We present several advances on neural operators by viewing the action of operator layers as the minimizers of Bregman regularized optimization problems over Banach function spaces. The proposed framework allows interpreting the activation operators as Bregman proximity operators from dual to primal space. This novel viewpoint is general enough to recover classical neural operators as well as a new variant, coined Bregman neural operators, which includes the inverse activation operator and features the same expressivity of standard neural operators. Numerical experiments support the added benefits of the Bregman variant of Fourier neural operators for training deeper and more accurate models.

We study the problem of efficiently computing the derivative of the fixed-point of a parametric nondifferentiable contraction map. This problem has wide applications in machine learning, including hyperparameter optimization, meta-learning and data poisoning attacks. We analyze two popular approaches: iterative differentiation (ITD) and approximate implicit differentiation (AID). A key challenge behind the nonsmooth setting is that the chain rule does not hold anymore. We build upon the work by Bolte et al. (2022), who prove linear convergence of nonsmooth ITD under a piecewise Lipschitz smooth assumption. In the deterministic case, we provide a linear rate for AID and an improved linear rate for ITD which closely match the ones for the smooth setting. We further introduce NSID, a new stochastic method to compute the implicit derivative when the contraction map is defined as the composition of an outer map and an inner map which is accessible only through a stochastic unbiased estimator. We establish rates for the convergence of NSID, encompassing the best available rates in the smooth setting. We also present illustrative experiments confirming our analysis.

We analyse a general class of bilevel problems, in which the upper-level problem consists in the minimization of a smooth objective function and the lower-level problem is to find the fixed point of a smooth contraction map. This type of problems include instances of meta-learning, equilibrium models, hyperparameter optimization and data poisoning adversarial attacks. Several recent works have proposed algorithms which warm-start the lower-level problem, i.e. they use the previous lower-level approximate solution as a staring point for the lower-level solver. This warm-start procedure allows one to improve the sample complexity in both the stochastic and deterministic settings, achieving in some cases the order-wise optimal sample complexity. However, there are situations, e.g., meta learning and equilibrium models, in which the warm-start procedure is not well-suited or ineffective. In this work we show that without warm-start, it is still possible to achieve order-wise (near) optimal sample complexity. In particular, we propose a simple method which uses (stochastic) fixed point iterations at the lower-level and projected inexact gradient descent at the upper-level, that reaches an $\epsilon$-stationary point using $O(\epsilon^{-2})$ and $\tilde{O}(\epsilon^{-1})$ samples for the stochastic and the deterministic setting, respectively. Finally, compared to methods using warm-start, our approach yields a simpler analysis that does not need to study the coupled interactions between the upper-level and lower-level iterates.

We present a framework based on bilevel optimization for learning multilayer, deep data representations. On the one hand, the lower-level problem finds a representation by successively minimizing layer-wise objectives made of the sum of a prescribed regularizer as well as a fidelity term and some linear function both depending on the representation found at the previous layer. On the other hand, the upper-level problem optimizes over the linear functions to yield a linearly separable final representation. We show that, by choosing the fidelity term as the quadratic distance between two successive layer-wise representations, the bilevel problem reduces to the training of a feed-forward neural network. Instead, by elaborating on Bregman distances, we devise a novel neural network architecture additionally involving the inverse of the activation function reminiscent of the skip connection used in ResNets. Numerical experiments suggest that the proposed Bregman variant benefits from better learning properties and more robust prediction performance.

In this work we propose a batch multimarginal version of the Greenkhorn algorithm for the entropic-regularized optimal transport problem. This framework is general enough to cover, as particular cases, existing Sinkhorn and Greenkhorn algorithms for the bi-marginal setting, and greedy MultiSinkhorn for the general multimarginal case. We provide a comprehensive convergence analysis based on the properties of the iterative Bregman projections method with greedy control. Linear rate of convergence as well as explicit bounds on the iteration complexity are obtained. When specialized to the above mentioned algorithms, our results give new convergence rates or provide key improvements over the state-of-the-art rates. We present numerical experiments showing that the flexibility of the batch can be exploited to improve performance of Sinkhorn algorithm both in bi-marginal and multimarginal settings.

Bilevel optimization problems are receiving increasing attention in machine learning as they provide a natural framework for hyperparameter optimization and meta-learning. A key step to tackle these problems is the efficient computation of the gradient of the upper-level objective (hypergradient). In this work, we study stochastic approximation schemes for the hypergradient, which are important when the lower-level problem is empirical risk minimization on a large dataset. The method that we propose is a stochastic variant of the approximate implicit differentiation approach in (Pedregosa, 2016). We provide bounds for the mean square error of the hypergradient approximation, under the assumption that the lower-level problem is accessible only through a stochastic mapping which is a contraction in expectation. In particular, our main bound is agnostic to the choice of the two stochastic solvers employed by the procedure. We provide numerical experiments to support our theoretical analysis and to show the advantage of using stochastic hypergradients in practice.

We study a general class of bilevel problems, consisting in the minimization of an upper-level objective which depends on the solution to a parametric fixed-point equation. Important instances arising in machine learning include hyperparameter optimization, meta-learning, and certain graph and recurrent neural networks. Typically the gradient of the upper-level objective (hypergradient) is hard or even impossible to compute exactly, which has raised the interest in approximation methods. We investigate some popular approaches to compute the hypergradient, based on reverse mode iterative differentiation and approximate implicit differentiation. Under the hypothesis that the fixed point equation is defined by a contraction mapping, we present a unified analysis which allows for the first time to quantitatively compare these methods, providing explicit bounds for their iteration complexity. This analysis suggests a hierarchy in terms of computational efficiency among the above methods, with approximate implicit differentiation based on conjugate gradient performing best. We present an extensive experimental comparison among the methods which confirm the theoretical findings.

A common approach in multi-task learning is to encourage the tasks to share a low dimensional representation. This has led to the popular method of trace norm regularization, which has proved effective in many applications. In this paper, we extend this approach by allowing the tasks to partition into different groups, within which trace norm regularization is separately applied. We propose a continuous bilevel optimization framework to simultaneously identify groups of related tasks and learn a low dimensional representation within each group. Hinging on recent results on the derivative of generalized matrix functions, we devise a smooth approximation of the upper-level objective via a dual forward-backward algorithm with Bregman distances. This allows us to solve the bilevel problem by a gradient-based scheme. Numerical experiments on synthetic and benchmark datasets support the effectiveness of the proposed method.

We present a novel algorithm to estimate the barycenter of arbitrary probability distributions with respect to the Sinkhorn divergence. Based on a Frank-Wolfe optimization strategy, our approach proceeds by populating the support of the barycenter incrementally, without requiring any pre-allocation. We consider discrete as well as continuous distributions, proving convergence rates of the proposed algorithm in both settings. Key elements of our analysis are a new result showing that the Sinkhorn divergence on compact domains has Lipschitz continuous gradient with respect to the Total Variation and a characterization of the sample complexity of Sinkhorn potentials. Experiments validate the effectiveness of our method in practice.

In this paper, we discuss how a suitable family of tensor kernels can be used to efficiently solve nonparametric extensions of lp regularized learning methods. Our main contribution is proposing a fast dual algorithm, and showing that it allows to solve the problem efficiently. Our results contrast recent findings suggesting kernel methods cannot be extended beyond Hilbert setting. Numerical experiments confirm the effectiveness of the method.

We introduce a framework based on bilevel programming that unifies gradient-based hyperparameter optimization and meta-learning. We show that an approximate version of the bilevel problem can be solved by taking into explicit account the optimization dynamics for the inner objective. Depending on the specific setting, the outer variables take either the meaning of hyperparameters in a supervised learning problem or parameters of a meta-learner. We provide sufficient conditions under which solutions of the approximate problem converge to those of the exact problem. We instantiate our approach for meta-learning in the case of deep learning where representation layers are treated as hyperparameters shared across a set of training episodes. In experiments, we confirm our theoretical findings, present encouraging results for few-shot learning and contrast the bilevel approach against classical approaches for learning-to-learn.

In many applications of finance, biology and sociology, complex systems involve entities interacting with each other. These processes have the peculiarity of evolving over time and of comprising latent factors, which influence the system without being explicitly measured. In this work we present latent variable time-varying graphical lasso (LTGL), a method for multivariate time-series graphical modelling that considers the influence of hidden or unmeasurable factors. The estimation of the contribution of the latent factors is embedded in the model which produces both sparse and low-rank components for each time point. In particular, the first component represents the connectivity structure of observable variables of the system, while the second represents the influence of hidden factors, assumed to be few with respect to the observed variables. Our model includes temporal consistency on both components, providing an accurate evolutionary pattern of the system. We derive a tractable optimisation algorithm based on alternating direction method of multipliers, and develop a scalable and efficient implementation which exploits proximity operators in closed form. LTGL is extensively validated on synthetic data, achieving optimal performance in terms of accuracy, structure learning and scalability with respect to ground truth and state-of-the-art methods for graphical inference. We conclude with the application of LTGL to real case studies, from biology and finance, to illustrate how our method can be successfully employed to gain insights on multivariate time-series data.

Regression with group-sparsity penalty plays a central role in high-dimensional prediction problems. Most of existing methods require the group structure to be known a priori. In practice, this may be a too strong assumption, potentially hampering the effectiveness of the regularization method. To circumvent this issue, we present a method to estimate the group structure by means of a continuous bilevel optimization problem where the data is split into training and validation sets. Our approach relies on an approximation scheme where the lower level problem is replaced by a smooth dual forward-backward algorithm with Bregman distances. We provide guarantees regarding the convergence of the approximate procedure to the exact problem and demonstrate the well behaviour of the proposed method on synthetic experiments. Finally, a preliminary application to genes expression data is tackled with the purpose of unveiling functional groups.

We present an algorithm for dictionary learning that is based on the alternating proximal algorithm studied by Attouch, Bolte, Redont, and Soubeyran (2010), coupled with a reliable and efficient dual algorithm for computation of the related proximity operators. This algorithm is suitable for a general dictionary learning model composed of a Bregman-type data fit term that accounts for the goodness of the representation and several convex penalization terms on the coefficients and atoms, explaining the prior knowledge at hand. As Attouch et al. recently proved, an alternating proximal scheme ensures better convergence properties than the simpler alternating minimization. We take care of the issue of inexactness in the computation of the involved proximity operators, giving a sound stopping criterion for the dual inner algorithm, which keeps under control the related errors, unavoidable for such a complex penalty terms, providing ultimately an overall effective procedure. Thanks to the generality of the proposed framework, we give an application in the context of genome-wide data understanding, revising the model proposed by Nowak, Hastie, Pollack, and Tibshirani (2011). The aim is to extract latent features (atoms) and perform segmentation on array-based comparative genomic hybridization (aCGH) data. We improve several important aspects that increase the quality and interpretability of the results. We show the effectiveness of the proposed model with two experiments on synthetic data, which highlight the enhancements over the original model.

We propose an algorithm for the construction of a nearly optimal integer to integer approximation of the Karhunen-Loeve Transform. The algorithm is based on the method of P. Hao and Q. Shi as described in [1] but-unlike described in the paper-we vary the pivoting in order to obtain a better approximation of the linear transform. We have then developed an algorithm for hyperspectral image lossless compression that uses first an Integer Wavelet Transform for the spatial decorrelation, then our Integer-KLT for the spectral decorrelation, and finally a 3D context-based adaptive arithmetic coding (3D-CBAC), which exploits the dependencies among symbols. The results on the AVIRIS images are better than those indicated in literature.