Silvia Villa

Vassilis Apidopoulos, Tomaso Poggio, Lorenzo Rosasco et al.

Iterative regularization is a classic idea in regularization theory, that has recently become popular in machine learning. On the one hand, it allows to design efficient algorithms controlling at the same time numerical and statistical accuracy. On the other hand it allows to shed light on the learning curves observed while training neural networks. In this paper, we focus on iterative regularization in the context of classification. After contrasting this setting with that of linear inverse problems, we develop an iterative regularization approach based on the use of the hinge loss function. More precisely we consider a diagonal approach for a family of algorithms for which we prove convergence as well as rates of convergence and stability results for a suitable classification noise model. Our approach compares favorably with other alternatives, as confirmed by numerical simulations.

Marco Rando, Cesare Molinari, Silvia Villa et al.

We introduce and analyze Structured Stochastic Zeroth order Descent (S-SZD), a finite difference approach that approximates a stochastic gradient on a set of $l\leq d$ orthogonal directions, where $d$ is the dimension of the ambient space. These directions are randomly chosen and may change at each step. For smooth convex functions we prove almost sure convergence of the iterates and a convergence rate on the function values of the form $O( (d/l) k^{-c})$ for every $c<1/2$, which is arbitrarily close to the one of Stochastic Gradient Descent (SGD) in terms of number of iterations. Our bound shows the benefits of using $l$ multiple directions instead of one. For non-convex functions satisfying the Polyak-Łojasiewicz condition, we establish the first convergence rates for stochastic structured zeroth order algorithms under such an assumption. We corroborate our theoretical findings with numerical simulations where the assumptions are satisfied and on the real-world problem of hyper-parameter optimization in machine learning, achieving competitive practical performance.

Saverio Salzo, Silvia Villa

An extension of the Gauss-Newton algorithm is proposed to find local minimizers of penalized nonlinear least squares problems, under generalized Lipschitz assumptions. Convergence results of local type are obtained, as well as an estimate of the radius of the convergence ball. Some applications for solving constrained nonlinear equations are discussed and the numerical performance of the method is assessed on some significant test problems.

Cheik Traoré, Vassilis Apidopoulos, Saverio Salzo et al.

In the context of finite sums minimization, variance reduction techniques are widely used to improve the performance of state-of-the-art stochastic gradient methods. Their practical impact is clear, as well as their theoretical properties. Stochastic proximal point algorithms have been studied as an alternative to stochastic gradient algorithms since they are more stable with respect to the choice of the step size. However, their variance-reduced versions are not as well studied as the gradient ones. In this work, we propose the first unified study of variance reduction techniques for stochastic proximal point algorithms. We introduce a generic stochastic proximal-based algorithm that can be specified to give the proximal version of SVRG, SAGA, and some of their variants. For this algorithm, in the smooth setting, we provide several convergence rates for the iterates and the objective function values, which are faster than those of the vanilla stochastic proximal point algorithm. More specifically, for convex functions, we prove a sublinear convergence rate of $O(1/k)$. In addition, under the Polyak-Łojasiewicz (PL) condition, we obtain linear convergence rates. Finally, our numerical experiments demonstrate the advantages of the proximal variance reduction methods over their gradient counterparts in terms of the stability with respect to the choice of the step size in most cases, especially for difficult problems.

Sofiane Tanji, Andrea Della Vecchia, François Glineur et al.

Kernel methods provide a powerful framework for non parametric learning. They are based on kernel functions and allow learning in a rich functional space while applying linear statistical learning tools, such as Ridge Regression or Support Vector Machines. However, standard kernel methods suffer from a quadratic time and memory complexity in the number of data points and thus have limited applications in large-scale learning. In this paper, we propose Snacks, a new large-scale solver for Kernel Support Vector Machines. Specifically, Snacks relies on a Nyström approximation of the kernel matrix and an accelerated variant of the stochastic subgradient method. We demonstrate formally through a detailed empirical evaluation, that it competes with other SVM solvers on a variety of benchmark datasets.

Christian Daniele, Silvia Villa, Samuel Vaiter et al.

Deep Equilibrium Models (DEQs) are implicit neural networks with fixed points, which have recently gained attention for learning image regularization functionals, particularly in settings involving Gaussian fidelities, where assumptions on the forward operator ensure contractiveness of standard (proximal) Gradient Descent operators. In this work, we extend the application of DEQs to Poisson inverse problems, where the data fidelity term is more appropriately modeled by the Kullback--Leibler divergence. To this end, we introduce a novel DEQ formulation based on Mirror Descent defined in terms of a tailored non-Euclidean geometry that naturally adapts with the structure of the data term. This enables the learning of neural regularizers within a principled training framework. We derive sufficient conditions and establish refined convergence results based on the Kurdyka--Lojasiewicz framework for subanalytic functions with non-closed domains to guarantee the convergence of the learned reconstruction scheme and propose computational strategies that enable both efficient training and parameter-free inference. Numerical experiments show that our method outperforms traditional model-based approaches and it is comparable to the performance of Bregman Plug-and-Play methods, while mitigating their typical drawbacks, such as time-consuming tuning of hyper-parameters. The code is publicly available at https://github.com/christiandaniele/DEQ-MD.

Hippolyte Labarrière, Cesare Molinari, Silvia Villa et al.

Black-Box Variational Inference (BBVI) typically relies on Stochastic Gradient Descent (SGD) to optimize the Evidence Lower Bound (ELBO). However, the stochastic gradients in BBVI inherently exhibit unbounded variance, violating standard assumptions and instead satisfying the weaker Blum-Gladyshev (BG) condition, where variance grows quadratically with distance from the optimum. In this paper, we bridge the gap between stochastic optimization theory and the practical instances of BBVI. Focusing on the broad elliptic location-scale family of parameterized distributions, we offer two main contributions. First, we prove the existence of an ELBO solution, a foundational property usually assumed a priori in the literature. Second, we establish comprehensive convergence guarantees spanning finite-time and asymptotic regimes for Minibatch Projected SGD (PSGD) equipped with dynamic batching and preconditioning under the BG condition. Our theoretical framework demonstrates that dynamic batching combined with preconditioning systematically enables rigorous guarantees even in complex settings. We illustrate our theoretical findings with numerical results, highlighting the efficacy of our approach for modern inference tasks.

Nathan Buskulic, Luca Calatroni, Lorenzo Rosasco et al.

Blind inverse problems arise in many experimental settings where the forward operator is partially or entirely unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward manner. Recently, data-driven approaches have been proposed to address blind inverse problems, demonstrating strong empirical performance and adaptability. However, these methods often lack interpretability and are not supported by rigorous theoretical guarantees, limiting their reliability in applied domains such as imaging inverse problems. In this work, we shed light on learning in blind inverse problems within the simplified yet insightful framework of Linear Minimum Mean Square Estimators (LMMSEs). We provide a theoretical analysis, deriving closed-form expressions for optimal estimators and extending classical results. In particular, we establish equivalences with suitably chosen Tikhonov-regularized formulations, where the regularization depends explicitly on the distributions of the unknown signal, the noise, and the random forward operators. We also prove convergence results of the reconstruction error under appropriate source condition assumptions. Furthermore, we derive finite-sample error bounds that characterize the performance of learned estimators as a function of the noise level, problem conditioning, and number of available samples. These bounds explicitly quantify the impact of operator randomness and reveal the associated convergence rates as this randomness vanishes. Finally, we validate our theoretical findings through illustrative numerical experiments that confirm the predicted convergence behavior.

Hippolyte Labarrière, Cesare Molinari, Lorenzo Rosasco et al.

Overparameterized models trained with (stochastic) gradient descent are ubiquitous in modern machine learning. These large models achieve unprecedented performance on test data, but their theoretical understanding is still limited. In this paper, we take a step towards filling this gap by adopting an optimization perspective. More precisely, we study the implicit regularization properties of the gradient flow "algorithm" for estimating the parameters of a deep diagonal neural network. Our main contribution is showing that this gradient flow induces a mirror flow dynamic on the model, meaning that it is biased towards a specific solution of the problem depending on the initialization of the network. Along the way, we prove several properties of the trajectory.

Mouna Gharbi, Silvia Villa, Emilie Chouzenoux et al.

Data restoration from degraded observations, of sparsity hypotheses, is an active field of study. Traditional iterative optimization methods are now complemented by deep learning techniques. The development of unfolded methods benefits from both families. We carry out a comparative study of three architectures on parameterized chromatographic signal databases, highlighting the performance of these approaches, especially when employing metrics adapted to physico-chemical peak signal characterization.

Cesare Molinari, Mathurin Massias, Lorenzo Rosasco et al.

Iterative regularization exploits the implicit bias of an optimization algorithm to regularize ill-posed problems. Constructing algorithms with such built-in regularization mechanisms is a classic challenge in inverse problems but also in modern machine learning, where it provides both a new perspective on algorithms analysis, and significant speed-ups compared to explicit regularization. In this work, we propose and study the first iterative regularization procedure able to handle biases described by non smooth and non strongly convex functionals, prominent in low-complexity regularization. Our approach is based on a primal-dual algorithm of which we analyze convergence and stability properties, even in the case where the original problem is unfeasible. The general results are illustrated considering the special case of sparse recovery with the $\ell_1$ penalty. Our theoretical results are complemented by experiments showing the computational benefits of our approach.

Marco Rando, Luigi Carratino, Silvia Villa et al.

Gaussian process optimization is a successful class of algorithms(e.g. GP-UCB) to optimize a black-box function through sequential evaluations. However, for functions with continuous domains, Gaussian process optimization has to rely on either a fixed discretization of the space, or the solution of a non-convex optimization subproblem at each evaluation. The first approach can negatively affect performance, while the second approach requires a heavy computational burden. A third option, only recently theoretically studied, is to adaptively discretize the function domain. Even though this approach avoids the extra non-convex optimization costs, the overall computational complexity is still prohibitive. An algorithm such as GP-UCB has a runtime of $O(T^4)$, where $T$ is the number of iterations. In this paper, we introduce Ada-BKB (Adaptive Budgeted Kernelized Bandit), a no-regret Gaussian process optimization algorithm for functions on continuous domains, that provably runs in $O(T^2 d_\text{eff}^2)$, where $d_\text{eff}$ is the effective dimension of the explored space, and which is typically much smaller than $T$. We corroborate our theoretical findings with experiments on synthetic non-convex functions and on the real-world problem of hyper-parameter optimization, confirming the good practical performances of the proposed approach.

Cesare Molinari, Mathurin Massias, Lorenzo Rosasco et al.

We study iterative regularization for linear models, when the bias is convex but not necessarily strongly convex. We characterize the stability properties of a primal-dual gradient based approach, analyzing its convergence in the presence of worst case deterministic noise. As a main example, we specialize and illustrate the results for the problem of robust sparse recovery. Key to our analysis is a combination of ideas from regularization theory and optimization in the presence of errors. Theoretical results are complemented by experiments showing that state-of-the-art performances can be achieved with considerable computational speed-ups.

Simon Matet, Lorenzo Rosasco, Silvia Villa et al.

We consider the problem of designing efficient regularization algorithms when regularization is encoded by a (strongly) convex functional. Unlike classical penalization methods based on a relaxation approach, we propose an iterative method where regularization is achieved via early stopping. Our results show that the proposed procedure achieves the same recovery accuracy as penalization methods, while naturally integrating computational considerations. An empirical analysis on a number of problems provides promising results with respect to the state of the art.

Guillaume Garrigos, Lorenzo Rosasco, Silvia Villa

We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those geometric notions over arbitrary sets. In turn, this allows us to present several new results as well as collect in a unified view a variety of results scattered in the literature. Our contributions include the analysis of infinite dimensional convex minimization problems, showing the first Łojasiewicz inequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach allows to establish unexpected connections between geometry and a priori conditions in inverse problems, such as source conditions, or restricted isometry properties.

Carlo Ciliberto, Lorenzo Rosasco, Silvia Villa

Multi-task learning is a natural approach for computer vision applications that require the simultaneous solution of several distinct but related problems, e.g. object detection, classification, tracking of multiple agents, or denoising, to name a few. The key idea is that exploring task relatedness (structure) can lead to improved performances. In this paper, we propose and study a novel sparse, non-parametric approach exploiting the theory of Reproducing Kernel Hilbert Spaces for vector-valued functions. We develop a suitable regularization framework which can be formulated as a convex optimization problem, and is provably solvable using an alternating minimization approach. Empirical tests show that the proposed method compares favorably to state of the art techniques and further allows to recover interpretable structures, a problem of interest in its own right.

Lorenzo Rosasco, Silvia Villa

Within a statistical learning setting, we propose and study an iterative regularization algorithm for least squares defined by an incremental gradient method. In particular, we show that, if all other parameters are fixed a priori, the number of passes over the data (epochs) acts as a regularization parameter, and prove strong universal consistency, i.e. almost sure convergence of the risk, as well as sharp finite sample bounds for the iterates. Our results are a step towards understanding the effect of multiple epochs in stochastic gradient techniques in machine learning and rely on integrating statistical and optimization results.

Silvia Villa, Lorenzo Rosasco, Tomaso Poggio

We consider the fundamental question of learnability of a hypotheses class in the supervised learning setting and in the general learning setting introduced by Vladimir Vapnik. We survey classic results characterizing learnability in term of suitable notions of complexity, as well as more recent results that establish the connection between learnability and stability of a learning algorithm.

Silvia Villa, Lorenzo Rosasco, Sofia Mosci et al.

We consider a regularized least squares problem, with regularization by structured sparsity-inducing norms, which extend the usual $\ell_1$ and the group lasso penalty, by allowing the subsets to overlap. Such regularizations lead to nonsmooth problems that are difficult to optimize, and we propose in this paper a suitable version of an accelerated proximal method to solve them. We prove convergence of a nested procedure, obtained composing an accelerated proximal method with an inner algorithm for computing the proximity operator. By exploiting the geometrical properties of the penalty, we devise a new active set strategy, thanks to which the inner iteration is relatively fast, thus guaranteeing good computational performances of the overall algorithm. Our approach allows to deal with high dimensional problems without pre-processing for dimensionality reduction, leading to better computational and prediction performances with respect to the state-of-the art methods, as shown empirically both on toy and real data.

Lorenzo Rosasco, Silvia Villa, Sofia Mosci et al.

In this work we are interested in the problems of supervised learning and variable selection when the input-output dependence is described by a nonlinear function depending on a few variables. Our goal is to consider a sparse nonparametric model, hence avoiding linear or additive models. The key idea is to measure the importance of each variable in the model by making use of partial derivatives. Based on this intuition we propose a new notion of nonparametric sparsity and a corresponding least squares regularization scheme. Using concepts and results from the theory of reproducing kernel Hilbert spaces and proximal methods, we show that the proposed learning algorithm corresponds to a minimization problem which can be provably solved by an iterative procedure. The consistency properties of the obtained estimator are studied both in terms of prediction and selection performance. An extensive empirical analysis shows that the proposed method performs favorably with respect to the state-of-the-art methods.