We introduce a principled way of computing the Wasserstein distance between two distributions in a federated manner. Namely, we show how to estimate the Wasserstein distance between two samples stored and kept on different devices/clients whilst a central entity/server orchestrates the computations (again, without having access to the samples). To achieve this feat, we take advantage of the geometric properties of the Wasserstein distance -- in particular, the triangle inequality -- and that of the associated {\em geodesics}: our algorithm, FedWad (for Federated Wasserstein Distance), iteratively approximates the Wasserstein distance by manipulating and exchanging distributions from the space of geodesics in lieu of the input samples. In addition to establishing the convergence properties of FedWad, we provide empirical results on federated coresets and federate optimal transport dataset distance, that we respectively exploit for building a novel federated model and for boosting performance of popular federated learning algorithms.

The Sliced-Wasserstein distance (SW) is a computationally efficient and theoretically grounded alternative to the Wasserstein distance. Yet, the literature on its statistical properties – or, more accurately, its generalization properties – with respect to the distribution of slices, beyond the uniform measure, is scarce. To bring new contributions to this line of research, we leverage the PAC-Bayesian theory and a central observation that SW may be interpreted as an average risk, the quantity PAC-Bayesian bounds have been designed to characterize. We provide three types of results: i) PAC-Bayesian generalization bounds that hold on what we refer as adaptive Sliced-Wasserstein distances, i.e. SW defined with respect to arbitrary distributions of slices (among which data-dependent distributions), ii) a principled procedure to learn the distribution of slices that yields maximally discriminative SW, by optimizing our theoretical bounds, and iii) empirical illustrations of our theoretical findings.

The Nyström method, known as an efficient technique for approximating Gram matrices, builds upon a small subset of the data called landmarks, whose choice impacts the quality of the approximated Gram matrix. Various sampling methods have been proposed in the literature to choose such a subset, among which some based on ridge Leverage scores, which come with good theoretical and practical results. Nevertheless, direct computation of ridge leverage scores has an Θ(n3) computation cost if n is the number of data, which is prohibitive when n is large. To tackle this problem, we here propose a Θ(n) divide-and-conquer (DAC) method to approximate ridge leverage scores and we provide theoretical guarantees and empirical results regarding their ability to blend with the Nyström approximation strategy. Our experimental results show that the proposed approximate leverage score sampling scheme achieves a good trade-off between predictive performance and running time.

Developing machine learning methods that are privacy preserving is today a central topic of research, with huge practical impacts. Among the numerous ways to address privacy-preserving learning, we here take the perspective of computing the divergences between distributions under the Differential Privacy (DP) framework — being able to compute divergences between distributions is pivotal for many machine learning problems, such as learning generative models or domain adaptation problems. Instead of resorting to the popular gradient-based sanitization method for DP, we tackle the problem at its roots by focusing on the Sliced Wasserstein Distance and seamlessly making it differentially private. Our main contribution is as follows: we analyze the property of adding a Gaussian perturbation to the intrinsic randomized mechanism of the Sliced Wasserstein Distance, and we establish the sensitivity of the resulting differentially private mechanism. One of our important findings is that this DP mechanism transforms the Sliced Wasserstein distance into another distance, that we call the Smoothed Sliced Wasserstein Distance. This new differentially private distribution distance can be plugged into generative models and domain adaptation algorithms in a transparent way, and we empirically show that it yields highly competitive performance compared with gradient-based DP approaches from the literature, with almost no loss in accuracy for the domain adaptation problems that we consider.

Optical Processing Units (OPUs) -- low-power photonic chips dedicated to large scale random projections -- have been used in previous work to train deep neural networks using Direct Feedback Alignment (DFA), an effective alternative to backpropagation. Here, we demonstrate how to leverage the intrinsic noise of optical random projections to build a differentially private DFA mechanism, making OPUs a solution of choice to provide a \emph{private-by-design} training. We provide a theoretical analysis of our adaptive privacy mechanism, carefully measuring how the noise of optical random projections propagates in the process and gives rise to provable Differential Privacy. Finally, we conduct experiments demonstrating the ability of our learning procedure to achieve solid end-task performance.

The trace regression model, a direct extension of the well-studied linear regression model, allows one to map matrices to real-valued outputs. We here introduce an even more general model, namely the partial-trace regression model, a family of linear mappings from matrix-valued inputs to matrix-valued outputs; this model subsumes the trace regression model and thus the linear regression model. Borrowing tools from quantum information theory, where partial trace operators have been extensively studied, we propose a framework for learning partial trace regression models from data by taking advantage of the so-called low-rank Kraus representation of completely positive maps. We show the relevance of our framework with synthetic and real-world experiments conducted for both i) matrix-to-matrix regression and ii) positive semidefinite matrix completion, two tasks which can be formulated as partial trace regression problems.

We study the properties of the Frank-Wolfe algorithm to solve the m-EXACT-SPARSE reconstruction problem, where a signal y must be expressed as a sparse linear combination of a predefined set of atoms, called dictionary. We prove that when the signal is sparse enough with respect to the coherence of the dictionary, then the iterative process implemented by the Frank-Wolfe algorithm only recruits atoms from the support of the signal, is the smallest set of atoms from the dictionary that allows for a perfect reconstruction of y. We also prove that under this same condition, there exists an iteration beyond which the algorithm converges exponentially.

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We address the problem of dueling bandits defined on partially ordered sets, or posets. In this setting, arms may not be comparable, and there may be several (incomparable) optimal arms. We propose an algorithm, UnchainedBandits, that efficiently finds the set of optimal arms, or Pareto front, of any poset even when pairs of comparable arms cannot be a priori distinguished from pairs of incomparable arms, with a set of minimal assumptions. This means that UnchainedBandits does not require information about comparability and can be used with limited knowledge of the poset. To achieve this, the algorithm relies on the concept of decoys, which stems from social psychology. We also provide theoretical guarantees on both the regret incurred and the number of comparison required by UnchainedBandits, and we report compelling empirical results.

Several sparsity-constrained algorithms, such as orthogonal matching pursuit (OMP) or the Frank-Wolfe (FW) algorithm, with sparsity constraints work by iteratively selecting a novel atom to add to the current nonzero set of variables. This selection step is usually performed by computing the gradient and then by looking for the gradient component with maximal absolute entry. This step can be computationally expensive especially for large-scale and high-dimensional data. In this paper, we aim at accelerating these sparsity-constrained optimization algorithms by exploiting the key observation that, for these algorithms to work, one only needs the coordinate of the gradient's top entry. Hence, we introduce algorithms based on greedy methods and randomization approaches that aim at cheaply estimating the gradient and its top entry. Another of our contribution is to cast the problem of finding the best gradient entry as a best-arm identification in a multiarmed bandit problem. Owing to this novel insight, we are able to provide a bandit-based algorithm that directly estimates the top entry in a very efficient way. Theoretical observations stating that the resulting inexact FW or OMP algorithms act, with high probability, similar to their exact versions are also given. We have carried out several experiments showing that the greedy deterministic and the bandit approaches we propose can achieve an acceleration of an order of magnitude while being as efficient as the exact gradient when used in algorithms, such as OMP, FW, or CoSaMP.

We provide new concentration inequalities for functions of dependent variables. The work extends that of Janson (2004), which proposes concentration inequalities using a combination of the Laplace transform and the idea of fractional graph coloring, as well as many works that derive concentration inequalities using the entropy method (see, e.g., (Boucheron et al., 2003)). We give inequalities for fractionally sub-additive and fractionally self-bounding functions. In the way, we prove a new Talagrand concentration inequality for fractionally sub-additive functions of dependent variables. The results allow us to envision the derivation of generalization bounds for various applications where dependent variables naturally appear, such as in bipartite ranking.

We adapt a policy gradient approach to the problem of reward-based online learning of a non-invasive EEG-based “P300”-speller. We first clarify the nature of the P300-speller classification problem and present a general regularized gradient ascent formula. We then show that when the reward is immediate and binary (namely “bad response” or “good response”), each update is expected to improve the classifier accuracy, whether the actual response is correct or not. We also estimate the robustness of the method to occasional mistaken rewards, i.e. show that the learning efficacy may only linearly decrease with the rate of invalid rewards. The effectiveness of our approach is tested in a series of simulations reproducing the conditions of real experiments. We show in a first experiment that a systematic improvement of the spelling rate is obtained for all subjects in the absence of initial calibration. In a second experiment, we consider the case of the online recovery that is expected to follow failed electrodes. Combined with a specific failure detection algorithm, the spelling error information (typically contained in a “backspace” hit) is shown useful for the policy gradient to adapt the P300 classifier to the new situation, provided the feedback is reliable enough (namely having a reliability greater than 70%).

Cutting-plane methods are well-studied localization (and optimization) algorithms. We show that they provide a natural framework to perform machine learning -and not just to solve optimization problems posed by machine learning- in addition to their intended optimization use. In particular, they allow one to learn sparse classifiers and provide good compression schemes. Moreover, we show that very little effort is required to turn them into effective active learning methods. This last property provides a generic way to design a whole family of active learning algorithms from existing passive methods. We present numerical simulations testifying of the relevance of cutting-plane methods for passive and active learning tasks.

We study the restless bandit problem where arms are associated with stationary $\varphi$-mixing processes and where rewards are therefore dependent: the question that arises from this setting is that of carefully recovering some independence by `ignoring' the values of some rewards. As we shall see, the bandit problem we tackle requires us to address the exploration/exploitation/independence trade-off, which we do by considering the idea of a {\em waiting arm} in the new Remix-UCB algorithm, a generalization of Improved-UCB for the problem at hand, that we introduce. We provide a regret analysis for this bandit strategy; two noticeable features of Remix-UCB are that i) it reduces to the regular Improved-UCB when the $\varphi$-mixing coefficients are all $0$, i.e. when the i.i.d scenario is recovered, and ii) when $\varphi(n)=O(n^{-\alpha})$, it is able to ensure a controlled regret of order $\Ot\left( \Delta_*^{(\alpha- 2)/\alpha} \log^{1/\alpha} T\right),$ where $\Delta_*$ encodes the distance between the best arm and the best suboptimal arm, even in the case when $\alpha<1$, i.e. the case when the $\varphi$-mixing coefficients {\em are not} summable.

We tackle the problem of learning linear classifiers from noisy datasets in a multiclass setting. The two-class version of this problem was studied a few years ago by, e.g. Bylander (1994) and Blum et al. (1996): in these contributions, the proposed approaches to fight the noise revolve around a Perceptron learning scheme fed with peculiar examples computed through a weighted average of points from the noisy training set. We propose to build upon these approaches and we introduce a new algorithm called \uma (for Unconfused Multiclass additive Algorithm) which may be seen as a generalization to the multiclass setting of the previous approaches. In order to characterize the noise we use the \em confusion matrix as a multiclass extension of the classification noise studied in the aforementioned literature. Theoretically well-founded, \uma furthermore displays very good empirical noise robustness, as evidenced by numerical simulations conducted on both synthetic and real data.

This paper provides the first ---to the best of our knowledge--- analysis of online learning algorithms for multiclass problems when the {\em confusion} matrix is taken as a performance measure. The work builds upon recent and elegant results on noncommutative concentration inequalities, i.e. concentration inequalities that apply to matrices, and more precisely to matrix martingales. We do establish generalization bounds for online learning algorithm and show how the theoretical study motivate the proposition of a new confusion-friendly learning procedure. This learning algorithm, called \copa (for COnfusion Passive-Aggressive) is a passive-aggressive learning algorithm; it is shown that the update equations for \copa can be computed analytically, thus allowing the user from having to recours to any optimization package to implement it.

We analyze the problem of call-type classification using data that is weakly labelled. The training data is not systematically annotated, but we consider we have a weak or lazy oracle able to answer the question “Is sample x of class q?” by a simple `yes' or `no' answer. This situation of learning might be encountered in many real-world problems where the cost of labelling data is very high. We prove that it is possible to learn linear classifiers in this setting, by estimating adequate expectations inspired by the Multiclass Bandit paradgim. We propose a learning strategy that builds on Kessler's construction to learn multiclass perceptrons. We test our learning procedure against two real-world datasets from spoken langage understanding and provide compelling results.

We introduce Multiclass Kernel Projection Machines (MKPM), a new formalism that extends the Kernel Projection Machine framework to the multiclass case. Our formulation is based on the use of output codes and it implements a co-regularization scheme by simultaneously constraining the projection dimensions associated with the individual predictors that constitute the global classifier. In order to solve the optimization problem posed by our formulation, we propose an efficient dynamic programming approach. Numerical simulations conducted on a few pattern recognition problems illustrate the soundness of our approach.

We present original empirical Bernstein inequalities for U-statistics with bounded symmetric kernels q. They are expressed with respect to empirical estimates of either the variance of q or the conditional variance that appears in the Bernstein-type inequality for U-statistics derived by Arcones [2]. Our result subsumes other existing empirical Bernstein inequalities, as it reduces to them when U-statistics of order 1 are considered. In addition, it is based on a rather direct argument using two applications of the same (non-empirical) Bernstein inequality for U-statistics. We discuss potential applications of our new inequalities, especially in the realm of learning ranking/scoring functions. In the process, we exhibit an efficient procedure to compute the variance estimates for the special case of bipartite ranking that rests on a sorting argument. We also argue that our results may provide test set bounds and particularly interesting empirical racing algorithms for the problem of online learning of scoring functions.

One of the main problems in probabilistic grammatical inference consists in inferring a stochastic language, i.e. a probability distribution, in some class of probabilistic models, from a sample of strings independently drawn according to a fixed unknown target distribution p. Here, we consider the class of rational stochastic languages composed of stochastic languages that can be computed by multiplicity automata, which can be viewed as a generalization of probabilistic automata. Rational stochastic languages p have a useful algebraic characterization: all the mappings up: v → p(uv) lie in a finite dimensional vector subspace Vp* of the vector space ℝ 〈〈Σ〉〉 composed of all real-valued functions defined over Σ*. Hence, a first step in the grammatical inference process can consist in identifying the subspace Vp*. In this paper, we study the possibility of using Principal Component Analysis to achieve this task. We provide an inference algorithm which computes an estimate of this space and then build a multiplicity automaton which computes an estimate of the target distribution. We prove some theoretical properties of this algorithm and we provide results from numerical simulations that confirm the relevance of our approach.

We address the problem of learning classifiers using several kernel functions. On the contrary to many contributions in the field of learning from different sources of information using kernels, we here do not assume that the kernels used are positive definite. The learning problem that we are interested in involves a misclassification loss term and a regularization term that is expressed by means of a mixed norm. The use of a mixed norm allows us to enforce some sparsity structure, a particular case of which is, for instance, the Group Lasso. We solve the convex problem by employing proximal minimization algorithms, which can be viewed as refined versions of gradient descent procedures capable of naturally dealing with nondifferentiability. A numerical simulation on a Uci dataset shows the modularity of our approach.

We address the problem of efficiently learning Naive Bayes classifiers under class-conditional classification noise (CCCN). Naive Bayes classifiers rely on the hypothesis that the distributions associated to each class are product distributions. When data is subject to CCC-noise, these conditional distributions are themselves mixtures of product distributions. We give analytical formulas which makes it possible to identify them from data subject to CCCN. Then, we design a learning algorithm based on these formulas able to learn Naive Bayes classifiers under CCCN. We present results on artificial datasets and datasets extracted from the UCI repository database. These results show that CCCN can be efficiently and successfully handled.

We address the issue of the learnability of concept classes under three classification noise models in the probably approximately correct framework. After introducing the Class-Conditional Classification Noise (CCCN) model, we investigate the problem of the learnability of concept classes under this particular setting and we show that concept classes that are learnable under the well-known uniform classification noise (CN) setting are also CCCN-learnable, which gives CN = CCCN. We then use this result to prove the equality between the set of concept classes that are CN-learnable and the set of concept classes that are learnable in the Constant Partition Classification Noise (CPCN) setting, or, in other words, we show that CN = CPCN.

In this paper, we propose a new model, the kernel Kalman Filter, to perform various nonlinear time series processing. This model is based on the use of Mercer kernel functions in the framework of the Kalman filter or linear dynamical systems. Thanks to the kernel trick, all the equations involved in our model to perform filtering, smoothing and learning tasks, only require matrix algebra calculus whilst providing the ability to model complex time series. In particular, it is possible to learn dynamics from some nonlinear noisy time series implementing an exact expectation-maximization procedure.

We consider the question of predicting nonlinear time series. Kernel Dy- namical Modeling (KDM), a new method based on kernels, is proposed as an extension to linear dynamical models. The kernel trick is used twice: first, to learn the parameters of the model, and second, to compute preimages of the time series predicted in the feature space by means of Support Vector Regression. Our model shows strong connection with the classic Kalman Filter model, with the kernel feature space as hidden state space. Kernel Dynamical Modeling is tested against two benchmark time series and achieves high quality predictions.