arXiv subjects

Silvia Villa

Publications and source records attributed to Silvia Villa.

At least 19 recordsLinked to original sources

arXiv2026

Langevin for Nonconvex Optimization: Exact, Inexact and Zeroth-Order

We study Langevin-based methods for non-convex optimization under smoothness and dissipativity assumptions. Our focus is on obtaining non-asymptotic bounds for the expected excess risk rather than sampling guarantees for the full target distribution. The key ingredient of our analysis is a direct passage from relative entropy to objective-value error, based on a weighted Csiszár--Kullback--Pinsker inequality and exponential-moment estimates. This avoids intermediate Wasserstein bounds and yields sharper dependence on the Log-Sobolev constant, a quantity that may scale exponentially with the inverse temperature and the dimension in non-convex problems. We first analyze the Unadjusted Langevin Algorithm with exact gradients and derive explicit bounds on $\mathbb{E}[F(x_k)]-\min F$ in terms of the inverse temperature, dimension, stepsize, smoothness and dissipativity parameters, and the Log-Sobolev constant. We then extend the result to an inexact-gradient version of ULA, allowing for biased and stochastic gradient surrogates whose mean-square error grows at most quadratically in the state. This framework covers stochastic gradients and zeroth-order estimators based only on function evaluations. In particular, we show that both Gaussian and spherical finite-difference estimators fit into the inexact-ULA theory and obtain explicit function-evaluation complexity bounds for zeroth-order Langevin optimization. To the best of our knowledge, these are the first non-asymptotic global non-convex optimization complexity bounds for zeroth-order ULA. We also provide numerical experiments illustrating the behavior of the proposed zeroth-order Langevin schemes.

math.OC↗

arXiv2026

Learning Ergodic Dynamical Systems from a Finite Trajectory

We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system. More precisely, we study discrete-time autonomous stochastic systems defining time-homogeneous Markov processes. We first focus on estimating the optimal one-step prediction function by nonlinear least squares, and derive high-probability guarantees measured with respect to the invariant measure of the process. These results make explicit how the non-independent and non-identically distributed nature of trajectory data modifies the classical statistical learning analysis. We then extend the framework to higher-order systems and finite-state spaces. Finally, we show that the same least squares and concentration arguments naturally extend to learning Koopman operators. Our approach combines tools from statistical learning theory and quantitative ergodic theory for Markov chains. It relies, in particular, on a concentration inequality for Hilbert-space-valued additive functionals of uniformly geometrically ergodic Markov chains.

stat.ML↗

arXiv2026

Convergence of zeroth-order proximal point algorithms in the high-temperature regime

Efficient methods for non-convex black-box optimization largely rely on sampling. In this context, the Zeroth-Order Proximal Operator (ZOPO) and the corresponding Zeroth-Order Proximal Point Algorithm (ZOPPA) have attracted significant interest, as they combine the advantage of requiring only objective evaluations with the powerful theoretical framework of proximal point algorithms. ZOPO depends on a temperature parameter which, when going to zero, reduces ZOPO to the exact proximal operator. By exploiting this property, the vanishing-temperature regime has been leveraged in several works to obtain theoretical guarantees via inexact proximal methods. However, this regime is computationally unsustainable when sampling is used to estimate ZOPO, since the corresponding Monte Carlo estimators suffer from severe variance issues. We therefore propose a comprehensive analysis of ZOPO for any fixed positive temperature, and prove convergence of ZOPPA under minimal assumptions on the objective function. We do so by demonstrating that ZOPPA can be interpreted as an exact proximal point method applied to an auxiliary smoothed objective, rather than an inexact method on the original function. Importantly, we further derive explicit guarantees connecting this smoothed problem back to the original objective and establish convergence results for the sampled method (S-ZOPPA) at a fixed temperature.

math.OC↗

arXiv2026

SGD for Variational Inference: Tackling Unbounded Variance via Preconditioning and Dynamic Batching

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.

cs.LG↗

arXiv2026

On the Sample Complexity of Learning for Blind Inverse Problems

Blind inverse problems arise in many experimental settings where both the signal of interest and the forward operator are (partially) unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward manner due to identifiability issues and symmetric solutions inherent to the blind setting. Recently, data-driven approaches have been proposed to address such problems, demonstrating strong empirical performance and adaptability. However, these methods often lack interpretability and are not supported by theoretical guarantees, limiting their reliability in domains such as applied imaging where a blind approach often relates to a calibration of the acquisition device. In this work, we shed light on learning in blind inverse problems within the insightful framework of Linear Minimum Mean Square Estimators (LMMSEs). We provide a theoretical analysis, deriving closed-form expressions for optimal estimators and extending classical recovery results to the blind setting. In particular, we establish equivalences with tailored Tikhonov-regularized formulations, where the regularization structure depends explicitly on the distributions of the unknown signal, of the noise, and of the random forward operator. We also show how the reconstruction error converges as the noise and the randomness of the operator diminish when we use a source condition assumption. Furthermore, we derive finite-sample error bounds that characterize the performance of the 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 show explicitly the dependence of the associated convergence rates to this randomness factors. Finally, we validate our theoretical findings through illustrative exemplar numerical experiments that confirm the predicted convergence behavior.

cs.LG↗

arXiv2026

Optimization Insights into Deep Diagonal Linear Networks

Gradient-based methods successfully train highly overparameterized models in practice, even though the associated optimization problems are markedly nonconvex. Understanding the mechanisms that make such methods effective has become a central problem in modern optimization. To investigate this question in a tractable setting, we study Deep Diagonal Linear Networks. These are multilayer architectures with a reparameterization that preserves convexity in the effective parameter, while inducing a nontrivial geometry in the optimization landscape. Under mild initialization conditions, we show that gradient flow on the layer parameters induces a mirror-flow dynamic in the effective parameter space. This structural insight yields explicit convergence guarantees, including exponential decay of the loss under a Polyak-Lojasiewicz condition, and clarifies how the parametrization and initialization scale govern the training speed. Overall, our results demonstrate that deep diagonal over parameterizations, despite their apparent complexity, can endow standard gradient methods with well-behaved and interpretable optimization dynamics.

cs.LG↗

arXiv2025

Deep Equilibrium models for Poisson Imaging Inverse problems via Mirror Descent

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.

math.OC↗

arXiv2025

Analyse comparative d'algorithmes de restauration en architecture dépliée pour des signaux chromatographiques parcimonieux

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.

eess.SP↗

arXiv2025

Model Consistency of the Iterative Regularization of Dual Ascent for Low-Complexity Regularization

Regularization is a core component of modern inverse problems, as it helps establish the well-posedness of the solution of interest. Popular regularization approaches include variational regularization and iterative regularization. The former involves solving a variational optimization problem, which consists of a data-fidelity term and a regularization term, balanced by an appropriate weighting parameter. The latter mitigates overfitting to noise by selecting a suitable stopping time during the iterative process. A key topic in the study of regularization is the relationship between the regularized solution and the original ground truth. When the ground truth possesses a low-complexity structure referred to as the "model" it can be shown that under appropriate regularization promoting the same structure, the solution to the regularized problem is robust to small perturbations. This property is called "model consistency". For variational regularization, model consistency in linear inverse problems has been studied in [1]. However, for iterative regularization, model consistency remains an open question. In this paper, building on recent developments in partial smoothness [1], we show that when the noise level is sufficiently small and an appropriate stopping criterion is used, iterative regularization is model consistent as well. Moreover, we show that the considered algorithm exhibits local linear behavior of the regularization. We provide numerical simulations to support our theoretical findings.

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arXiv2025

A Structured Proximal Stochastic Variance Reduced Zeroth-order Algorithm

Minimizing finite sums of functions is a central problem in optimization, arising in numerous practical applications. Such problems are commonly addressed using first-order optimization methods. However, these procedures cannot be used in settings where gradient information is unavailable. Finite-difference methods provide an alternative by approximating gradients through function evaluations along a set of directions. For finite-sum minimization problems, it was shown that incorporating variance-reduction techniques into finite-difference methods can improve convergence rates. Additionally, recent studies showed that imposing structure on the directions (e.g., orthogonality) enhances performance. However, the impact of structured directions on variance-reduced finite-difference methods remains unexplored. In this work, we close this gap by proposing a structured variance-reduced finite-difference algorithm for non-smooth finite-sum minimization. We analyze the proposed method, establishing convergence rates for non-convex functions and those satisfying the Polyak-Łojasiewicz condition. Our results show that our algorithm achieves state-of-the-art convergence rates while incurring lower per-iteration costs. Finally, numerical experiments highlight the strong practical performance of our method.

math.OC↗

arXiv2025

Inexact JKO and proximal-gradient algorithms in the Wasserstein space

This paper studies the convergence properties of the inexact Jordan-Kinderlehrer-Otto (JKO) scheme and proximal-gradient algorithm in the context of Wasserstein spaces. The JKO scheme, a widely-used method for approximating solutions to gradient flows in Wasserstein spaces, typically assumes exact solutions to iterative minimization problems. However, practical applications often require approximate solutions due to computational limitations. This work focuses on the convergence of the scheme to minimizers for the underlying functional and addresses these challenges by analyzing two types of inexactness: errors in Wasserstein distance and errors in energy functional evaluations. The paper provides rigorous convergence guarantees under controlled error conditions, demonstrating that weak convergence can still be achieved with inexact steps. The analysis is further extended to proximal-gradient algorithms, showing that convergence is preserved under inexact evaluations.

math.OC↗

arXiv2025

Preconditioned primal-dual dynamics in convex optimization: non-ergodic convergence rates

We introduce and analyze a continuous primal-dual dynamical system in the context of the minimization problem $f(x)+g(Ax)$, where $f$ and $g$ are convex functions and $A$ is a linear operator. In this setting, the trajectories of the Arrow-Hurwicz continuous flow may not converge, accumulating at points that are not solutions. Our proposal is inspired by the primal-dual algorithm of Chambolle and Pock (2011), where convergence and splitting on the primal-dual variable are ensured by adequately preconditioning the proximal-point algorithm. We consider a family of preconditioners, which are allowed to depend on time and on the operator $A$, but not on the functions $f$ and $g$, and analyze asymptotic properties of the corresponding preconditioned flow. Fast convergence rates for the primal-dual gap and optimality of its (weak) limit points are obtained, in the general case, for asymptotically antisymmetric preconditioners, and, in the case of linearly constrained optimization problems, under milder hypotheses. Numerical examples support our theoretical findings, especially in favor of the antisymmetric preconditioners.

math.OC↗

arXiv2025

A Structured Tour of Optimization with Finite Differences

Finite-difference methods are widely used for zeroth-order optimization in settings where gradient information is unavailable or expensive to compute. These procedures mimic first-order strategies by approximating gradients through function evaluations along a set of random directions. From a theoretical perspective, recent studies indicate that imposing structure (such as orthogonality) on the chosen directions allows for the derivation of convergence rates comparable to those achieved with unstructured random directions (i.e., directions sampled independently from a distribution). Empirically, although structured directions are expected to enhance performance, they often introduce additional computational costs, which can limit their applicability in high-dimensional settings. In this work, we examine the impact of structured direction selection in finite-difference methods. We review and extend several strategies for constructing structured direction matrices and compare them with unstructured approaches in terms of computational cost, gradient approximation quality, and convergence behavior. Our evaluation spans both synthetic tasks and real-world applications such as adversarial perturbation. The results demonstrate that structured directions can be generated with computational costs comparable to unstructured ones while significantly improving gradient estimation accuracy and optimization performance.

math.OC↗

arXiv2024

Iterative regularization in classification via hinge loss diagonal descent

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.

stat.ML↗

arXiv2024

Stochastic Zeroth order Descent with Structured Directions

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.

math.OC↗

arXiv2024

Variance reduction techniques for stochastic proximal point algorithms

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.

math.OC↗

arXiv2024

On Learning the Optimal Regularization Parameter in Inverse Problems

Selecting the best regularization parameter in inverse problems is a classical and yet challenging problem. Recently, data-driven approaches have become popular to tackle this challenge. These approaches are appealing since they do require less a priori knowledge, but their theoretical analysis is limited. In this paper, we propose and study a statistical machine learning approach, based on empirical risk minimization. Our main contribution is a theoretical analysis, showing that, provided with enough data, this approach can reach sharp rates while being essentially adaptive to the noise and smoothness of the problem. Numerical simulations corroborate and illustrate the theoretical findings. Our results are a step towards grounding theoretically data-driven approaches to inverse problems.

math.ST↗

arXiv2024

On the convergence of proximal gradient methods for convex simple bilevel optimization

This paper studies proximal gradient iterations for solving simple bilevel optimization problems where both the upper and the lower level cost functions are split as the sum of differentiable and (possibly nonsmooth) proximable functions. We develop a novel convergence recipe for iteration varying stepsizes that relies on Barzilai-Borwein type local estimates for the differentiable terms. Leveraging the convergence recipe, under global Lipschitz gradient continuity, we establish convergence for a nonadaptive stepsize sequence, without requiring any strong convexity or linesearch. In the locally Lipschitz differentiable setting, we develop an adaptive linesearch method that introduces a systematic adaptive scheme enabling large and nonmonotonic stepsize sequences while being insensitive to the choice of hyperparameters and initialization. Numerical simulations are provided showcasing favorable convergence speed of our methods.

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