduce the annotation cost. However, there has been no wide consensus on the function to be used for uncertainty estimation in binary classification tasks and convergence guarantees of the corresponding active learning algorithms are not well understood. The situation is even more challenging for multi-category classification. In this work, we propose an efficient uncertainty estimator for binary classification which we also extend to multiple classes, and provide a non-asymptotic rate of convergence for our uncertainty sampling based active learning algorithm in both cases under no-noise conditions (i.e., linearly separable data). We also extend our analysis to the noisy case and provide theoretical guarantees for our algorithm under the influence of noise in the task of binary and multi-class classification.

Temporal-difference learning is a popular algorithm for policy evaluation. In this paper, we study the convergence of the regularized non-parametric TD(0) algorithm, in both the independent and Markovian observation settings. In particular, when TD is performed in a universal reproducing kernel Hilbert space (RKHS), we prove convergence of the averaged iterates to the optimal value function, even when it does not belong to the RKHS. We provide explicit convergence rates that depend on a source condition relating the regularity of the optimal value function to the RKHS. We illustrate this convergence numerically on a simple continuous-state Markov reward process.

The workhorse of machine learning is stochastic gradient descent.To access stochastic gradients, it is common to consider iteratively input/output pairs of a training dataset.Interestingly, it appears that one does not need full supervision to access stochastic gradients, which is the main motivation of this paper.After formalizing the "active labeling" problem, which focuses on active learning with partial supervision, we provide a streaming technique that provably minimizes the ratio of generalization error over the number of samples.We illustrate our technique in depth for robust regression.

We consider the problem of minimizing the sum of two convex functions. One of those functions has Lipschitz-continuous gradients, and can be accessed via stochastic oracles, whereas the other is ``simple''. We provide a Bregman-type algorithm with accelerated convergence in function values to a ball containing the minimum. The radius of this ball depends on problem-dependent constants, including the variance of the stochastic oracle. We further show that this algorithmic setup naturally leads to a variant of Frank-Wolfe achieving acceleration under parallelization. More precisely, when minimizing a smooth convex function on a bounded domain, we show that one can achieve an $\epsilon$ primal-dual gap (in expectation) in $\tilde{O}(1 /\sqrt{\epsilon})$ iterations, by only accessing gradients of the original function and a linear maximization oracle with $O(1 / \sqrt{\epsilon})$ computing units in parallel. We illustrate this fast convergence on synthetic numerical experiments.

Along with Markov chain Monte Carlo (MCMC) methods, variational inference (VI) has emerged as a central computational approach to large-scale Bayesian inference. Rather than sampling from the true posterior $\pi$, VI aims at producing a simple but effective approximation $\hat \pi$ to $\pi$ for which summary statistics are easy to compute. However, unlike the well-studied MCMC methodology, algorithmic guarantees for VI are still relatively less well-understood. In this work, we propose principled methods for VI, in which $\hat \pi$ is taken to be a Gaussian or a mixture of Gaussians, which rest upon the theory of gradient flows on the Bures--Wasserstein space of Gaussian measures. Akin to MCMC, it comes with strong theoretical guarantees when $\pi$ is log-concave.

Studying the properties of stochastic noise to optimize complex non-convex functions has been an active area of research in the field of machine learning. Prior work~\citep{zhou2019pgd, wei2019noise} has shown that the noise of stochastic gradient descent improves optimization by overcoming undesirable obstacles in the landscape. Moreover, injecting artificial Gaussian noise has become a popular idea to quickly escape saddle points. Indeed, in the absence of reliable gradient information, the noise is used to explore the landscape, but it is unclear what type of noise is optimal in terms of exploration ability. In order to narrow this gap in our knowledge, we study a general type of continuous-time non-Markovian process, based on fractional Brownian motion, that allows for the increments of the process to be correlated. This generalizes processes based on Brownian motion, such as the Ornstein-Uhlenbeck process. We demonstrate how to discretize such processes which gives rise to the new algorithm ``fPGD''. This method is a generalization of the known algorithms PGD and Anti-PGD~\citep{orvieto2022anti}. We study the properties of fPGD both theoretically and empirically, demonstrating that it possesses exploration abilities that, in some cases, are favorable over PGD and Anti-PGD. These results open the field to novel ways to exploit noise for training machine learning models.

The existing analysis of asynchronous stochastic gradient descent (SGD) degrades dramatically when any delay is large, giving the impression that performance depends primarily on the delay. On the contrary, we prove much better guarantees for the same asynchronous SGD algorithm regardless of the delays in the gradients, depending instead just on the number of parallel devices used to implement the algorithm. Our guarantees are strictly better than the existing analyses, and we also argue that asynchronous SGD outperforms synchronous minibatch SGD in the settings we consider. For our analysis, we introduce a novel recursion based on ``virtual iterates'' and delay-adaptive stepsizes, which allow us to derive state-of-the-art guarantees for both convex and non-convex objectives.

Abstract We propose AriaNN, a low-interaction privacy-preserving framework for private neural network training and inference on sensitive data. Our semi-honest 2-party computation protocol (with a trusted dealer) leverages function secret sharing, a recent lightweight cryptographic protocol that allows us to achieve an efficient online phase. We design optimized primitives for the building blocks of neural networks such as ReLU, MaxPool and BatchNorm. For instance, we perform private comparison for ReLU operations with a single message of the size of the input during the online phase, and with preprocessing keys close to 4× smaller than previous work. Last, we propose an extension to support n-party private federated learning. We implement our framework as an extensible system on top of PyTorch that leverages CPU and GPU hardware acceleration for cryptographic and machine learning operations. We evaluate our end-to-end system for private inference between distant servers on standard neural networks such as AlexNet, VGG16 or ResNet18, and for private training on smaller networks like LeNet. We show that computation rather than communication is the main bottleneck and that using GPUs together with reduced key size is a promising solution to overcome this barrier.

We consider stochastic gradient methods under the interpolation regime where a perfect fit can be obtained (minimum loss at each observation). While previous work highlighted the implicit regularization of such algorithms, we consider an explicit regularization framework as a minimum Bregman divergence convex feasibility problem. Using convex duality, we propose randomized Dykstra-style algorithms based on randomized dual coordinate ascent. For non-accelerated coordinate descent, we obtain an algorithm which bears strong similarities with (non-averaged) stochastic mirror descent on specific functions, as it is equivalent for quadratic objectives, and equivalent in the early iterations for more general objectives. It comes with the benefit of an explicit convergence theorem to a minimum norm solution. For accelerated coordinate descent, we obtain a new algorithm that has better convergence properties than existing stochastic gradient methods in the interpolating regime. This leads to accelerated versions of the perceptron for generic $\ell_p$-norm regularizers, which we illustrate in experiments.

Discrete supervised learning problems such as classification are often tackled by introducing a continuous surrogate problem akin to regression. Bounding the original error, between estimate and solution, by the surrogate error endows discrete problems with convergence rates already shown for continuous instances. Yet, current approaches do not leverage the fact that discrete problems are essentially predicting a discrete output when continuous problems are predicting a continuous value. In this paper, we tackle this issue for general structured prediction problems, opening the way to “super fast” rates, that is, convergence rates for the excess risk faster than $n^{-1}$, where $n$ is the number of observations, with even exponential rates with the strongest assumptions. We first illustrate it for predictors based on nearest neighbors, generalizing rates known for binary classification to any discrete problem within the framework of structured prediction. We then consider kernel ridge regression where we improve known rates in $n^{-1/4}$ to arbitrarily fast rates, depending on a parameter characterizing the hardness of the problem, thus allowing, under smoothness assumptions, to bypass the curse of dimensionality.

It is well-known that plug-in statistical estimation of optimal transport suffers from the curse of dimensionality. Despite recent efforts to improve the rate of estimation with the smoothness of the problem, the computational complexity of these recently proposed methods still degrade exponentially with the dimension. In this paper, thanks to an infinite-dimensional sum-of-squares representation, we derive a statistical estimator of smooth optimal transport which achieves a precision $\varepsilon$ from $\tilde{O}(\varepsilon^{-2})$ independent and identically distributed samples from the distributions, for a computational cost of $\tilde{O}(\varepsilon^{-4})$ when the smoothness increases, hence yielding dimension-free statistical \emph{and} computational rates, with potentially exponentially dimension-dependent constants.

Machine learning approached through supervised learning requires expensive annotation of data. This motivates weakly supervised learning, where data are annotated with incomplete yet discriminative information. In this paper, we focus on partial labelling, an instance of weak supervision where, from a given input, we are given a set of potential targets. We review a disambiguation principle to recover full supervision from weak supervision, and propose an empirical disambiguation algorithm. We prove exponential convergence rates of our algorithm under classical learnability assumptions, and we illustrate the usefulness of our method on practical examples.

As annotations of data can be scarce in large-scale practical problems, leveraging unlabelled examples is one of the most important aspects of machine learning. This is the aim of semi-supervised learning. To benefit from the access to unlabelled data, it is natural to diffuse smoothly knowledge of labelled data to unlabelled one. This induces to the use of Laplacian regularization. Yet, current implementations of Laplacian regularization suffer from several drawbacks, notably the well-known curse of dimensionality. In this paper, we design a new class of algorithms overcoming this issue, unveiling a large body of spectral filtering methods. Additionally, we provide a statistical analysis showing that our estimators exhibit desirable behaviors. They are implemented through (reproducing) kernel methods, for which we provide realistic computational guidelines in order to make our method usable with large amounts of data.

This paper underlines an elegant property of batch-normalization (BN): Successive batch normalizations with random linear updates make samples increasingly orthogonal. We establish a non-asymptotic characterization of the interplay between depth, width, and the orthogonality of deep representations. More precisely, we prove, under a mild assumption, the deviation of the representations from orthogonality rapidly decays with depth up to a term inversely proportional to the network width. This result has two main theoretical and practical implications: 1) Theoretically, as the depth grows, the distribution of the outputs contracts to a Wasserstein-2 ball around an isotropic normal distribution. Furthermore, the radius of this Wasserstein ball shrinks with the width of the network. 2) Practically, the orthogonality of the representations directly influences the performance of stochastic gradient descent (SGD). When representations are initially aligned, we observe SGD wastes many iterations to disentangle representations before the classification. Nevertheless, we experimentally show that starting optimization from orthogonal representations is sufficient to accelerate SGD, with no need for BN.

We introduce the ``continuized'' Nesterov acceleration, a close variant of Nesterov acceleration whose variables are indexed by a continuous time parameter. The two variables continuously mix following a linear ordinary differential equation and take gradient steps at random times. This continuized variant benefits from the best of the continuous and the discrete frameworks: as a continuous process, one can use differential calculus to analyze convergence and obtain analytical expressions for the parameters; but a discretization of the continuized process can be computed exactly with convergence rates similar to those of Nesterov original acceleration. We show that the discretization has the same structure as Nesterov acceleration, but with random parameters. We provide continuized Nesterov acceleration under deterministic as well as stochastic gradients, with either additive or multiplicative noise. Finally, using our continuized framework and expressing the gossip averaging problem as the stochastic minimization of a certain energy function, we provide the first rigorous acceleration of asynchronous gossip algorithms.

Poincaré inequalities are ubiquitous in probability and analysis and have various applications in statistics (concentration of measure, rate of convergence of Markov chains). The Poincaré constant, for which the inequality is tight, is related to the typical convergence rate of diffusions to their equilibrium measure. In this paper, we show both theoretically and experimentally that, given sufficiently many samples of a measure, we can estimate its Poincaré constant. As a by-product of the estimation of the Poincaré constant, we derive an algorithm that captures a low dimensional representation of the data by finding directions which are difficult to sample. These directions are of crucial importance for sampling or in fields like molecular dynamics, where they are called reaction coordinates. Their knowledge can leverage, with a simple conditioning step, computational bottlenecks by using importance sampling techniques.

Neural networks trained to minimize the logistic (a.k.a. cross-entropy) loss with gradient-based methods are observed to perform well in many supervised classification tasks. Towards understanding this phenomenon, we analyze the training and generalization behavior of infinitely wide two-layer neural networks with homogeneous activations. We show that the limits of the gradient flow on exponentially tailed losses can be fully characterized as a max-margin classifier in a certain non-Hilbertian space of functions. In presence of hidden low-dimensional structures, the resulting margin is independent of the ambiant dimension, which leads to strong generalization bounds. In contrast, training only the output layer implicitly solves a kernel support vector machine, which a priori does not enjoy such an adaptivity. Our analysis of training is non-quantitative in terms of running time but we prove computational guarantees in simplified settings by showing equivalences with online mirror descent. Finally, numerical experiments suggest that our analysis describes well the practical behavior of two-layer neural networks with ReLU activation and confirm the statistical benefits of this implicit bias.

Optimal transport is a foundational problem in optimization, that allows to compare probability distributions while taking into account geometric aspects. Its optimal objective value, the Wasserstein distance, provides an important loss between distributions that has been used in many applications throughout machine learning and statistics. Recent algorithmic progress on this problem and its regularized versions have made these tools increasingly popular. However, existing techniques require solving an optimization problem to obtain a single gradient of the loss, thus slowing down first-order methods to minimize the sum of losses, that require many such gradient computations. In this work, we introduce an algorithm to solve a regularized version of this problem of Wasserstein estimators, with a time per step which is sublinear in the natural dimensions of the problem. We introduce a dual formulation, and optimize it with stochastic gradient steps that can be computed directly from samples, without solving additional optimization problems at each step. Doing so, the estimation and computation tasks are performed jointly. We show that this algorithm can be extended to other tasks, including estimation of Wasserstein barycenters. We provide theoretical guarantees and illustrate the performance of our algorithm with experiments on synthetic data.

Annotating datasets is one of the main costs in nowadays supervised learning. The goal of weak supervision is to enable models to learn using only forms of labelling which are cheaper to collect, as partial labelling. This is a type of incomplete annotation where, for each datapoint, supervision is cast as a set of labels containing the real one. The problem of supervised learning with partial labelling has been studied for specific instances such as classification, multi-label, ranking or segmentation, but a general framework is still missing. This paper provides a unified framework based on structured prediction and on the concept of \emph{infimum loss} to deal with partial labelling over a wide family of learning problems and loss functions. The framework leads naturally to explicit algorithms that can be easily implemented and for which proved statistical consistency and learning rates. Experiments confirm the superiority of the proposed approach over commonly used baselines.

We consider the setting of distributed empirical risk minimization where multiple machines compute the gradients in parallel and a centralized server updates the model parameters. In order to reduce the number of communications required to reach a given accuracy, we propose a preconditioned accelerated gradient method where the preconditioning is done by solving a local optimization problem over a subsampled dataset at the server. The convergence rate of the method depends on the square root of the relative condition number between the global and local loss functions. We estimate the relative condition number for linear prediction models by studying uniform concentration of the Hessians over a bounded domain, which allows us to derive improved convergence rates for existing preconditioned gradient methods and our accelerated method. Experiments on real-world datasets illustrate the benefits of acceleration in the ill-conditioned regime.

Max-margin methods for binary classification such as the support vector machine (SVM) have been extended to the structured prediction setting under the name of max-margin Markov networks ($M^3N$), or more generally structural SVMs. Unfortunately, these methods are statistically inconsistent when the relationship between inputs and labels is far from deterministic. We overcome such limitations by defining the learning problem in terms of a “max-min” margin formulation, naming the resulting method max-min margin Markov networks ($M^4N$). We prove consistency and finite sample generalization bounds for $M^4N$ and provide an explicit algorithm to compute the estimator. The algorithm achieves a generalization error of $O(1/\sqrt{n})$ for a total cost of $O(n)$ projection-oracle calls (which have at most the same cost as the max-oracle from $M^3N$). Experiments on multi-class classification, ordinal regression, sequence prediction and ranking demonstrate the effectiveness of the proposed method.

Subquadratic norms have been studied recently in the context of structured sparsity, which has been shown to be more beneficial than conventional regularizers in applications such as image denoising, compressed sensing, banded covariance estimation, etc. While existing works have been successful in learning structured sparse models such as trees, graphs, their associated optimization procedures have been inefficient because of hard-to-evaluate proximal operators of the norms. In this paper, we study the computational aspects of learning with subquadratic norms in a general setup. Our main contributions are two proximal-operator based algorithms ADMM-η and CP-η, which generically apply to these learning problems with convex loss functions, and achieve a proven rate of convergence of O(1/T) after T iterations. These algorithms are derived in a primal-dual framework, which have not been examined for subquadratic norms. We illustrate the efficiency of the algorithms developed in the context of tree-structured sparsity, where they comprehensively outperform relevant baselines.

Machine learning pipelines often rely on optimizers procedures to make discrete decisions (e.g., sorting, picking closest neighbors, or shortest paths). Although these discrete decisions are easily computed in a forward manner, they break the back-propagation of computational graphs. In order to expand the scope of learning problems that can be solved in an end-to-end fashion, we propose a systematic method to transform optimizers into operations that are differentiable and never locally constant. Our approach relies on stochastically perturbed optimizers, and can be used readily within existing solvers. Their derivatives can be evaluated efficiently, and smoothness tuned via the chosen noise amplitude. We also show how this framework can be connected to a family of losses developed in structured prediction, and give theoretical guarantees for their use in learning tasks. We demonstrate experimentally the performance of our approach on various tasks.

In the context of statistical supervised learning, the noiseless linear model assumes that there exists a deterministic linear relation $Y = \langle \theta_*, \Phi(U) \rangle$ between the random output $Y$ and the random feature vector $\Phi(U)$, a potentially non-linear transformation of the inputs~$U$. We analyze the convergence of single-pass, fixed step-size stochastic gradient descent on the least-square risk under this model. The convergence of the iterates to the optimum $\theta_*$ and the decay of the generalization error follow polynomial convergence rates with exponents that both depend on the regularities of the optimum $\theta_*$ and of the feature vectors $\Phi(U)$. We interpret our result in the reproducing kernel Hilbert space framework. As a special case, we analyze an online algorithm for estimating a real function on the unit hypercube from the noiseless observation of its value at randomly sampled points; the convergence depends on the Sobolev smoothness of the function and of a chosen kernel. Finally, we apply our analysis beyond the supervised learning setting to obtain convergence rates for the averaging process (a.k.a. gossip algorithm) on a graph depending on its spectral dimension.

Randomly initialized neural networks are known to become harder to train with increasing depth, unless architectural enhancements like residual connections and batch normalization are used. We here investigate this phenomenon by revisiting the connection between random initialization in deep networks and spectral instabilities in products of random matrices. Given the rich literature on random matrices, it is not surprising to find that the rank of the intermediate representations in unnormalized networks collapses quickly with depth. In this work we highlight the fact that batch normalization is an effective strategy to avoid rank collapse for both linear and ReLU networks. Leveraging tools from Markov chain theory, we derive a meaningful lower rank bound in deep linear networks. Empirically, we also demonstrate that this rank robustness generalizes to ReLU nets. Finally, we conduct an extensive set of experiments on real-world data sets, which confirm that rank stability is indeed a crucial condition for training modern-day deep neural architectures.

We consider the problem of training machine learning models on distributed data in a decentralized way. For finite-sum problems, fast single-machine algorithms for large datasets rely on stochastic updates combined with variance reduction. Yet, existing decentralized stochastic algorithms either do not obtain the full speedup allowed by stochastic updates, or require oracles that are more expensive than regular gradients. In this work, we introduce a Decentralized stochastic algorithm with Variance Reduction called DVR. DVR only requires computing stochastic gradients of the local functions, and is computationally as fast as a standard stochastic variance-reduced algorithms run on a $1/n$ fraction of the dataset, where $n$ is the number of nodes. To derive DVR, we use Bregman coordinate descent on a well-chosen dual problem, and obtain a dual-free algorithm using a specific Bregman divergence. We give an accelerated version of DVR based on the Catalyst framework, and illustrate its effectiveness with simulations on real data.

Linear models have shown great effectiveness and flexibility in many fields such as machine learning, signal processing and statistics. They can represent rich spaces of functions while preserving the convexity of the optimization problems where they are used, and are simple to evaluate, differentiate and integrate. However, for modeling non-negative functions, which are crucial for unsupervised learning, density estimation, or non-parametric Bayesian methods, linear models are not applicable directly. Moreover, current state-of-the-art models like generalized linear models either lead to non-convex optimization problems, or cannot be easily integrated. In this paper we provide the first model for non-negative functions which benefits from the same good properties of linear models. In particular, we prove that it admits a representer theorem and provide an efficient dual formulation for convex problems. We study its representation power, showing that the resulting space of functions is strictly richer than that of generalized linear models. Finally we extend the model and the theoretical results to functions with outputs in convex cones. The paper is complemented by an experimental evaluation of the model showing its effectiveness in terms of formulation, algorithmic derivation and practical results on the problems of density estimation, regression with heteroscedastic errors, and multiple quantile regression.

Stochastic optimization lies at the heart of machine learning, and its cornerstone is stochastic gradient descent (SGD), a method introduced over 60 years ago. The last eight years have seen an exciting new development: variance reduction for stochastic optimization methods. These variance-reduced (VR) methods excel in settings where more than one pass through the training data is allowed, achieving a faster convergence than SGD in theory and practice. These speedups underline the surge of interest in VR methods and the fast-growing body of work on this topic. This review covers the key principles and main developments behind VR methods for optimization with finite data sets and is aimed at nonexpert readers. We focus mainly on the convex setting and leave pointers to readers interested in extensions for minimizing nonconvex functions.

Independent component analysis (ICA) is a widespread data exploration technique, where observed signals are modeled as linear mixtures of independent components. From a machine learning point of view, it amounts to a matrix factorization problem with a statistical independence criterion. Infomax is one of the most used ICA algorithms. It is based on a loss function which is a non-convex log-likelihood. We develop a new majorization-minimization framework adapted to this loss function. We derive an online algorithm for the streaming setting, and an incremental algorithm for the finite sum setting, with the following benefits. First, unlike most algorithms found in the literature, the proposed methods do not rely on any critical hyper-parameter like a step size, nor do they require a line-search technique. Second, the algorithm for the finite sum setting, although stochastic, guarantees a decrease of the loss function at each iteration. Experiments demonstrate progress on the state-of-the-art for large scale datasets, without the necessity for any manual parameter tuning.

Optimal transport (OT) and maximum mean discrepancies (MMD) are now routinely used in machine learning to compare probability measures. We focus in this paper on Sinkhorn divergences (SDs), a regularized variant of OT distances which can interpolate, depending on the regularization strength $\varepsilon$, between OT ($\varepsilon=0$) and MMD ($\varepsilon=\infty$). Although the tradeoff induced by that regularization is now well understood computationally (OT, SDs and MMD require respectively $O(n^3\log n)$, $O(n^2)$ and $n^2$ operations given a sample size $n$), much less is known in terms of their sample complexity, namely the gap between these quantities, when evaluated using finite samples vs. their respective densities. Indeed, while the sample complexity of OT and MMD stand at two extremes, $1/n^{1/d}$ for OT in dimension $d$ and $1/\sqrt{n}$ for MMD, that for SDs has only been studied empirically. In this paper, we (i) derive a bound on the approximation error made with SDs when approximating OT as a function of the regularizer $\varepsilon$, (ii) prove that the optimizers of regularized OT are bounded in a Sobolev (RKHS) ball independent of the two measures and (iii) provide the first sample complexity bound for SDs, obtained,by reformulating SDs as a maximization problem in a RKHS. We thus obtain a scaling in $1/\sqrt{n}$ (as in MMD), with a constant that depends however on $\varepsilon$, making the bridge between OT and MMD complete.

In this paper, we study the problem of minimizing a sum of smooth and strongly convex functions split over the nodes of a network in a decentralized fashion. We propose the algorithm ESDACD, a decentralized accelerated algorithm that only requires local synchrony. Its rate depends on the condition number $\kappa$ of the local functions as well as the network topology and delays. Under mild assumptions on the topology of the graph, ESDACD takes a time $O((\tau_{\max} + \Delta_{\max})\sqrt{{\kappa}/{\gamma}}\ln(\epsilon^{-1}))$ to reach a precision $\epsilon$ where $\gamma$ is the spectral gap of the graph, $\tau_{\max}$ the maximum communication delay and $\Delta_{\max}$ the maximum computation time. Therefore, it matches the rate of SSDA, which is optimal when $\tau_{\max} = \Omega\left(\Delta_{\max}\right)$. Applying ESDACD to quadratic local functions leads to an accelerated randomized gossip algorithm of rate $O( \sqrt{\theta_{\rm gossip}/n})$ where $\theta_{\rm gossip}$ is the rate of the standard randomized gossip. To the best of our knowledge, it is the first asynchronous algorithm with a provably improved rate of convergence of the second moment of the error. We illustrate these results with experiments in idealized settings.

The problem of devising learning strategies for discrete losses (e.g., multilabeling, ranking) is currently addressed with methods and theoretical analyses ad-hoc for each loss. In this paper we study a least-squares framework to systematically design learning algorithms for discrete losses, with quantitative characterizations in terms of statistical and computational complexity. In particular, we improve existing results by providing explicit dependence on the number of labels for a wide class of losses and faster learning rates in conditions of low-noise. Theoretical results are complemented with experiments on real datasets, showing the effectiveness of the proposed general approach.

We present a novel algorithm for overcomplete independent components analysis (ICA), where the number of latent sources k exceeds the dimension p of observed variables. Previous algorithms either suffer from high computational complexity or make strong assumptions about the form of the mixing matrix. Our algorithm does not make any sparsity assumption yet enjoys favorable computational and theoretical properties. Our algorithm consists of two main steps: (a) estimation of the Hessians of the cumulant generating function (as opposed to the fourth and higher order cumulants used by most algorithms) and (b) a novel semi-definite programming (SDP) relaxation for recovering a mixing component. We show that this relaxation can be efficiently solved with a projected accelerated gradient descent method, which makes the whole algorithm computationally practical. Moreover, we conjecture that the proposed program recovers a mixing component at the rate $k < p^2/4$ and prove that a mixing component can be recovered with high probability when $k <(2 - \epsilon)p\log p$ when the original components are sampled uniformly at random on the hyper sphere. Experiments are provided on synthetic data and the CIFAR-10 dataset of real images.

Modern machine learning focuses on highly expressive models that are able to fit or interpolate the data completely, resulting in zero training loss. For such models, we show that the stochastic gradients of common loss functions satisfy a strong growth condition. Under this condition, we prove that constant step-size stochastic gradient descent (SGD) with Nesterov acceleration matches the convergence rate of the deterministic accelerated method for both convex and strongly-convex functions. We also show that this condition implies that SGD can find a first-order stationary point as efficiently as full gradient descent in non-convex settings. Under interpolation, we further show that all smooth loss functions with a finite-sum structure satisfy a weaker growth condition. Given this weaker condition, we prove that SGD with a constant step-size attains the deterministic convergence rate in both the strongly-convex and convex settings. Under additional assumptions, the above results enable us to prove an $O(1/k^2)$ mistake bound for $k$ iterations of a stochastic perceptron algorithm using the squared-hinge loss. Finally, we validate our theoretical findings with experiments on synthetic and real datasets.

We consider variational inequalities coming from monotone operators, a setting that includes convex minimization and convex-concave saddle-point problems. We assume an access to potentially noisy unbiased values of the monotone operators and assess convergence through a compatible gap function which corresponds to the standard optimality criteria in the aforementioned subcases. We present a universal algorithm for these inequalities based on the Mirror-Prox algorithm. Concretely, our algorithm \emph{simultaneously} achieves the optimal rates for the smooth/non-smooth, and noisy/noiseless settings. This is done without any prior knowledge of these properties, and in the general set-up of arbitrary norms and compatible Bregman divergences. For convex minimization and convex-concave saddle-point problems, this leads to new adaptive algorithms. Our method relies on a novel yet simple adaptive choice of the step-size, which can be seen as the appropriate extension of AdaGrad to handle constrained problems.

We consider learning methods based on the regularization of a convex empirical risk by a squared Hilbertian norm, a setting that includes linear predictors and non-linear predictors through positive-definite kernels. In order to go beyond the generic analysis leading to convergence rates of the excess risk as $O(1/\sqrt{n})$ from $n$ observations, we assume that the individual losses are self-concordant, that is, their third-order derivatives are bounded by their second-order derivatives. This setting includes least-squares, as well as all generalized linear models such as logistic and softmax regression. For this class of losses, we provide a bias-variance decomposition and show that the assumptions commonly made in least-squares regression, such as the source and capacity conditions, can be adapted to obtain fast non-asymptotic rates of convergence by improving the bias terms, the variance terms or both.

We provide a novel computer-assisted technique for systematically analyzing first-order methods for optimization. In contrast with previous works, the approach is particularly suited for handling sublinear convergence rates and stochastic oracles. The technique relies on semidefinite programming and potential functions. It allows simultaneously obtaining worst-case guarantees on the behavior of those algorithms, and assisting in choosing appropriate parameters for tuning their worst-case performances. The technique also benefits from comfortable tightness guarantees, meaning that unsatisfactory results can be improved only by changing the setting. We use the approach for analyzing deterministic and stochastic first-order methods under different assumptions on the nature of the stochastic noise. Among others, we treat unstructured noise with bounded variance, different noise models arising in over-parametrized expectation minimization problems, and randomized block-coordinate descent schemes.

Learning with complete or partial supervision is power- ful but relies on ever-growing human annotation efforts. As a way to mitigate this serious problem, as well as to serve specific applications, unsupervised learning has emerged as an important field of research. In computer vision, unsu- pervised learning comes in various guises. We focus here on the unsupervised discovery and matching of object cate- gories among images in a collection, following the work of Cho et al. [12]. We show that the original approach can be reformulated and solved as a proper optimization problem. Experiments on several benchmarks establish the merit of our approach.

In this paper, we consider the estimation of hyperparameters for regularization terms commonly used for obtaining structured sparse parameters in signal estimation problems, such as signal denoising. By considering the convex regularization terms as negative log-densities, we propose approximate maximum likelihood estimation for estimating parameters for continuous log-supermodular distributions, which is a key property that many sparse priors have. We then show how "perturb-and-MAP" ideas based on the Gumbel distribution and efficient discretization can be used to approximate the log-partition function for these models, which is a crucial step for approximate maximum likelihood estimation. We illustrate our estimation procedure on a set of experiments with flow-based priors and signal denoising.

The Sinkhorn "distance," a variant of the Wasserstein distance with entropic regularization, is an increasingly popular tool in machine learning and statistical inference. However, the time and memory requirements of standard algorithms for computing this distance grow quadratically with the size of the data, rendering them prohibitively expensive on massive data sets. In this work, we show that this challenge is surprisingly easy to circumvent: combining two simple techniques—the Nyström method and Sinkhorn scaling—provably yields an accurate approximation of the Sinkhorn distance with significantly lower time and memory requirements than other approaches. We prove our results via new, explicit analyses of the Nyström method and of the stability properties of Sinkhorn scaling. We validate our claims experimentally by showing that our approach easily computes Sinkhorn distances on data sets hundreds of times larger than can be handled by other techniques.

In a series of recent theoretical works, it was shown that strongly over-parameterized neural networks trained with gradient-based methods could converge exponentially fast to zero training loss, with their parameters hardly varying. In this work, we show that this lazy training'' phenomenon is not specific to over-parameterized neural networks, and is due to a choice of scaling, often implicit, that makes the model behave as its linearization around the initialization, thus yielding a model equivalent to learning with positive-definite kernels. Through a theoretical analysis, we exhibit various situations where this phenomenon arises in non-convex optimization and we provide bounds on the distance between the lazy and linearized optimization paths. Our numerical experiments bring a critical note, as we observe that the performance of commonly used non-linear deep convolutional neural networks in computer vision degrades when trained in the lazy regime. This makes it unlikely thatlazy training'' is behind the many successes of neural networks in difficult high dimensional tasks.

Key to structured prediction is exploiting the problem's structure to simplify the learning process. A major challenge arises when data exhibit a local structure (i.e., are made ``by parts'') that can be leveraged to better approximate the relation between (parts of) the input and (parts of) the output. Recent literature on signal processing, and in particular computer vision, shows that capturing these aspects is indeed essential to achieve state-of-the-art performance. However, in this context algorithms are typically derived on a case-by-case basis. In this work we propose the first theoretical framework to deal with part-based data from a general perspective and study a novel method within the setting of statistical learning theory. Our analysis is novel in that it explicitly quantifies the benefits of leveraging the part-based structure of a problem on the learning rates of the proposed estimator.

When optimizing over-parameterized models, such as deep neural networks, a large set of parameters can achieve zero training error. In such cases, the choice of the optimization algorithm and its respective hyper-parameters introduces biases that will lead to convergence to specific minimizers of the objective. Consequently, this choice can be considered as an implicit regularization for the training of over-parametrized models. In this work, we push this idea further by studying the discrete gradient dynamics of the training of a two-layer linear network with the least-squares loss. Using a time rescaling, we show that, with a vanishing initialization and a small enough step size, this dynamics sequentially learns the solutions of a reduced-rank regression with a gradually increasing rank.

Modern large-scale finite-sum optimization relies on two key aspects: distribution and stochastic updates. For smooth and strongly convex problems, existing decentralized algorithms are slower than modern accelerated variance-reduced stochastic algorithms when run on a single machine, and are therefore not efficient. Centralized algorithms are fast, but their scaling is limited by global aggregation steps that result in communication bottlenecks. In this work, we propose an efficient \textbf{A}ccelerated \textbf{D}ecentralized stochastic algorithm for \textbf{F}inite \textbf{S}ums named ADFS, which uses local stochastic proximal updates and randomized pairwise communications between nodes. On $n$ machines, ADFS learns from $nm$ samples in the same time it takes optimal algorithms to learn from $m$ samples on one machine. This scaling holds until a critical network size is reached, which depends on communication delays, on the number of samples $m$, and on the network topology. We provide a theoretical analysis based on a novel augmented graph approach combined with a precise evaluation of synchronization times and an extension of the accelerated proximal coordinate gradient algorithm to arbitrary sampling. We illustrate the improvement of ADFS over state-of-the-art decentralized approaches with experiments.

We consider the problem of minimizing the sum of submodular set functions assuming minimization oracles of each summand function. Most existing approaches reformulate the problem as the convex minimization of the sum of the corresponding Lov\'asz extensions and the squared Euclidean norm, leading to algorithms requiring total variation oracles of the summand functions; without further assumptions, these more complex oracles require many calls to the simpler minimization oracles often available in practice. In this paper, we consider a modified convex problem requiring constrained version of the total variation oracles that can be solved with significantly fewer calls to the simple minimization oracles. We support our claims by showing results on graph cuts for 2D and 3D graphs.

We propose a novel adaptive, accelerated algorithm for the stochastic constrained convex optimization setting.Our method, which is inspired by the Mirror-Prox method, \emph{simultaneously} achieves the optimal rates for smooth/non-smooth problems with either deterministic/stochastic first-order oracles. This is done without any prior knowledge of the smoothness nor the noise properties of the problem. To the best of our knowledge, this is the first adaptive, unified algorithm that achieves the optimal rates in the constrained setting. We demonstrate the practical performance of our framework through extensive numerical experiments.

In this paper, we study large-scale convex optimization algorithms based on the Newton method applied to regularized generalized self-concordant losses, which include logistic regression and softmax regression. We first prove that our new simple scheme based on a sequence of problems with decreasing regularization parameters is provably globally convergent, that this convergence is linear with a constant factor which scales only logarithmically with the condition number. In the parametric setting, we obtain an algorithm with the same scaling than regular first-order methods but with an improved behavior, in particular in ill-conditioned problems. Second, in the non parametric machine learning setting, we provide an explicit algorithm combining the previous scheme with Nystr\"om projections techniques, and prove that it achieves optimal generalization bounds with a time complexity of order O(n df), a memory complexity of order O(df^2) and no dependence on the condition number, generalizing the results known for least squares regression. Here n is the number of observations and df is the associated degrees of freedom. In particular, this is the first large-scale algorithm to solve logistic and softmax regressions in the non-parametric setting with large condition numbers and theoretical guarantees.

Machine learning on encrypted data has received a lot of attention thanks to recent breakthroughs in homomorphic encryption and secure multi-party computation. It allows outsourcing computation to untrusted servers without sacrificing privacy of sensitive data. We propose a practical framework to perform partially encrypted and privacy-preserving predictions which combines adversarial training and functional encryption. We first present a new functional encryption scheme to efficiently compute quadratic functions so that the data owner controls what can be computed but is not involved in the calculation: it provides a decryption key which allows one to learn a specific function evaluation of some encrypted data. We then show how to use it in machine learning to partially encrypt neural networks with quadratic activation functions at evaluation time and we provide a thorough analysis of the information leaks based on indistinguishability of data items of the same label. Last, since several encryption schemes cannot deal with the last thresholding operation used for classification, we propose a training method to prevent selected sensitive features from leaking which adversarially optimizes the network against an adversary trying to identify these features. This is of great interest for several existing works using partially encrypted machine learning as it comes with almost no cost on the model's accuracy and significantly improves data privacy.

Amongst the very first variance reduced stochastic methods for solving the empirical risk minimization problem was the SVRG method. SVRG is an inner-outer loop based method, where in the outer loop a reference full gradient is evaluated, after which $m \in \N$ steps of an inner loop are executed where the reference gradient is used to build a variance reduced estimate of the current gradient. The simplicity of the SVRG method and its analysis have lead to multiple extensions and variants for even non-convex optimization. Yet there is a significant gap between the parameter settings that the analysis suggests and what is known to work well in practice. Our first contribution is that we take several steps towards closing this gap. In particular, the current analysis shows that $m$ should be of the order of the condition number so that the resulting method has a favorable complexity. Yet in practice $m=n$ works well regardless of the condition number, where $n$ is the number of data points. Furthermore, the current analysis shows that the inner iterates have to be reset using averaging after every outer loop. Yet in practice SVRG works best when the inner iterates are updated continuously and not reset. We provide an analysis of these aforementioned practical settings and show that they achieve the same favorable complexity as the original analysis (with slightly better constants). Our second contribution is to provide a more general analysis than had been previously done by using arbitrary sampling, which allows us to analyze virtually all forms of mini-batching through a single theorem. Since our setup and analysis reflect what is done in practice, we are able to set the parameters such as the mini-batch size and step size using our theory in such a way that produces a more efficient algorithm in practice, as we show in extensive numerical experiments.

This work proposes a theoretical analysis of distributed optimization of convex functions using a network of computing units. We investigate this problem under two communication schemes (centralized and decentralized) and four classical regularity assumptions: Lipschitz continuity, strong convexity, smoothness, and a combination of strong convexity and smoothness. Under the decentralized communication scheme, we provide matching upper and lower bounds of complexity along with algorithms achieving this rate up to logarithmic constants. For non-smooth objective functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly convex objective functions. Such a convergence rate is achieved by the novel multi-step primal-dual (MSPD) algorithm. Under the centralized communication scheme, we show that the naive distribution of standard optimization algorithms is optimal for smooth objective functions, and provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function for non-smooth functions. We then show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

We propose a new class of determinantal point processes (DPPs) which can be manipulated for inference and parameter learning in potentially sublinear time in the number of items. This class, based on a specific low-rank factorization of the marginal kernel, is particularly suited to a subclass of continuous DPPs and DPPs defined on exponentially many items. We apply this new class to modelling text documents as sampling a DPP of sentences, and propose a conditional maximum likelihood formulation to model topic proportions, which is made possible with no approximation for our class of DPPs. We present an application to document summarization with a DPP on 2 to the power 500 items.

Our goal is to improve variance reducing stochastic methods through better control variates. We first propose a modification of SVRG which uses the Hessian to track gradients over time, rather than to recondition, increasing the correlation of the control variates and leading to faster theoretical convergence close to the optimum. We then propose accurate and computationally efficient approximations to the Hessian, both using a diagonal and a low-rank matrix. Finally, we demonstrate the effectiveness of our method on a wide range of problems.

We consider the homogeneous and the non-homogeneous convex relaxations for combinatorial penalty functions defined on support sets. Our study identifies key differences in the tightness of the resulting relaxations through the notion of the lower combinatorial envelope of a set-function along with new necessary conditions for support identification. We then propose a general adaptive estimator for convex monotone regularizers, and derive new sufficient conditions for support recovery in the asymptotic setting.

We consider the problem of minimizing a convex function over the intersection of finitely many simple sets which are easy to project onto. This is an important problem arising in various domains such as machine learning. The main difficulty lies in finding the projection of a point in the intersection of many sets. Existing approaches yield an infeasible point with an iteration-complexity of $O(1/ε^2)$ for nonsmooth problems with no guarantees on the in-feasibility. By reformulating the problem through exact penalty functions, we derive first-order algorithms which not only guarantees that the distance to the intersection is small but also improve the complexity to $O(1/ε)$ and $O(1/\sqrt{ε})$ for smooth functions. For composite and smooth problems, this is achieved through a saddle-point reformulation where the proximal operators required by the primal-dual algorithms can be computed in closed form. We illustrate the benefits of our approach on a graph transduction problem and on graph matching.

A central challenge to using first-order methods for optimizing nonconvex problems is the presence of saddle points. First-order methods often get stuck at saddle points, greatly deteriorating their performance. Typically, to escape from saddles one has to use second-order methods. However, most works on second-order methods rely extensively on expensive Hessian-based computations, making them impractical in large-scale settings. To tackle this challenge, we introduce a generic framework that minimizes Hessian-based computations while at the same time provably converging to second-order critical points. Our framework carefully alternates between a first-order and a second-order subroutine, using the latter only close to saddle points, and yields convergence results competitive to the state-of-the-art. Empirical results suggest that our strategy also enjoys a good practical performance.

We consider binary classification problems with positive definite kernels and square loss, and study the convergence rates of stochastic gradient methods. We show that while the excess testing \emph{loss} (squared loss) converges slowly to zero as the number of observations (and thus iterations) goes to infinity, the testing \emph{error} (classification error) converges exponentially fast if low-noise conditions are assumed. To achieve these rates of convergence we show sharper high-probability bounds with respect to the number of observations for stochastic gradient descent.

We consider the minimization of a function defined on a Riemannian manifold $\mathcal{M}$ accessible only through unbiased estimates of its gradients. We develop a geometric framework to transform a sequence of slowly converging iterates generated from stochastic gradient descent (SGD) on $\mathcal{M}$ to an averaged iterate sequence with a robust and fast $O(1/n)$ convergence rate. We then present an application of our framework to geodesically-strongly-convex (and possibly Euclidean non-convex) problems. Finally, we demonstrate how these ideas apply to the case of streaming $k$-PCA, where we show how to accelerate the slow rate of the randomized power method (without requiring knowledge of the eigengap) into a robust algorithm achieving the optimal rate of convergence.

We consider the minimization of submodular functions subject to ordering constraints. We show that this potentially non-convex optimization problem can be cast as a convex optimization problem on a space of uni-dimensional measures, with ordering constraints corresponding to first-order stochastic dominance. We propose new discretization schemes that lead to simple and efficient algorithms based on zero-th, first, or higher order oracles; these algorithms also lead to improvements without isotonic constraints. Finally, our experiments show that non-convex loss functions can be much more robust to outliers for isotonic regression, while still being solvable in polynomial time.

Many tasks in machine learning and signal processing can be solved by minimizing a convex function of a measure. This includes sparse spikes deconvolution or training a neural network with a single hidden layer. For these problems, we study a simple minimization method: the unknown measure is discretized into a mixture of particles and a continuous-time gradient descent is performed on their weights and positions. This is an idealization of the usual way to train neural networks with a large hidden layer. We show that, when initialized correctly and in the many-particle limit, this gradient flow, although non-convex, converges to global minimizers. The proof involves Wasserstein gradient flows, a by-product of optimal transport theory. Numerical experiments show that this asymptotic behavior is already at play for a reasonable number of particles, even in high dimension.

Recent progress in deep learning for audio synthesis opens the way to models that directly produce the waveform, shifting away from the traditional paradigm of relying on vocoders or MIDI synthesizers for speech or music generation. Despite their successes, current state-of-the-art neural audio synthesizers such as WaveNet and SampleRNN suffer from prohibitive training and inference times because they are based on autoregressive models that generate audio samples one at a time at a rate of 16kHz. In this work, we study the more computationally efficient alternative of generating the waveform frame-by-frame with large strides. We present a lightweight neural audio synthesizer for the original task of generating musical notes given desired instrument, pitch and velocity. Our model is trained end-to-end to generate notes from nearly 1000 instruments with a single decoder, thanks to a new loss function that minimizes the distances between the log spectrograms of the generated and target waveforms. On the generalization task of synthesizing notes for pairs of pitch and instrument not seen during training, SING produces audio with significantly improved perceptual quality compared to a state-of-the-art autoencoder based on WaveNet as measured by a Mean Opinion Score (MOS), and is about 32 times faster for training and 2, 500 times faster for inference.

Statistical leverage scores emerged as a fundamental tool for matrix sketching and column sampling with applications to low rank approximation, regression, random feature learning and quadrature. Yet, the very nature of this quantity is barely understood. Borrowing ideas from the orthogonal polynomial literature, we introduce the regularized Christoffel function associated to a positive definite kernel. This uncovers a variational formulation for leverage scores for kernel methods and allows to elucidate their relationships with the chosen kernel as well as population density. Our main result quantitatively describes a decreasing relation between leverage score and population density for a broad class of kernels on Euclidean spaces. Numerical simulations support our findings.

We consider stochastic gradient descent (SGD) for least-squares regression with potentially several passes over the data. While several passes have been widely reported to perform practically better in terms of predictive performance on unseen data, the existing theoretical analysis of SGD suggests that a single pass is statistically optimal. While this is true for low-dimensional easy problems, we show that for hard problems, multiple passes lead to statistically optimal predictions while single pass does not; we also show that in these hard models, the optimal number of passes over the data increases with sample size. In order to define the notion of hardness and show that our predictive performances are optimal, we consider potentially infinite-dimensional models and notions typically associated to kernel methods, namely, the decay of eigenvalues of the covariance matrix of the features and the complexity of the optimal predictor as measured through the covariance matrix. We illustrate our results on synthetic experiments with non-linear kernel methods and on a classical benchmark with a linear model.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

We propose a structure-adaptive variant of the state-of-the-art stochastic variance-reduced gradient algorithm Katyusha for regularized empirical risk minimization. The proposed method is able to exploit the intrinsic low-dimensional structure of the solution, such as sparsity or low rank which is enforced by a non-smooth regularization, to achieve even faster convergence rate. This provable algorithmic improvement is done by restarting the Katyusha algorithm according to restricted strong-convexity constants. We demonstrate the effectiveness of our approach via numerical experiments.

The failure of LASSO to identify groups of correlated predictors in linear regression has sparked significant research interest. Recently, various norms were proposed, which can be best described as instances of ordered weighted $\ell_1$ norms (OWL), as an alternative to $\ell_1$ regularization used in LASSO. OWL can identify groups of correlated variables but it forces the model to be constant within a group. This artifact induces unnecessary bias in the model estimation. In this paper we take a submodular perspective and show that OWL can be posed as the Lovász extension of a suitably defined submodular function. The submodular perspective not only explains the group-wise constant behavior of OWL, but also suggests alternatives. The main contribution of this paper is smoothed OWL (SOWL), a new family of norms, which not only identifies the groups but also allows the model to be flexible inside a group. We establish several algorithmic and theoretical properties of SOWL including group identification and model consistency. We also provide algorithmic tools to compute the SOWL norm and its proximal operator, whose computational complexity $O(d\log d)$ is significantly better than that of general purpose solvers in $O(d^2\log d)$. In our experiments, SOWL compares favorably with respect to OWL in the regimes of interest.

We consider the minimization of composite objective functions composed of the expectation of quadratic functions and an arbitrary convex function. We study the stochastic dual averaging algorithm with a constant step-size, showing that it leads to a convergence rate of O(1/n) without strong convexity assumptions. This thus extends earlier results on least-squares regression with the Euclidean geometry to (a) all convex regularizers and constraints, and (b) all geometries represented by a Bregman divergence. This is achieved by a new proof technique that relates stochastic and deterministic recursions

Zepeda and Perez [41] have recently demonstrated the promise of the exemplar SVM (ESVM) as a feature encoder for image retrieval. This paper extends this approach in several directions: We first show that replacing the hinge loss by the square loss in the ESVM cost function significantly reduces encoding time with negligible effect on accuracy. We call this model square-loss exemplar machine, or SLEM. We then introduce a kernelized SLEM which can be implemented efficiently through low-rank matrix decomposition, and displays improved performance. Both SLEM variants exploit the fact that the negative examples are fixed, so most of the SLEM computational complexity is relegated to an offline process independent of the positive examples. Our experiments establish the performance and computational advantages of our approach using a large array of base features and standard image retrieval datasets.

Non-negative matrix factorization (NMF) approximates a given matrix as a product of two non-negative matrix factors. Multiplicative algorithms deliver reliable results, but they show slow convergence for high-dimensional data and may be stuck away from local minima. Gradient descent methods have better behavior, but only apply to smooth losses. For non-smooth losses such as the Kullback-Leibler (KL) loss, surprisingly, these methods are lacking. In this paper, we propose a first-order primal-dual algorithm for non-negative decomposition problems (one of the two factors is fixed) with the KL distance. All required computations may be obtained in closed form and we provide an efficient heuristic way to select step-sizes. By using alternating optimization, our algorithm readily extends to NMF and, on synthetic or real world data, it is either faster than existing algorithms, or leads to improved local optima, or both.

In this paper, we determine the optimal convergence rates for strongly convex and smooth distributed optimization in two settings: centralized and decentralized communications over a network. For centralized (i.e. master/slave) algorithms, we show that distributing Nesterov’s accelerated gradient descent is optimal and achieves a precision $\varepsilon > 0$ in time $O(\sqrt{\kappa_g}(1+\Delta\tau)\ln(1/\varepsilon))$, where $\kappa_g$ is the condition number of the (global) function to optimize, $\Delta$ is the diameter of the network, and $\tau$ (resp. $1$) is the time needed to communicate values between two neighbors (resp. perform local computations). For decentralized algorithms based on gossip, we provide the first optimal algorithm, called the multi-step dual accelerated (MSDA) method, that achieves a precision $\varepsilon > 0$ in time $O(\sqrt{\kappa_l}(1+\frac{\tau}{\sqrt{\gamma}})\ln(1/\varepsilon))$, where $\kappa_l$ is the condition number of the local functions and $\gamma$ is the (normalized) eigengap of the gossip matrix used for communication between nodes. We then verify the efficiency of MSDA against state-of-the-art methods for two problems: least-squares regression and classification by logistic regression.

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We provide novel theoretical insights on structured prediction in the context of efficient convex surrogate loss minimization with consistency guarantees. For any task loss, we construct a convex surrogate that can be optimized via stochastic gradient descent and we prove tight bounds on the so-called "calibration function" relating the excess surrogate risk to the actual risk. In contrast to prior related work, we carefully monitor the effect of the exponential number of classes in the learning guarantees as well as on the optimization complexity. As an interesting consequence, we formalize the intuition that some task losses make learning harder than others, and that the classical 0-1 loss is ill-suited for structured prediction.

Extrapolation methods use the last few iterates of an optimization algorithm to produce a better estimate of the optimum. They were shown to achieve optimal convergence rates in a deterministic setting using simple gradient iterates. Here, we study extrapolation methods in a stochastic setting, where the iterates are produced by either a simple or an accelerated stochastic gradient algorithm. We first derive convergence bounds for arbitrary, potentially biased perturbations, then produce asymptotic bounds using the ratio between the variance of the noise and the accuracy of the current point. Finally, we apply this acceleration technique to stochastic algorithms such as SGD, SAGA, SVRG and Katyusha in different settings, and show significant performance gains.

We show that accelerated optimization methods can be seen as particular instances of multi-step integration schemes from numerical analysis, applied to the gradient flow equation. Compared with recent advances in this vein, the differential equation considered here is the basic gradient flow, and we derive a class of multi-step schemes which includes accelerated algorithms, using classical conditions from numerical analysis. Multi-step schemes integrate the differential equation using larger step sizes, which intuitively explains the acceleration phenomenon.

We consider neural networks with a single hidden layer and non- decreasing positively homogeneous activation functions like the rectified linear units. By letting the number of hidden units grow unbounded and using classical non-Euclidean regularization tools on the output weights, they lead to a convex optimization problem and we provide a detailed theoretical analysis of their generalization performance, with a study of both the approximation and the estimation errors. We show in particular that they are adaptive to unknown underlying linear structures, such as the dependence on the projection of the input variables onto a low-dimensional subspace. Moreover, when using sparsity- inducing norms on the input weights, we show that high- dimensional non-linear variable selection may be achieved, without any strong assumption regarding the data and with a total number of variables potentially exponential in the number of observations. However, solving this convex optimization problem in infinite dimensions is only possible if the non- convex subproblem of addition of a new unit can be solved efficiently. We provide a simple geometric interpretation for our choice of activation functions and describe simple conditions for convex relaxations of the finite-dimensional non- convex subproblem to achieve the same generalization error bounds, even when constant-factor approximations cannot be found. We were not able to find strong enough convex relaxations to obtain provably polynomial-time algorithms and leave open the existence or non-existence of such tractable algorithms with non-exponential sample complexities.

We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a theoretical analysis of the number of required samples for a given approximation error, leading to both upper and lower bounds that are based solely on the eigenvalues of the associated integral operator and match up to logarithmic terms. In particular, we show that the upper bound may be obtained from independent and identically distributed samples from a specific non-uniform distribution, while the lower bound if valid for any set of points. Applying our results to kernel-based quadrature, while our results are fairly general, we recover known upper and lower bounds for the special cases of Sobolev spaces. Moreover, our results extend to the more general problem of full function approximations (beyond simply computing an integral), with results in $L_2$- and $L_\infty$-norm that match known results for special cases. Applying our results to random features, we show an improvement of the number of random features needed to preserve the generalization guarantees for learning with Lipshitz-continuous losses.

We consider the optimization of a quadratic objective function whose gradients are only accessible through a stochastic oracle that returns the gradient at any given point plus a zero-mean finite variance random error. We present the first algorithm that achieves jointly the optimal prediction error rates for least-squares regression, both in terms of forgetting the initial conditions in $O(1/n^2)$, and in terms of dependence on the noise and dimension $d$ of the problem, as $O(d/n)$. Our new algorithm is based on averaged accelerated regularized gradient descent, and may also be analyzed through finer assumptions on initial conditions and the Hessian matrix, leading to dimension- free quantities that may still be small in some distances while the “optimal” terms above are large. In order to characterize the tightness of these new bounds, we consider an application to non-parametric regression and use the known lower bounds on the statistical performance (without computational limits), which happen to match our bounds obtained from a single pass on the data and thus show optimality of our algorithm in a wide variety of particular trade-offs between bias and variance.

We study parameter inference in large-scale latent variable models. We first propose a unified treatment of online inference for latent variable models from a non-canonical exponential family, and draw explicit links between several previously proposed frequentist or Bayesian methods. We then propose a novel inference method for the frequentist estimation of parameters, that adapts MCMC methods to online inference of latent variable models with the proper use of local Gibbs sampling. Then, for latent Dirichlet allocation,we provide an extensive set of experiments and comparisons with existing work, where our new approach outperforms all previously proposed methods. In particular, using Gibbs sampling for latent variable inference is superior to variational inference in terms of test log-likelihoods. Moreover, Bayesian inference through variational methods perform poorly, sometimes leading to worse fits with latent variables of higher dimensionality.

Clustering high-dimensional data often requires some form of dimensionality reduction, where clustered variables are separated from noise-looking variables. We cast this problem as finding a low-dimensional projection of the data which is well-clustered. This yields a one-dimensional projection in the simplest situation with two clusters, and extends naturally to a multi-label scenario for more than two clusters. In this paper, (a) we first show that this joint clustering and dimension reduction formulation is equivalent to previously proposed discriminative clustering frameworks, thus leading to convex relaxations of the problem; (b) we propose a novel sparse extension, which is still cast as a convex relaxation and allows estimation in higher dimensions; (c) we propose a natural extension for the multi-label scenario; (d) we provide a new theoretical analysis of the performance of these formulations with a simple probabilistic model, leading to scalings over the form $d=O(\sqrt{n})$ for the affine invariant case and $d=O(n)$ for the sparse case, where $n$ is the number of examples and $d$ the ambient dimension; and finally, (e) we propose an efficient iterative algorithm with running-time complexity proportional to $O(nd^2)$, improving on earlier algorithms for discriminative clustering with the square loss, which had quadratic complexity in the number of examples.

We consider the submodular function minimization (SFM) and the quadratic minimization problems regularized by the Lovasz extension of the submodular function. These optimization problems are intimately related; for example, min-cut problems and total variation denoising problems, where the cut function is submodular and its Lovasz extension is given by the associated total variation. When a quadratic loss is regularized by the total variation of a cut function, it thus becomes a total variation denoising problem and we use the same terminology in this paper for general submodular functions. We propose a new active-set algorithm for total variation denoising with the assumption of an oracle that solves the corresponding SFM problem. This can be seen as local descent algorithm over ordered partitions with explicit convergence guarantees. It is more flexible than the existing algorithms with the ability for warm-restarts using the solution of a closely related problem. Further, we also consider the case when a submodular function can be decomposed into the sum of two submodular functions $F_1$ and $F_2$ and assume SFM oracles for these two functions. We propose a new active-set algorithm for total variation denoising (and hence SFM by thresholding the solution at zero). This algorithm also performs local descent over ordered partitions and its ability to warm start considerably improves the performance of the algorithm. In the experiments, we compare the performance of the proposed algorithms with state-of-the-art algorithms, showing that it reduces the calls to SFM oracles.

Many of the ordinal regression models that have been proposed in the literature can be seen as methods that minimize a convex surrogate of the zero-one, absolute, or squared loss functions. A key property that allows to study the statistical implications of such approximations is that of Fisher consistency. Fisher consistency is a desirable property for surrogate loss functions and implies that in the population setting, i.e., if the probability distribution that generates the data were available, then optimization of the surrogate would yield the best possible model. In this paper we will characterize the Fisher consistency of a rich family of surrogate loss functions used in the context of ordinal regression, including support vector ordinal regression, ORBoosting and least absolute deviation. We will see that, for a family of surrogate loss functions that subsumes support vector ordinal regression and ORBoosting, consistency can be fully characterized by the derivative of a real-valued function at zero, as happens for convex margin-based surrogates in binary classification. We also derive excess risk bounds for a surrogate of the absolute error that generalize existing risk bounds for binary classification. Finally, our analysis suggests a novel surrogate of the squared error loss. We compare this novel surrogate with competing approaches on 9 different datasets. Our method shows to be highly competitive in practice, outperforming the least squares loss on 7 out of 9 datasets.

The minimization of convex functions which are only available through partial and noisy information is a key methodological problem in many disciplines. In this paper we consider convex optimization with noisy zero-th order information, that is noisy function evaluations at any desired point. We focus on problems with high degrees of smoothness, such as logistic regression. We show that as opposed to gradient-based algorithms, high-order smoothness may be used to improve estimation rates, with a precise dependence of our upper-bounds on the degree of smoothness. In particular, we show that for infinitely differentiable functions, we recover the same dependence on sample size as gradient-based algorithms, with an extra dimension-dependent factor. This is done for both convex and strongly-convex functions, with finite horizon and anytime algorithms. Finally, we also recover similar results in the online optimization setting.

In this paper, we consider a new discriminative approach to the problem of audio-to-score alignment. We consider two distinct informations provided by music scores: (i) an exact ordered list of musical events and (ii) an approximate prior information about relative duration of events. We extend the basic dynamic time warping algorithm to a convex problem that learns optimal classifiers for all events while jointly aligning files, using only weak supervision. We show that the relative duration between events can be easily used as a penalization of our cost function and allows us to drastically improve performances of our approach. We demonstrate the validity of our approach on a large and realistic dataset.

We introduce three novel semi-parametric extensions of probabilistic canonical correlation analysis with identifiability guarantees. We consider moment matching techniques for estimation in these models. For that, by drawing explicit links between the new models and a discrete version of independent component analysis (DICA), we first extend the DICA cumulant tensors to the new discrete version of CCA. By further using a close connection with independent component analysis, we introduce generalized covariance matrices, which can replace the cumulant tensors in the moment matching framework, and, therefore, improve sample complexity and simplify derivations and algorithms significantly. As the tensor power method or orthogonal joint diagonalization are not applicable in the new setting, we use non-orthogonal joint diagonalization techniques for matching the cumulants. We demonstrate performance of the proposed models and estimation techniques on experiments with both synthetic and real datasets.

Optimal transport (OT) defines a powerful framework to compare probability distributions in a geometrically faithful way. However, the practical impact of OT is still limited because of its computational burden. We propose a new class of stochastic optimization algorithms to cope with large-scale problems routinely encountered in machine learning applications. These methods are able to manipulate arbitrary distributions (either discrete or continuous) by simply requiring to be able to draw samples from them, which is the typical setup in high-dimensional learning problems. This alleviates the need to discretize these densities, while giving access to provably convergent methods that output the correct distance without discretization error. These algorithms rely on two main ideas: (a) the dual OT problem can be re-cast as the maximization of an expectation; (b) entropic regularization of the primal OT problem results in a smooth dual optimization optimization which can be addressed with algorithms that have a provably faster convergence. We instantiate these ideas in three different computational setups: (i) when comparing a discrete distribution to another, we show that incremental stochastic optimization schemes can beat the current state of the art finite dimensional OT solver (Sinkhorn's algorithm) ; (ii) when comparing a discrete distribution to a continuous density, a re-formulation (semi-discrete) of the dual program is amenable to averaged stochastic gradient descent, leading to better performance than approximately solving the problem by discretization ; (iii) when dealing with two continuous densities, we propose a stochastic gradient descent over a reproducing kernel Hilbert space (RKHS). This is currently the only known method to solve this problem, and is more efficient than discretizing beforehand the two densities. We backup these claims on a set of discrete, semi-discrete and continuous benchmark problems.

We exhibit a strong link between frequentist PAC-Bayesian bounds and the Bayesian marginal likelihood. That is, for the negative log-likelihood loss function, we show that the minimization of PAC-Bayesian generalization bounds maximizes the Bayesian marginal likelihood. This provides an alternative explanation to the Bayesian Occam's razor criteria, under the assumption that the data is generated by an i.i.d. distribution. Moreover, as the negative log-likelihood is an unbounded loss function, we motivate and propose a PAC-Bayesian theorem tailored for the sub-gamma loss family, and we show that our approach is sound on classical Bayesian linear regression tasks.

We consider convex-concave saddle-point problems where the objective functions may be split in many components, and extend recent stochastic variance reduction methods (such as SVRG or SAGA) to provide the first large-scale linearly convergent algorithms for this class of problems which are common in machine learning. While the algorithmic extension is straightforward, it comes with challenges and opportunities: (a) the convex minimization analysis does not apply and we use the notion of monotone operators to prove convergence, showing in particular that the same algorithm applies to a larger class of problems, such as variational inequalities, (b) there are two notions of splits, in terms of functions, or in terms of partial derivatives, (c) the split does need to be done with convex-concave terms, (d) non-uniform sampling is key to an efficient algorithm, both in theory and practice, and (e) these incremental algorithms can be easily accelerated using a simple extension of the "catalyst" framework, leading to an algorithm which is always superior to accelerated batch algorithms.

We describe a convergence acceleration technique for generic optimization problems. Our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method. The weights in this average are computed via a simple and small linear system, whose solution can be updated online. This acceleration scheme runs in parallel to the base algorithm, providing improved estimates of the solution on the fly, while the original optimization method is running. Numerical experiments are detailed on classical classification problems.

We consider log-supermodular models on binary variables, which are probabilistic models with negative log-densities which are submodular. These models provide probabilistic interpretations of common combinatorial optimization tasks such as image segmentation. In this paper, we focus primarily on parameter estimation in the models from known upper-bounds on the intractable log-partition function. We show that the bound based on separable optimization on the base polytope of the submodular function is always inferior to a bound based on ``perturb-and-MAP'' ideas. Then, to learn parameters, given that our approximation of the log-partition function is an expectation (over our own randomization), we use a stochastic subgradient technique to maximize a lower-bound on the log-likelihood. This can also be extended to conditional maximum likelihood. We illustrate our new results in a set of experiments in binary image denoising, where we highlight the flexibility of a probabilistic model to learn with missing data.

We consider the least-squares regression problem and provide a detailed asymptotic analysis of the performance of averaged constant-step-size stochastic gradient descent. In the strongly-convex case, we provide an asymptotic expansion up to explicit exponentially decaying terms. Our analysis leads to new insights into stochastic approximation algorithms: (a) it gives a tighter bound on the allowed step-size; (b) the generalization error may be divided into a variance term which is decaying as O(1/n), independently of the step-size g, and a bias term that decays as O(1/g^2 n^2); (c) when allowing non-uniform sampling of examples over a dataset, the choice of a good sampling density depends on the trade-off between bias and variance: when the variance term dominates, optimal sampling densities do not lead to much gain, while when the bias term dominates, we can choose larger step-sizes that lead to significant improvements.

Recently, the Frank-Wolfe optimization algorithm was suggested as a procedure to obtain adaptive quadrature rules for integrals of functions in a reproducing kernel Hilbert space (RKHS) with a potentially faster rate of convergence than Monte Carlo integration (and “kernel herding” was shown to be a special case of this procedure). In this paper, we propose to replace the random sampling step in a particle filter by Frank-Wolfe optimization. By optimizing the position of the particles, we can obtain better accuracy than random or quasi-Monte Carlo sampling. In applications where the evaluation of the emission probabilities is expensive (such as in robot localization), the additional computational cost to generate the particles through optimization can be justified. Experiments on standard synthetic examples as well as on a robot localization task indicate indeed an improvement of accuracy over random and quasi-Monte Carlo sampling.

We show that accelerated gradient descent, averaged gradient descent and the heavy-ball method for quadratic non-strongly-convex problems may be reformulated as constant parameter second-order difference equation algorithms, where stability of the system is equivalent to convergence at rate O(1/n^2), where n is the number of iterations. We provide a detailed analysis of the eigenvalues of the corresponding linear dynamical system, showing various oscillatory and non-oscillatory behaviors, together with a sharp stability result with explicit constants. We also consider the situation where noisy gradients are available, where we extend our general convergence result, which suggests an alternative algorithm (i.e., with different step sizes) that exhibits the good aspects of both averaging and acceleration.

Audio segmentation is an essential problem in many audio signal processing tasks, which tries to segment an audio signal into homogeneous chunks. Rather than separately finding change points and computing similarities between segments, we focus on joint segmentation and clustering, using the framework of hidden Markov and semi-Markov models. We introduce a new incremental EM algorithm for hidden Markov models (HMMs) and show that it compares favorably to existing online EM algorithms for HMMs. We present results for real-time segmentation of musical notes and acoustic scenes.

Suppose that we are given a set of videos, along with natural language descriptions in the form of multiple sentences (e.g., manual annotations, movie scripts, sport summaries etc.), and that these sentences appear in the same temporal order as their visual counterparts. We propose in this paper a method for aligning the two modalities, i.e., automatically providing a time (frame) stamp for every sentence. Given vectorial features for both video and text, this can be cast as a temporal assignment problem, with an implicit linear mapping between the two feature modalities. We formulate this problem as an integer quadratic program, and solve its continuous convex relaxation using an efficient conditional gradient algorithm. Several rounding procedures are proposed to construct the final integer solution. After demonstrating significant improvements over the state of the art on the related task of aligning video with symbolic labels [7], we evaluate our method on a challenging dataset of videos with associated textual descriptions [37], and explore bag-of-words and continuous representations for text.

We consider moment matching techniques for estimation in Latent Dirichlet Allocation (LDA). By drawing explicit links between LDA and discrete versions of independent component analysis (ICA), we first derive a new set of cumulant-based tensors, with an improved sample complexity. Moreover, we reuse standard ICA techniques such as joint diagonalization of tensors to improve over existing methods based on the tensor power method. In an extensive set of experiments on both synthetic and real datasets, we show that our new combination of tensors and orthogonal joint diagonalization techniques outperforms existing moment matching methods.

Recent literature~\cite{ando} suggests that embedding a graph on an unit sphere leads to better generalization for graph transduction. However, the choice of optimal embedding and an efficient algorithm to compute the same remains open. In this paper, we show that orthonormal representations, a class of unit-sphere graph embeddings are PAC learnable. Existing PAC-based analysis do not apply as the VC dimension of the function class is infinite. We propose an alternative PAC-based bound, which do not depend on the VC dimension of the underlying function class, but is related to the famous Lov\'{a}sz~$\vartheta$ function. The main contribution of the paper is SPORE, a SPectral regularized ORthonormal Embedding for graph transduction, derived from the PAC bound. SPORE is posed as a non-smooth convex function over an \emph{elliptope}. These problems are usually solved as semi-definite programs (SDPs) with time complexity $O(n^6)$. We present, Infeasible Inexact proximal~(IIP): an Inexact proximal method which performs subgradient procedure on an approximate projection, not necessarily feasible. IIP is more scalable than SDP, has an $O(\frac{1}{\sqrt{T}})$ convergence, and is generally applicable whenever a suitable approximate projection is available. We use IIP to compute SPORE where the approximate projection step is computed by FISTA, an accelerated gradient descent procedure. We show that the method has a convergence rate of $O(\frac{1}{\sqrt{T}})$. The proposed algorithm easily scales to 1000's of vertices, while the standard SDP computation does not scale beyond few hundred vertices. Furthermore, the analysis presented here easily extends to the multiple graph setting.

A popular approach within the signal processing and machine learning communities consists in modeling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various fields ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. In this paper, we consider a probabilistic model of sparse signals, and show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. This paper considers the case of over-complete dictionaries, noisy signals, and possible outliers, thus extending the previous work limited to noiseless settings and/or undercomplete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity, and the number of observations.

Many modern tools in machine learning and signal processing, such as sparse dictionary learning, principal component analysis, non-negative matrix factorization, K-means clustering, and so on, rely on the factorization of a matrix obtained by concatenating high-dimensional vectors from a training collection. While the idealized task would be to optimize the expected quality of the factors over the underlying distribution of training vectors, it is achieved in practice by minimizing an empirical average over the considered collection. The focus of this paper is to provide sample complexity estimates to uniformly control how much the empirical average deviates from the expected cost function. Standard arguments imply that the performance of the empirical predictor also exhibit such guarantees. The level of genericity of the approach encompasses several possible constraints on the factors (tensor product structure, shift-invariance, sparsity...), thus providing a unified perspective on the sample complexity of several widely used matrix factorization schemes. The derived generalization bounds behave proportional to (log (n)/n)1/2with respect to the number of samples n for the considered matrix factorization techniques.

We consider unsupervised partitioning problems based explicitly or implicitly on the minimization of Euclidean distortions, such as clustering, image or video segmentation, and other change-point detection problems. We emphasize on cases with specific structure, which include many practical situations ranging from mean-based change-point detection to image segmentation problems. We aim at learning a Mahalanobis metric for these unsupervised problems, leading to feature weighting and/or selection. This is done in a supervised way by assuming the availability of several (partially) labeled datasets that share the same metric. We cast the metric learning problem as a large-margin structured prediction problem, with proper definition of regularizers and losses, leading to a convex optimization problem which can be solved efficiently. Our experiments show how learning the metric can significantly improve performance on bioinformatics, video or image segmentation problems.

The Minimal-Pair ABX (MP-ABX) paradigm has been pro-posed as a method for evaluating speech features for zero-resource/unsupervised speech technologies. We apply it in a phoneme discrimination task on the Articulation Index corpus to evaluate the resistance to noise of various speech features. In Experiment 1, we evaluate the robustness to additive noise at different signal-to-noise ratios, using car and babble noise from the Aurora-4 database and white noise. In Experiment 2, we ex-amine the robustness to different kinds of convolutional noise. In both experiments we consider two classes of techniques to induce noise resistance: smoothing of the time-frequency rep-resentation and short-term adaptation in the time-domain. We consider smoothing along the spectral axis (as in PLP) and along the time axis (as in FDLP). For short-term adaptation in the time-domain, we compare the use of a static compressive non-linearity followed by RASTA filtering to an adaptive com-pression scheme. Index Terms: noise resistance, zero-resource, speech features, evaluation framework, minimal-pair ABX task

MOTIVATION: DNA copy number profiles characterize regions of chromosome gains, losses and breakpoints in tumor genomes. Although many models have been proposed to detect these alterations, it is not clear which model is appropriate before visual inspection the signal, noise and models for a particular profile. RESULTS: We propose SegAnnDB, a Web-based computer vision system for genomic segmentation: first, visually inspect the profiles and manually annotate altered regions, then SegAnnDB determines the precise alteration locations using a mathematical model of the data and annotations. SegAnnDB facilitates collaboration between biologists and bioinformaticians, and uses the University of California, Santa Cruz genome browser to visualize copy number alterations alongside known genes. AVAILABILITY AND IMPLEMENTATION: The breakpoints project on INRIA GForge hosts the source code, an Amazon Machine Image can be launched and a demonstration Web site is http://bioviz.rocq.inria.fr.

We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practical limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n^2). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p^2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the \emphdegrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such methods, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same \emphpredictive performance than existing algorithms, for any given problem instance, and not only for worst-case situations.

Language models can be formalized as loglinear regression models where the input features represent previously observed contexts up to a certain length m.The complexity of existing algorithms to learn the parameters by maximum likelihood scale linearly in nd, where n is the length of the training corpus and d is the number of observed features.We present a model that grows logarithmically in d, making it possible to efficiently leverage longer contexts.We account for the sequential structure of natural language using treestructured penalized objectives to avoid overfitting and achieve better generalization.

We address the problem of learning a joint model of actors and actions in movies using weak supervision provided by scripts. Specifically, we extract actor/action pairs from the script and use them as constraints in a discriminative clustering framework. The corresponding optimization problem is formulated as a quadratic program under linear constraints. People in video are represented by automatically extracted and tracked faces together with corresponding motion features. First, we apply the proposed framework to the task of learning names of characters in the movie and demonstrate significant improvements over previous methods used for this task. Second, we explore the joint actor/action constraint and show its advantage for weakly supervised action learning. We validate our method in the challenging setting of localizing and recognizing characters and their actions in feature length movies Casablanca and American Beauty.

In segmentation models, the number of change-points is typically chosen using a penalized cost function. In this work, we propose to learn the penalty and its constants in databases of signals with weak change-point annotations. We propose a convex relaxation for the resulting interval regression problem, and solve it using accelerated proximal gradient methods. We show that this method achieves state-of-the-art change-point detection in a database of annotated DNA copy number profiles from neuroblastoma tumors.

We consider the problem of learning the structure of undirected graphical models with bounded treewidth, within the maximum likelihood framework. This is an NP-hard problem and most approaches consider local search techniques. In this paper, we pose it as a combinatorial optimization problem, which is then relaxed to a convex optimization problem that involves searching over the forest and hyperforest polytopes with special structures. A supergradient method is used to solve the dual problem, with a run-time complexity of O(k^3 n^k+2 \log n) for each iteration, where n is the number of variables and k is a bound on the treewidth. We compare our approach to state-of-the-art methods on synthetic datasets and classical benchmarks, showing the gains of the novel convex approach.

We address the problem of designing a convex nonsmooth regularizer encouraging multiple structural effects simultaneously. Focusing on the inference of sparse and low-rank matrices we suggest a new complexity index and a convex penalty approximating it. The new penalty term can be written as the trace norm of a linear function of the matrix. By analyzing theoretical properties of this family of regularizers we come up with oracle inequalities and compressed sensing results ensuring the quality of our regularized estimator. We also provide algorithms and supporting numerical experiments.

We present a new framework for the evaluation of speech representations in zero-resource settings, that extends and complements previous work by Carlin, Jansen and Hermansky [1].In particular, we replace their Same/Different discrimination task by several Minimal-Pair ABX (MP-ABX) tasks.We explain the analytical advantages of this new framework and apply it to decompose the standard signal processing pipelines for computing PLP and MFC coefficients.This method enables us to confirm and quantify a variety of well-known and not-so-well-known results in a single framework.

We consider the stochastic approximation problem where a convex function has to be minimized, given only the knowledge of unbiased estimates of its gradients at certain points, a framework which includes machine learning methods based on the minimization of the empirical risk. We focus on problems without strong convexity, for which all previously known algorithms achieve a convergence rate for function values of $O(1/\sqrt{n})$. We consider and analyze two algorithms that achieve a rate of $O(1/n)$ for classical supervised learning problems. For least-squares regression, we show that averaged stochastic gradient descent with constant step-size achieves the desired rate. For logistic regression, this is achieved by a simple novel stochastic gradient algorithm that (a) constructs successive local quadratic approximations of the loss functions, while (b) preserving the same running time complexity as stochastic gradient descent. For these algorithms, we provide a non-asymptotic analysis of the generalization error (in expectation, and also in high probability for least-squares), and run extensive experiments showing that they often outperform existing approaches.

Seriation seeks to reconstruct a linear order between variables using unsorted similarity information. It has direct applications in archeology and shotgun gene sequencing for example. We prove the equivalence between the seriation and the combinatorial 2-sum problem (a quadratic minimization problem over permutations) over a class of similarity matrices. The seriation problem can be solved exactly by a spectral algorithm in the noiseless case and we produce a convex relaxation for the 2-sum problem to improve the robustness of solutions in a noisy setting. This relaxation also allows us to impose additional structural constraints on the solution, to solve semi-supervised seriation problems. We present numerical experiments on archeological data, Markov chains and gene sequences.

Recently, it has become evident that submodularity naturally captures widely occurring concepts in machine learning, signal processing and computer vision. In consequence, there is need for efficient optimization procedures for submodular functions, in particular for minimization problems. While general submodular minimization is challenging, we propose a new approach that exploits existing decomposability of submodular functions. In contrast to previous approaches, our method is neither approximate, nor impractical, nor does it need any cumbersome parameter tuning. Moreover, it is easy to implement and parallelize. A key component of our approach is a formulation of the discrete submodular minimization problem as a continuous best approximation problem. It is solved through a sequence of reflections and its solution can be automatically thresholded to obtain an optimal discrete solution. Our method solves both the continuous and discrete formulations of the problem, and therefore has applications in learning, inference, and reconstruction. In our experiments, we show the benefits of our new algorithms for two image segmentation tasks.

BACKGROUND: Many models have been proposed to detect copy number alterations in chromosomal copy number profiles, but it is usually not obvious to decide which is most effective for a given data set. Furthermore, most methods have a smoothing parameter that determines the number of breakpoints and must be chosen using various heuristics. RESULTS: We present three contributions for copy number profile smoothing model selection. First, we propose to select the model and degree of smoothness that maximizes agreement with visual breakpoint region annotations. Second, we develop cross-validation procedures to estimate the error of the trained models. Third, we apply these methods to compare 17 smoothing models on a new database of 575 annotated neuroblastoma copy number profiles, which we make available as a public benchmark for testing new algorithms. CONCLUSIONS: Whereas previous studies have been qualitative or limited to simulated data, our annotation-guided approach is quantitative and suggests which algorithms are fastest and most accurate in practice on real data. In the neuroblastoma data, the equivalent pelt.n and cghseg.k methods were the best breakpoint detectors, and exhibited reasonable computation times.

Bottom-up, fully unsupervised segmentation remains a daunting challenge for computer vision. In the cosegmentation context, on the other hand, the availability of multiple images assumed to contain instances of the same object classes provides a weak form of supervision that can be exploited by discriminative approaches. Unfortunately, most existing algorithms are limited to a very small number of images and/or object classes (typically two of each). This paper proposes a novel energy-minimization approach to cosegmentation that can handle multiple classes and a significantly larger number of images. The proposed cost function combines spectral- and discriminative-clustering terms, and it admits a probabilistic interpretation. It is optimized using an efficient EM method, initialized using a convex quadratic approximation of the energy. Comparative experiments show that the proposed approach matches or improves the state of the art on several standard datasets.

Positive definite operator-valued kernels generalize the well-known notion of reproducing kernels, and are naturally adapted to multi-output learning situations. This paper addresses the problem of learning a finite linear combination of infinite-dimensional operator-valued kernels which are suitable for extending functional data analysis methods to nonlinear contexts. We study this problem in the case of kernel ridge regression for functional responses with an lr-norm constraint on the combination coefficients. The resulting optimization problem is more involved than those of multiple scalar-valued kernel learning since operator-valued kernels pose more technical and theoretical issues. We propose a multiple operator-valued kernel learning algorithm based on solving a system of linear operator equations by using a block coordinate-descent procedure. We experimentally validate our approach on a functional regression task in the context of finger movement prediction in brain-computer interfaces.

We propose a new stochastic gradient method for optimizing the sum of a finite set of smooth functions, where the sum is strongly convex. While standard stochastic gradient methods converge at sublinear rates for this problem, the proposed method incorporates a memory of previous gradient values in order to achieve a linear convergence rate. In a machine learning context, numerical experiments indicate that the new algorithm can dramatically outperform standard algorithms, both in terms of optimizing the training error and reducing the test error quickly.

Modeling data with linear combinations of a few elements from a learned dictionary has been the focus of much recent research in machine learning, neuroscience, and signal processing. For signals such as natural images that admit such sparse representations, it is now well established that these models are well suited to restoration tasks. In this context, learning the dictionary amounts to solving a large-scale matrix factorization problem, which can be done efficiently with classical optimization tools. The same approach has also been used for learning features from data for other purposes, e.g., image classification, but tuning the dictionary in a supervised way for these tasks has proven to be more difficult. In this paper, we present a general formulation for supervised dictionary learning adapted to a wide variety of tasks, and present an efficient algorithm for solving the corresponding optimization problem. Experiments on handwritten digit classification, digital art identification, nonlinear inverse image problems, and compressed sensing demonstrate that our approach is effective in large-scale settings, and is well suited to supervised and semi-supervised classification, as well as regression tasks for data that admit sparse representations.

Reverse inference, or brain reading, is a recent paradigm for analyzing functional magnetic resonance imaging (fMRI) data based on pattern recognition and statistical learning. By predicting some cognitive variables related to brain activation maps, this approach aims at decoding brain activity. Reverse inference takes into account the multivariate information between voxels and is currently the only way to assess how precisely some cognitive information is encoded by the activity of neural populations within the whole brain. However, it relies on a prediction function that is plagued by the curse of dimensionality, since there are far more features than samples, i.e., more voxels than fMRI volumes. To address this problem, different methods have been proposed, including univariate feature selection, feature agglomeration, and regularization techniques. In this paper, we consider a sparse hierarchical structured regularization. Specifically, the penalization we use is constructed from a tree that is obtained by spatially constrained agglomerative clustering. This approach encodes the spatial structure of the data at different scales into the regularization, which makes the overall prediction procedure more robust to intersubject variability. The regularization used induces the selection of spatially coherent predictive brain regions simultaneously at different scales. We test our algorithm on real data acquired to study the mental representation of objects, and we show that the proposed algorithm not only delineates meaningful brain regions but also yields better prediction accuracy than reference methods.

Sparse coding, which is the decomposition of a vector using only a few basis elements, is widely used in machine learning and image processing. The basis set, also called dictionary, is learned to adapt to specific data. This approach has proven to be very effective in many image processing tasks. Traditionally, the dictionary is an unstructured “flat” set of atoms. In this paper, we study structured dictionaries which are obtained from an epitome, or a set of epitomes. The epitome is itself a small image, and the atoms are all the patches of a chosen size inside this image. This considerably reduces the number of parameters to learn and provides sparse image decompositions with shift-invariance properties. We propose a new formulation and an algorithm for learning the structured dictionaries associated with epitomes, and illustrate their use in image de-noising tasks.

We propose an unsupervised inference procedure for audio source separation. Components in nonnegative matrix factorization (NMF) are grouped automatically in audio sources via a penalized maximum likelihood approach. The penalty term we introduce favors sparsity at the group level, and is motivated by the assumption that the local amplitude of the sources are independent. Our algorithm extends multiplicative updates for NMF ; moreover we propose a test statistic to tune hyperparameters in our model, and illustrate its adequacy on synthetic data. Results on real audio tracks show that our sparsity prior allows to identify audio sources without knowledge on their spectral properties.

Invariant representations in object recognition systems are generally obtained by pooling feature vectors over spatially local neighborhoods. But pooling is not local in the feature vector space, so that widely dissimilar features may be pooled together if they are in nearby locations. Recent approaches rely on sophisticated encoding methods and more specialized codebooks (or dictionaries), e.g., learned on subsets of descriptors which are close in feature space, to circumvent this problem. In this work, we argue that a common trait found in much recent work in image recognition or retrieval is that it leverages locality in feature space on top of purely spatial locality. We propose to apply this idea in its simplest form to an object recognition system based on the spatial pyramid framework, to increase the performance of small dictionaries with very little added engineering. State-of-the-art results on several object recognition benchmarks show the promise of this approach.

We consider a class of sparsity-inducing regularization terms based on submodular functions. While previous work has focused on non-decreasing functions, we explore symmetric submodular functions and their \lova extensions. We show that the Lovasz extension may be seen as the convex envelope of a function that depends on level sets (i.e., the set of indices whose corresponding components of the underlying predictor are greater than a given constant): this leads to a class of convex structured regularization terms that impose prior knowledge on the level sets, and not only on the supports of the underlying predictors. We provide a unified set of optimization algorithms, such as proximal operators, and theoretical guarantees (allowed level sets and recovery conditions). By selecting specific submodular functions, we give a new interpretation to known norms, such as the total variation; we also define new norms, in particular ones that are based on order statistics with application to clustering and outlier detection, and on noisy cuts in graphs with application to change point detection in the presence of outliers.

We consider the minimization of a convex objective function defined on a Hilbert space, which is only available through unbiased estimates of its gradients. This problem includes standard machine learning algorithms such as kernel logistic regression and least-squares regression, and is commonly referred to as a stochastic approximation problem in the operations research community. We provide a non-asymptotic analysis of the convergence of two well-known algorithms, stochastic gradient descent (a.k.a.~Robbins-Monro algorithm) as well as a simple modification where iterates are averaged (a.k.a.~Polyak-Ruppert averaging). Our analysis suggests that a learning rate proportional to the inverse of the number of iterations, while leading to the optimal convergence rate in the strongly convex case, is not robust to the lack of strong convexity or the setting of the proportionality constant. This situation is remedied when using slower decays together with averaging, robustly leading to the optimal rate of convergence. We illustrate our theoretical results with simulations on synthetic and standard datasets.

Using the $\ell_1$-norm to regularize the estimation of the parameter vector of a linear model leads to an unstable estimator when covariates are highly correlated. In this paper, we introduce a new penalty function which takes into account the correlation of the design matrix to stabilize the estimation. This norm, called the trace Lasso, uses the trace norm of the selected covariates, which is a convex surrogate of their rank, as the criterion of model complexity. We analyze the properties of our norm, describe an optimization algorithm based on reweighted least-squares, and illustrate the behavior of this norm on synthetic data, showing that it is more adapted to strong correlations than competing methods such as the elastic net.

We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the second term. We show that the basic proximal-gradient method, the basic proximal-gradient method with a strong convexity assumption, and the accelerated proximal-gradient method achieve the same convergence rates as in the error-free case, provided the errors decrease at an appropriate rate. Our experimental results on a structured sparsity problem indicate that sequences of errors with these appealing theoretical properties can lead to practical performance improvements.

This paper addresses the problem of establishing correspondences between two sets of visual features using higher order constraints instead of the unary or pairwise ones used in classical methods. Concretely, the corresponding hypergraph matching problem is formulated as the maximization of a multilinear objective function over all permutations of the features. This function is defined by a tensor representing the affinity between feature tuples. It is maximized using a generalization of spectral techniques where a relaxed problem is first solved by a multidimensional power method and the solution is then projected onto the closest assignment matrix. The proposed approach has been implemented, and it is compared to state-of-the-art algorithms on both synthetic and real data.

Many successful models for scene or object recognition transform low-level descriptors (such as Gabor filter responses, or SIFT descriptors) into richer representations of intermediate complexity. This process can often be broken down into two steps: (1) a coding step, which performs a pointwise transformation of the descriptors into a representation better adapted to the task, and (2) a pooling step, which summarizes the coded features over larger neighborhoods. Several combinations of coding and pooling schemes have been proposed in the literature. The goal of this paper is threefold. We seek to establish the relative importance of each step of mid-level feature extraction through a comprehensive cross evaluation of several types of coding modules (hard and soft vector quantization, sparse coding) and pooling schemes (by taking the average, or the maximum), which obtains state-of-the-art performance or better on several recognition benchmarks. We show how to improve the best performing coding scheme by learning a supervised discriminative dictionary for sparse coding. We provide theoretical and empirical insight into the remarkable performance of max pooling. By teasing apart components shared by modern mid-level feature extractors, our approach aims to facilitate the design of better recognition architectures.

Purely bottom-up, unsupervised segmentation of a single image into foreground and background regions remains a challenging task for computer vision. Co-segmentation is the problem of simultaneously dividing multiple images into regions (segments) corresponding to different object classes. In this paper, we combine existing tools for bottom-up image segmentation such as normalized cuts, with kernel methods commonly used in object recognition. These two sets of techniques are used within a discriminative clustering framework: the goal is to assign foreground/background labels jointly to all images, so that a supervised classifier trained with these labels leads to maximal separation of the two classes. In practice, we obtain a combinatorial optimization problem which is relaxed to a continuous convex optimization problem, that can itself be solved efficiently for up to dozens of images. We illustrate the proposed method on images with very similar foreground objects, as well as on more challenging problems with objects with higher intra-class variations.

Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turned into a convex optimization problem by replacing the cardinality function by its convex envelope (tightest convex lower bound), in this case the L1-norm. In this paper, we investigate more general set-functions than the cardinality, that may incorporate prior knowledge or structural constraints which are common in many applications: namely, we show that for nondecreasing submodular set-functions, the corresponding convex envelope can be obtained from its Lovasz extension, a common tool in submodular analysis. This defines a family of polyhedral norms, for which we provide generic algorithmic tools (subgradients and proximal operators) and theoretical results (conditions for support recovery or high-dimensional inference). By selecting specific submodular functions, we can give a new interpretation to known norms, such as those based on rank-statistics or grouped norms with potentially overlapping groups; we also define new norms, in particular ones that can be used as non-factorial priors for supervised learning.

We develop an online variational Bayes (VB) algorithm for Latent Dirichlet Allocation (LDA). Online LDA is based on online stochastic optimization with a natural gradient step, which we show converges to a local optimum of the VB objective function. It can handily analyze massive document collections, including those arriving in a stream. We study the performance of online LDA in several ways, including by fitting a 100-topic topic model to 3.3M articles from Wikipedia in a single pass. We demonstrate that online LDA finds topic models as good or better than those found with batch VB, and in a fraction of the time.

Dimensionality reduction is commonly used in the setting of multi-label supervised classification to control the learning capacity and to provide a meaningful representation of the data. We introduce a simple forward probabilistic model which is a multinomial extension of reduced rank regression; we show that this model provides a probabilistic interpretation of discriminative clustering methods with added benefits in terms of number of hyperparameters and optimization. While expectation-maximization (EM) algorithm is commonly used to learn these models, its optimization usually leads to local minimum because it relies on a non-convex cost function with many such local minima. To avoid this problem, we introduce a local approximation of this cost function, which leads to a quadratic non-convex optimization problem over a product of simplices. In order to minimize such functions, we propose an efficient algorithm based on convex relaxation and low-rank representation of our data, which allows to deal with large instances. Experiments on text document classification show that the new model outperforms other supervised dimensionality reduction methods, while simulations on unsupervised clustering show that our probabilistic formulation has better properties than existing discriminative clustering methods.

We consider a class of learning problems that involve a structured sparsity-inducing norm defined as the sum of $\ell_\infty$-norms over groups of variables. Whereas a lot of effort has been put in developing fast optimization methods when the groups are disjoint or embedded in a specific hierarchical structure, we address here the case of general overlapping groups. To this end, we show that the corresponding optimization problem is related to network flow optimization. More precisely, the proximal problem associated with the norm we consider is dual to a quadratic min-cost flow problem. We propose an efficient procedure which computes its solution exactly in polynomial time. Our algorithm scales up to millions of groups and variables, and opens up a whole new range of applications for structured sparse models. We present several experiments on image and video data, demonstrating the applicability and scalability of our approach for various problems.

This paper addresses the problem of establishing correspondences between two sets of visual features using higher-order constraints instead of the unary or pairwise ones used in classical methods. Concretely, the corresponding hypergraph matching problem is formulated as the maximization of a multilinear objective function over all permutations of the features. This function is defined by a tensor representing the affinity between feature tuples. It is maximized using a generalization of spectral techniques where a relaxed problem is first solved by a multi-dimensional power method, and the solution is then projected onto the closest assignment matrix. The proposed approach has been implemented, and it is compared to state-of-the-art algorithms on both synthetic and real data.

This paper addresses the problem of automatic temporal annotation of realistic human actions in video using minimal manual supervision. To this end we consider two associated problems: (a) weakly-supervised learning of action models from readily available annotations, and (b) temporal localization of human actions in test videos. To avoid the prohibitive cost of manual annotation for training, we use movie scripts as a means of weak supervision. Scripts, however, provide only implicit, noisy, and imprecise information about the type and location of actions in video. We address this problem with a kernel-based discriminative clustering algorithm that locates actions in the weakly-labeled training data. Using the obtained action samples, we train temporal action detectors and apply them to locate actions in the raw video data. Our experiments demonstrate that the proposed method for weakly-supervised learning of action models leads to significant improvement in action detection. We present detection results for three action classes in four feature length movies with challenging and realistic video data.

We propose in this paper to unify two different approaches to image restoration: On the one hand, learning a basis set (dictionary) adapted to sparse signal descriptions has proven to be very effective in image reconstruction and classification tasks. On the other hand, explicitly exploiting the self-similarities of natural images has led to the successful non-local means approach to image restoration. We propose simultaneous sparse coding as a framework for combining these two approaches in a natural manner. This is achieved by jointly decomposing groups of similar signals on subsets of the learned dictionary. Experimental results in image denoising and demosaicking tasks with synthetic and real noise show that the proposed method outperforms the state of the art, making it possible to effectively restore raw images from digital cameras at a reasonable speed and memory cost.

Sparse coding---that is, modelling data vectors as sparse linear combinations of basis elements---is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on learning the basis set, also called dictionary, to adapt it to specific data, an approach that has recently proven to be very effective for signal reconstruction and classification in the audio and image processing domains. This paper proposes a new online optimization algorithm for dictionary learning, based on stochastic approximations, which scales up gracefully to large datasets with millions of training samples. A proof of convergence is presented, along with experiments with natural images demonstrating that it leads to faster performance and better dictionaries than classical batch algorithms for both small and large datasets.

This paper tackles the problem of selecting among several linear estimators in non-parametric regression; this includes model selection for linear regression, the choice of a regularization parameter in kernel ridge regression or spline smoothing, and the choice of a kernel in multiple kernel learning. We propose a new algorithm which first estimates consistently the variance of the noise, based upon the concept of minimal penalty which was previously introduced in the context of model selection. Then, plugging our variance estimate in Mallows $C_L$ penalty is proved to lead to an algorithm satisfying an oracle inequality. Simulation experiments with kernel ridge regression and multiple kernel learning show that the proposed algorithm often improves significantly existing calibration procedures such as 10-fold cross-validation or generalized cross-validation.

Many types of regularization schemes have been employed in statistical learning, each one motivated by some assumption about the problem domain. In this paper, we present a unified asymptotic analysis of smooth regularizers, which allows us to see how the validity of these assumptions impacts the success of a particular regularizer. In addition, our analysis motivates an algorithm for optimizing regularization parameters, which in turn can be analyzed within our framework. We apply our analysis to several examples, including hybrid generative-discriminative learning and multi-task learning.

MOTIVATION: Aligning protein-protein interaction (PPI) networks of different species has drawn a considerable interest recently. This problem is important to investigate evolutionary conserved pathways or protein complexes across species, and to help in the identification of functional orthologs through the detection of conserved interactions. It is, however, a difficult combinatorial problem, for which only heuristic methods have been proposed so far. RESULTS: We reformulate the PPI alignment as a graph matching problem, and investigate how state-of-the-art graph matching algorithms can be used for that purpose. We differentiate between two alignment problems, depending on whether strict constraints on protein matches are given, based on sequence similarity, or whether the goal is instead to find an optimal compromise between sequence similarity and interaction conservation in the alignment. We propose new methods for both cases, and assess their performance on the alignment of the yeast and fly PPI networks. The new methods consistently outperform state-of-the-art algorithms, retrieving in particular 78% more conserved interactions than IsoRank for a given level of sequence similarity. AVAILABILITY: All data and codes are freely and publicly available upon request.

We propose a convex-concave programming approach for the labeled weighted graph matching problem. The convex-concave programming formulation is obtained by rewriting the weighted graph matching problem as a least-square problem on the set of permutation matrices and relaxing it to two different optimization problems: a quadratic convex and a quadratic concave optimization problem on the set of doubly stochastic matrices. The concave relaxation has the same global minimum as the initial graph matching problem, but the search for its global minimum is also a hard combinatorial problem. We, therefore, construct an approximation of the concave problem solution by following a solution path of a convex-concave problem obtained by linear interpolation of the convex and concave formulations, starting from the convex relaxation. This method allows to easily integrate the information on graph label similarities into the optimization problem, and therefore, perform labeled weighted graph matching. The algorithm is compared with some of the best performing graph matching methods on four data sets: simulated graphs, QAPLib, retina vessel images, and handwritten Chinese characters. In all cases, the results are competitive with the state of the art.

Sparse signal models have been the focus of much recent research, leading to (or improving upon) state-of-the-art results in signal, image, and video restoration. This article extends this line of research into a novel framework for local image discrimination tasks, proposing an energy formulation with both sparse reconstruction and class discrimination components, jointly optimized during dictionary learning. This approach improves over the state of the art in texture segmentation experiments using the Brodatz database, and it paves the way for a novel scene analysis and recognition framework based on simultaneously learning discriminative and reconstructive dictionaries. Preliminary results in this direction using examples from the Pascal VOC06 and Graz02 datasets are presented as well.

Point clouds are sets of points in two or three dimensions. Most kernel methods for learning on sets of points have not yet dealt with the specific geometrical invariances and practical constraints associated with point clouds in computer vision and graphics. In this paper, we present extensions of graph kernels for point clouds, which allow one to use kernel methods for such objects as shapes, line drawings, or any three-dimensional point clouds. In order to design rich and numerically efficient kernels with as few free parameters as possible, we use kernels between covariance matrices and their factorizations on probabilistic graphical models. We derive polynomial time dynamic programming recursions and present applications to recognition of handwritten digits and Chinese characters from few training examples.

We consider the least-square linear regression problem with regularization by the l1-norm, a problem usually referred to as the Lasso. In this paper, we present a detailed asymptotic analysis of model consistency of the Lasso. For various decays of the regularization parameter, we compute asymptotic equivalents of the probability of correct model selection (i.e., variable selection). For a specific rate decay, we show that the Lasso selects all the variables that should enter the model with probability tending to one exponentially fast, while it selects all other variables with strictly positive probability. We show that this property implies that if we run the Lasso for several bootstrapped replications of a given sample, then intersecting the supports of the Lasso bootstrap estimates leads to consistent model selection. This novel variable selection algorithm, referred to as the Bolasso, is compared favorably to other linear regression methods on synthetic data and datasets from the UCI machine learning repository.

We present a generative model for performing sparse probabilistic projections, which includes sparse principal component analysis and sparse canonical correlation analysis as special cases. Sparsity is enforced by means of automatic relevance determination or by imposing appropriate prior distributions, such as generalised hyperbolic distributions. We derive a variational Expectation-Maximisation algorithm for the estimation of the hyperparameters and show that our novel probabilistic approach compares favourably to existing techniques. We illustrate how the proposed method can be applied in the context of cryptoanalysis as a pre-processing tool for the construction of template attacks.

For supervised and unsupervised learning, positive definite kernels allow to use large and potentially infinite dimensional feature spaces with a computational cost that only depends on the number of observations. This is usually done through the penalization of predictor functions by Euclidean or Hilbertian norms. In this paper, we explore penalizing by sparsity-inducing norms such as the L1-norm or the block L1-norm. We assume that the kernel decomposes into a large sum of individual basis kernels which can be embedded in a directed acyclic graph; we show that it is then possible to perform kernel selection through a hierarchical multiple kernel learning framework, in polynomial time in the number of selected kernels. This framework is naturally applied to non linear variable selection; our extensive simulations on synthetic datasets and datasets from the UCI repository show that efficiently exploring the large feature space through sparsity-inducing norms leads to state-of-the-art predictive performance.

We introduce a kernel-based method for change-point analysis within a sequence of temporal observations. Change-point analysis of an (unlabelled) sample of observations consists in, first, testing whether a change in the distribution occurs within the sample, and second, if a change occurs, estimating the change-point instant after which the distribution of the observations switches from one distribution to another different distribution. We propose a test statistics based upon the maximum kernel Fisher discriminant ratio as a measure of homogeneity between segments. We derive its limiting distribution under the null hypothesis (no change occurs), and establish the consistency under the alternative hypothesis (a change occurs). This allows to build a statistical hypothesis testing procedure for testing the presence of change-point, with a prescribed false-alarm probability and detection probability tending to one in the large-sample setting. If a change actually occurs, the test statistics also yields an estimator of the change-point location. Promising experimental results in temporal segmentation of mental tasks from BCI data and pop song indexation are presented.

In multi-task learning several related tasks are considered simultaneously, with the hope that by an appropriate sharing of information across tasks, each task may benefit from the others. In the context of learning linear functions for supervised classification or regression, this can be achieved by including a priori information about the weight vectors associated with the tasks, and how they are expected to be related to each other. In this paper, we assume that tasks are clustered into groups, which are unknown beforehand, and that tasks within a group have similar weight vectors. We design a new spectral norm that encodes this a priori assumption, without the prior knowledge of the partition of tasks into groups, resulting in a new convex optimization formulation for multi-task learning. We show in simulations on synthetic examples and on the iedb MHC-I binding dataset, that our approach outperforms well-known convex methods for multi-task learning, as well as related non convex methods dedicated to the same problem.

It is now well established that sparse signal models are well suited to restoration tasks and can effectively be learned from audio, image, and video data. Recent research has been aimed at learning discriminative sparse models instead of purely reconstructive ones. This paper proposes a new step in that direction with a novel sparse representation for signals belonging to different classes in terms of a shared dictionary and multiple decision functions. It is shown that the linear variant of the model admits a simple probabilistic interpretation, and that its most general variant also admits a simple interpretation in terms of kernels. An optimization framework for learning all the components of the proposed model is presented, along with experiments on standard handwritten digit and texture classification tasks.

We propose a family of kernels between images, defined as kernels between their respective segmentation graphs. The kernels are based on soft matching of subtree-patterns of the respective graphs, leveraging the natural structure of images while remaining robust to the associated segmentation process uncertainty. Indeed, output from morphological segmentation is often represented by a labelled graph, each vertex corresponding to a segmented region, with edges joining neighboring regions. However, such image representations have mostly remained underused for learning tasks, partly because of the observed instability of the segmentation process and the inherent hardness of inexact graph matching with uncertain graphs. Our kernels count common virtual substructures amongst images, which enables to perform efficient supervised classification of natural images with a support vector machine. Moreover, the kernel machinery allows us to take advantage of recent advances in kernel-based learning: (i) semi-supervised learning reduces the required number of labelled images, while (ii) multiple kernel learning algorithms efficiently select the most relevant similarity measures between images within our family.

An efficient and general multiple kernel learning (MKL) algorithm has been recently proposed by Sonnenburg et al. (2006). This approach has opened new perspectives since it makes the MKL approach tractable for large-scale problems, by iteratively using existing support vector machine code. However, it turns out that this iterative algorithm needs several iterations before converging towards a reasonable solution. In this paper, we address the MKL problem through an adaptive 2-norm regularization formulation. Weights on each kernel matrix are included in the standard SVM empirical risk minimization problem with a l1 constraint to encourage sparsity. We propose an algorithm for solving this problem and provide an new insight on MKL algorithms based on block 1-norm regularization by showing that the two approaches are equivalent. Experimental results show that the resulting algorithm converges rapidly and its efficiency compares favorably to other MKL algorithms.

Given a sample covariance matrix, we examine the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This is known as sparse principal component analysis and has a wide array of applications in machine learning and engineering. We formulate a new semidefinite relaxation to this problem and derive a greedy algorithm that computes a full set of good solutions for all numbers of non zero coefficients, with complexity O(n3), where n is the number of variables. We then use the same relaxation to derive sufficient conditions for global optimality of a solution, which can be tested in O(n3). We show on toy examples and biological data that our algorithm does provide globally optimal solutions in many cases.

We present a novel linear clustering framework (Diffrac) which relies on a linear discriminative cost function and a convex relaxation of a combinatorial optimization problem. The large convex optimization problem is solved through a sequence of lower dimensional singular value decompositions. This framework has several attractive properties: (1) although apparently similar to K-means, it exhibits superior clustering performance than K-means, in particular in terms of robustness to noise. (2) It can be readily extended to non linear clustering if the discriminative cost function is based on positive definite kernels, and can then be seen as an alternative to spectral clustering. (3) Prior information on the partition is easily incorporated, leading to state-of-the-art performance for semi-supervised learning, for clustering or classification. We present empirical evaluations of our algorithms on synthetic and real medium-scale datasets.

We propose to test for the homogeneity of two samples by using Kernel Fisher discriminant Analysis. This provides us with a consistent nonparametric test statistic, for which we derive the asymptotic distribution under the null hypothesis. We give experimental evidence of the relevance of our method on both artificial and real datasets.

MOTIVATION: Glycans are covalent assemblies of sugar that play crucial roles in many cellular processes. Recently, comprehensive data about the structure and function of glycans have been accumulated, therefore the need for methods and algorithms to analyze these data is growing fast. RESULTS: This article presents novel methods for classifying glycans and detecting discriminative glycan motifs with support vector machines (SVM). We propose a new class of tree kernels to measure the similarity between glycans. These kernels are based on the comparison of tree substructures, and take into account several glycan features such as the sugar type, the sugar bound type or layer depth. The proposed methods are tested on their ability to classify human glycans into four blood components: leukemia cells, erythrocytes, plasma and serum. They are shown to outperform a previously published method. We also applied a feature selection approach to extract glycan motifs which are characteristic of each blood component. We confirmed that some leukemia-specific glycan motifs detected by our method corresponded to several results in the literature. AVAILABILITY: Softwares are available upon request. SUPPLEMENTARY INFORMATION: Datasets are available at the following website: http://web.kuicr.kyoto-u.ac.jp/supp/yoshi/glycankernel/

The generalized linear discriminant sequence (GLDS) kernel has been shown to provide very good performance and efficiency at the NIST Speaker Recognition Evaluations (SRE) in the last few years. This kernel is based on an explicit map of polynomial expansions of input frames which, because of practical limitations, have to be of a degree less or equal to three. In this paper, we consider an extension of the GLDS kernel to allow not only any polynomial degree but also any embedding, including infinite-dimensional ones associated with Mercer kernels (such as Gaussian kernels). It turns out that the resulting kernels belong to the family of posterior covariance kernels. However, their exact ldquokernelizedrdquo form involves the computation of the Gram matrix on background data, and may be intractable when the background corpus is very large (which is the case in speaker verification). To overcome this problem, we use a low-rank approximation of the Gram matrix to provide an approximate but tractable form of these kernels. We then present comparative experiments on NIST SRE 2005. The results show that our sequence kernel outperforms the GLDS one, and gives similar (individual) performances to the traditional universal background model-Gaussiam mixture model (UBM-GMM) system. As expected, the fusion of both improves the scores.

Active learning refers to algorithmic frameworks aimed at selecting training data points in order to reduce the number of required training data points and/or im- prove the generalization performance of a learning method. In this paper, we present an asymptotic analysis of active learning for generalized linear models. Our analysis holds under the common practical situation of model misspecifica- tion, and is based on realistic assumptions regarding the nature of the sampling distributions, which are usually neither independent nor identical. We derive un- biased estimators of generalization performance, as well as estimators of expected reduction in generalization error after adding a new training data point, that allow us to optimize its sampling distribution through a convex optimization problem. Our analysis naturally leads to an algorithm for sequential active learning which is applicable for all tasks supported by generalized linear models (e.g., binary clas- sification, multi-class classification, regression) and can be applied in non-linear settings through the use of Mercer kernels.

Receiver Operating Characteristic (ROC) curves are a standard way to display the performance of a set of binary classifiers for all feasible ratios of the costs associated with false positives and false negatives. For linear classifiers, the set of classifiers is typically obtained by training once, holding constant the estimated slope and then varying the intercept to obtain a parameterized set of classifiers whose performances can be plotted in the ROC plane. We consider the alternative of varying the asymmetry of the cost function used for training. We show that the ROC curve obtained by varying both the intercept and the asymmetry, and hence the slope, always outperforms the ROC curve obtained by varying only the intercept. In addition, we present a path-following algorithm for the support vector machine (SVM) that can compute efficiently the entire ROC curve, and that has the same computational complexity as training a single classifier. Finally, we provide a theoretical analysis of the relationship between the asymmetric cost model assumed when training a classifier and the cost model assumed in applying the classifier. In particular, we show that the mismatch between the step function used for testing and its convex upper bounds, usually used for training, leads to a provable and quantifiable difference around extreme asymmetries.

Spectral clustering refers to a class of techniques which rely on the eigenstructure of a similarity matrix to partition points into disjoint clusters, with points in the same cluster having high similarity and points in different clusters having low similarity. In this paper, we derive new cost functions for spectral clustering based on measures of error between a given partition and a solution of the spectral relaxation of a minimum normalized cut problem. Minimizing these cost functions with respect to the partition leads to new spectral clustering algorithms. Minimizing with respect to the similarity matrix leads to algorithms for learning the similarity matrix from fully labelled data sets. We apply our learning algorithm to the blind one-microphone speech separation problem, casting the problem as one of segmentation of the spectrogram.

We present a multiple pitch tracking algorithm that is based on direct probabilistic modeling of the spectrogram of the signal. The model is a factorial hidden Markov model whose parameters are learned discriminatively from the Keele pitch database. Our algorithm can track several pitches and determines the number of pitches that are active at any given time. We present simulation results on mixtures of several speech signals and noise, showing the robustness of our approach.

Low-rank matrix decompositions are essential tools in the application of kernel methods to large-scale learning problems. These decompositions have generally been treated as black boxes---the decomposition of the kernel matrix that they deliver is independent of the specific learning task at hand---and this is a potentially significant source of inefficiency. In this paper, we present an algorithm that can exploit side information (e.g., classification labels, regression responses) in the computation of low-rank decompositions for kernel matrices. Our algorithm has the same favorable scaling as state-of-the-art methods such as incomplete Cholesky decomposition---it is linear in the number of data points and quadratic in the rank of the approximation. We present simulation results that show that our algorithm yields decompositions of significantly smaller rank than those found by incomplete Cholesky decomposition.

While kernel canonical correlation analysis (kernel CCA) has been applied in many problems, the asymptotic convergence of the functions estimated from a finite sample to the true functions has not yet been established. This paper gives a rigorous proof of the statistical convergence of kernel CCA and a related method (NOCCO), which provides a theoretical justification for these methods. The result also gives a sufficient condition on the decay of the regularization coefficient in the methods to ensure convergence.

While classical kernel-based classifiers are based on a single kernel, in practice it is often desirable to base classifiers on combinations of multiple kernels. Lanckriet et al. (2004) considered conic combinations of kernel matrices for the support vector machine (SVM), and showed that the optimization of the coefficients of such a combination reduces to a convex optimization problem known as a quadratically-constrained quadratic program (QCQP). Unfortunately, current convex optimization toolboxes can solve this problem only for a small number of kernels and a small number of data points; moreover, the sequential minimal optimization (SMO) techniques that are essential in large-scale implementations of the SVM cannot be applied because the cost function is non-differentiable. We propose a novel dual formulation of the QCQP as a second-order cone programming problem, and show how to exploit the technique of Moreau-Yosida regularization to yield a formulation to which SMO techniques can be applied. We present experimental results that show that our SMO-based algorithm is significantly more efficient than the general-purpose interior point methods available in current optimization toolboxes.

We present an algorithm to perform blind, one-microphone speech sep- aration. Our algorithm separates mixtures of speech without modeling individual speakers. Instead, we formulate the problem of speech sep- aration as a problem in segmenting the spectrogram of the signal into two or more disjoint sets. We build feature sets for our segmenter using classical cues from speech psychophysics. We then combine these fea- tures into parameterized affinity matrices. We also take advantage of the fact that we can generate training examples for segmentation by artifi- cially superposing separately-recorded signals. Thus the parameters of the affinity matrices can be tuned using recent work on learning spectral clustering [1]. This yields an adaptive, speech-specific segmentation al- gorithm that can successfully separate one-microphone speech mixtures.

The problem of learning a sparse conic combination of kernel functions or kernel matrices for classification or regression can be achieved via the regularization by a block 1-norm [1]. In this paper, we present an al- gorithm that computes the entire regularization path for these problems. The path is obtained by using numerical continuation techniques, and involves a running time complexity that is a constant times the complex- ity of solving the problem for one value of the regularization parameter. Working in the setting of kernel linear regression and kernel logistic re- gression, we show empirically that the effect of the block 1-norm reg- ularization differs notably from the (non-block) 1-norm regularization commonly used for variable selection, and that the regularization path is of particular value in the block case.

We present a class of algorithms for independent component analysis (ICA) which use contrast functions based on canonical correlations in a reproducing kernel Hilbert space. On the one hand, we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence. On the other hand, building on recent developments in kernel methods, we show that these criteria can be computed efficiently. Minimizing these criteria leads to flexible and robust algorithms for ICA. We illustrate with simulations involving a wide variety of source distributions, showing that our algorithms outperform many of the presently known algorithms.

Spectral clustering refers to a class of techniques which rely on the eigen- structure of a similarity matrix to partition points into disjoint clusters with points in the same cluster having high similarity and points in dif- ferent clusters having low similarity. In this paper, we derive a new cost function for spectral clustering based on a measure of error between a given partition and a solution of the spectral relaxation of a minimum normalized cut problem. Minimizing this cost function with respect to the partition leads to a new spectral clustering algorithm. Minimizing with respect to the similarity matrix leads to an algorithm for learning the similarity matrix. We develop a tractable approximation of our cost function that is based on the power method of computing eigenvectors.

We propose a novel method of dimensionality reduction for supervised learning. Given a regression or classification problem in which we wish to predict a variable Y from an explanatory vector X, we treat the prob- lem of dimensionality reduction as that of finding a low-dimensional “ef- fective subspace” of X which retains the statistical relationship between X and Y . We show that this problem can be formulated in terms of conditional independence. To turn this formulation into an optimization problem, we characterize the notion of conditional independence using covariance operators on reproducing kernel Hilbert spaces; this allows us to derive a contrast function for estimation of the effective subspace. Un- like many conventional methods, the proposed method requires neither assumptions on the marginal distribution of X, nor a parametric model of the conditional distribution of Y .

We present a class of algorithms for learning the structure of graphical models from data. The algorithms are based on a measure known as the kernel generalized variance (KGV), which essentially allows us to treat all variables on an equal footing as Gaussians in a feature space obtained from Mercer kernels. Thus we are able to learn hybrid graphs involving discrete and continuous variables of arbitrary type. We explore the computational properties of our approach, showing how to use the kernel trick to compute the relevant statistics in linear time. We illustrate our framework with experiments involving discrete and continuous data.

We present an algorithm that induces a class of models with thin junction trees—models that are characterized by an upper bound on the size of the maximal cliques of their triangulated graph. By ensuring that the junction tree is thin, inference in our models remains tractable throughout the learning process. This allows both an efficient implementation of an iterative scaling parameter estimation algorithm and also ensures that inference can be performed efficiently with the final model. We illustrate the approach with applications in handwritten digit recognition and DNA splice site detection.