arXiv subjects

Cesare Molinari

Publications and source records attributed to Cesare Molinari.

At least 19 recordsLinked to original sources

arXiv2026

Learning to control switching nonlinear systems with Koopman operator regression

In this work, we consider the identification and control of nonlinear systems with finite action spaces. The unknown dynamics are estimated from finite samples with Koopman operator regression in a reproducing kernel Hilbert space, yielding a linear switching predictive model, the switches governed by the value of the control variable. In order to perform control in closed-loop, the learned dynamics are employed in an infinite-horizon optimal control problem with time-varying stage cost, which is solved by means of model predictive control. In a theoretical analysis, we derive learning rates for the Koopman dynamics approximation. We further quantify, under suitable assumptions, the sub-optimality of the model predictive control strategy, both in the case of exact Koopman dynamics, and in the case of learned ones. Numerical simulations on the Duffing oscillator complement our theoretical findings.

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arXiv2026

Frank-Wolfe with Moreau Envelope Smoothing for Nonsmooth Nonconvex Problems

We present and analyze Frank-Wolfe with Moreau Envelope Smoothing (FRAMES) for solving nonsmooth nonconvex constrained optimization problems, taking advantage of iterative smoothing via the Moreau envelope followed by one Frank-Wolfe step per iteration. The problem template we consider encompasses splitting problems with multiple convex constraint sets as well as problems with nonsmooth weakly convex regularizers like the MCP or SCAD penalties. We prove convergence, with rates, for both of these cases under a variety of mild assumptions, including inconsistent problems. Additionally, we highlight a new relationship between the Frank-Wolfe gap for a problem with nonsmooth objective and the Frank-Wolfe gap for a smoothed surrogate problem, demonstrating suboptimality of prior analyses. Numerical experiments are performed for matrix factorization problems and nonconvex quadratic splitting over multiple convex constraint sets, where the improvements in analysis are empirically observed.

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arXiv2026

Proximal basin hopping: global optimization with guarantees

Global optimization is a challenging problem, with plenty of algorithms displaying empirical success, but scarce theoretical backing. In this work, we propose a new theoretical framework called Proximal Basin Hopping (PBH), carefully tailored to combine proximal optimization and local minimization. We use it to construct a practical algorithm that converges to the global minimizer with high probability, when using a finite amount of samples. Proximal Basin Hopping outperforms well known algorithms with theoretical backing on standard synthetic hard functions, and real problems such as fitting scaling laws for deep learning. Furthermore, the higher the dimension, the better the performance gap.

cs.LG↗

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.

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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.

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arXiv2025

Existence and optimisation of the partial correlation graphical lasso

The partial correlation graphical LASSO (PCGLASSO) is a penalised likelihood method for Gaussian graphical models which provides scale invariant sparse estimation of the precision matrix and improves upon the popular graphical LASSO method. However, the PCGLASSO suffers from computational challenges due to the non-convexity of its associated optimisation problem. This paper provides some important breakthroughs in the computation of the PCGLASSO. First, the existence of the PCGLASSO estimate is proven when the sample size is smaller than the dimension - a case in which the maximum likelihood estimate does not exist. This means that the PCGLASSO can be used with any Gaussian data. Second, a new alternating algorithm for computing the PCGLASSO is proposed and implemented in the R package PCGLASSO available at https://github.com/JackStorrorCarter/PCGLASSO. This was the first publicly available implementation of the PCGLASSO and provides competitive computation time for moderate dimension size.

stat.ME↗

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-{\L}ojasiewicz 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.

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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.

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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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arXiv2024

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↗

arXiv2024

Automatic Gain Tuning for Humanoid Robots Walking Architectures Using Gradient-Free Optimization Techniques

Developing sophisticated control architectures has endowed robots, particularly humanoid robots, with numerous capabilities. However, tuning these architectures remains a challenging and time-consuming task that requires expert intervention. In this work, we propose a methodology to automatically tune the gains of all layers of a hierarchical control architecture for walking humanoids. We tested our methodology by employing different gradient-free optimization methods: Genetic Algorithm (GA), Covariance Matrix Adaptation Evolution Strategy (CMA-ES), Evolution Strategy (ES), and Differential Evolution (DE). We validated the parameter found both in simulation and on the real ergoCub humanoid robot. Our results show that GA achieves the fastest convergence (10 x 10^3 function evaluations vs 25 x 10^3 needed by the other algorithms) and 100% success rate in completing the task both in simulation and when transferred on the real robotic platform. These findings highlight the potential of our proposed method to automate the tuning process, reducing the need for manual intervention.

cs.RO↗

arXiv2024

Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr\"om method

In this paper, we study how the Koopman operator framework can be combined with kernel methods to effectively control nonlinear dynamical systems. While kernel methods have typically large computational requirements, we show how random subspaces (Nystr\"om approximation) can be used to achieve huge computational savings while preserving accuracy. Our main technical contribution is deriving theoretical guarantees on the effect of the Nystr\"om approximation. More precisely, we study the linear quadratic regulator problem, showing that the approximated Riccati operator converges at the rate $m^{-1/2}$, and the regulator objective, for the associated solution of the optimal control problem, converges at the rate $m^{-1}$, where $m$ is the random subspace size. Theoretical findings are complemented by numerical experiments corroborating our results.

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arXiv2023

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↗

arXiv2023

An Optimal Structured Zeroth-order Algorithm for Non-smooth Optimization

Finite-difference methods are a class of algorithms designed to solve black-box optimization problems by approximating a gradient of the target function on a set of directions. In black-box optimization, the non-smooth setting is particularly relevant since, in practice, differentiability and smoothness assumptions cannot be verified. To cope with nonsmoothness, several authors use a smooth approximation of the target function and show that finite difference methods approximate its gradient. Recently, it has been proved that imposing a structure in the directions allows improving performance. However, only the smooth setting was considered. To close this gap, we introduce and analyze O-ZD, the first structured finite-difference algorithm for non-smooth black-box optimization. Our method exploits a smooth approximation of the target function and we prove that it approximates its gradient on a subset of random {\em orthogonal} directions. We analyze the convergence of O-ZD under different assumptions. For non-smooth convex functions, we obtain the optimal complexity. In the non-smooth non-convex setting, we characterize the number of iterations needed to bound the expected norm of the smoothed gradient. For smooth functions, our analysis recovers existing results for structured zeroth-order methods for the convex case and extends them to the non-convex setting. We conclude with numerical simulations where assumptions are satisfied, observing that our algorithm has very good practical performances.

math.OC↗

arXiv2023

Be greedy and learn: efficient and certified algorithms for parametrized optimal control problems

We consider parametrized linear-quadratic optimal control problems and provide their online-efficient solutions by combining greedy reduced basis methods and machine learning algorithms. To this end, we first extend the greedy control algorithm, which builds a reduced basis for the manifold of optimal final time adjoint states, to the setting where the objective functional consists of a penalty term measuring the deviation from a desired state and a term describing the control energy. Afterwards, we apply machine learning surrogates to accelerate the online evaluation of the reduced model. The error estimates proven for the greedy procedure are further transferred to the machine learning models and thus allow for efficient a posteriori error certification. We discuss the computational costs of all considered methods in detail and show by means of two numerical examples the tremendous potential of the proposed methodology.

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arXiv2023

Regularization properties of dual subgradient flow

Dual gradient descent combined with early stopping represents an efficient alternative to the Tikhonov variational approach when the regularizer is strongly convex. However, for many relevant applications, it is crucial to deal with regularizers which are only convex. In this setting, the dual problem is non smooth, and dual gradient descent cannot be used. In this paper, we study the regularization properties of a subgradient dual flow, and we show that the proposed procedure achieves the same recovery accuracy as penalization methods, while being more efficient from the computational perspective.

math.OC↗

arXiv2022

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-{\L}ojasiewicz 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.

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