arXiv subjects

Lorenzo Rosasco

Publications and source records attributed to Lorenzo Rosasco.

At least 37 records · Page 2Linked to original sources

arXiv2025

Learning functions, operators and dynamical systems with kernels

This expository article presents the approach to statistical machine learning based on reproducing kernel Hilbert spaces. The basic framework is introduced for scalar-valued learning and then extended to operator learning. Finally, learning dynamical systems is formulated as a suitable operator learning problem, leveraging Koopman operator theory. The manuscript collects the supporting material for the corresponding course taught at the CIME school "Machine Learning: From Data to Mathematical Understanding" in Cetraro.

cs.LG↗

arXiv2025

Neural reproducing kernel Banach spaces and representer theorems for deep networks

Characterizing the function spaces defined by neural networks helps understanding the corresponding learning models and their inductive bias. While in some limits neural networks correspond to function spaces that are Hilbert spaces, these regimes do not capture the properties of the networks used in practice. Indeed, several results have shown that shallow networks can be better characterized in terms of suitable Banach spaces. However, analogous results for deep networks are limited. In this paper we show that deep neural networks define suitable reproducing kernel Banach spaces. These spaces are equipped with norms that enforce a form of sparsity, enabling them to adapt to potential latent structures within the input data and their representations. In particular, by leveraging the theory of reproducing kernel Banach spaces, combined with variational results, we derive representer theorems that justify the finite architectures commonly employed in applications. Our study extends analogous results for shallow networks and represents a step towards understanding the function spaces induced by neural architectures used in practice.

stat.ML↗

arXiv2025

Efficient Numerical Integration in Reproducing Kernel Hilbert Spaces via Leverage Scores Sampling

In this work we consider the problem of numerical integration, i.e., approximating integrals with respect to a target probability measure using only pointwise evaluations of the integrand. We focus on the setting in which the target distribution is only accessible through a set of $n$ i.i.d. observations, and the integrand belongs to a reproducing kernel Hilbert space. We propose an efficient procedure which exploits a small i.i.d. random subset of $m<n$ samples drawn either uniformly or using approximate leverage scores from the initial observations. Our main result is an upper bound on the approximation error of this procedure for both sampling strategies. It yields sufficient conditions on the subsample size to recover the standard (optimal) $n^{-1/2}$ rate while reducing drastically the number of functions evaluations, and thus the overall computational cost. Moreover, we obtain rates with respect to the number $m$ of evaluations of the integrand which adapt to its smoothness, and match known optimal rates for instance for Sobolev spaces. We illustrate our theoretical findings with numerical experiments on real datasets, which highlight the attractive efficiency-accuracy tradeoff of our method compared to existing randomized and greedy quadrature methods. We note that, the problem of numerical integration in RKHS amounts to designing a discrete approximation of the kernel mean embedding of the target distribution. As a consequence, direct applications of our results also include the efficient computation of maximum mean discrepancies between distributions and the design of efficient kernel-based tests.

stat.ML↗

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↗

arXiv2025

Computational Efficiency under Covariate Shift in Kernel Ridge Regression

This paper addresses the covariate shift problem in the context of nonparametric regression within reproducing kernel Hilbert spaces (RKHSs). Covariate shift arises in supervised learning when the input distributions of the training and test data differ, presenting additional challenges for learning. Although kernel methods have optimal statistical properties, their high computational demands in terms of time and, particularly, memory, limit their scalability to large datasets. To address this limitation, the main focus of this paper is to explore the trade-off between computational efficiency and statistical accuracy under covariate shift. We investigate the use of random projections where the hypothesis space consists of a random subspace within a given RKHS. Our results show that, even in the presence of covariate shift, significant computational savings can be achieved without compromising learning performance.

stat.ML↗

arXiv2025

Fast and Fourier Features for Transfer Learning of Interatomic Potentials

Training machine learning interatomic potentials that are both computationally and data-efficient is a key challenge for enabling their routine use in atomistic simulations. To this effect, we introduce franken, a scalable and lightweight transfer learning framework that extracts atomic descriptors from pretrained graph neural networks and transfer them to new systems using random Fourier features-an efficient and scalable approximation of kernel methods. Franken enables fast and accurate adaptation of general-purpose potentials to new systems or levels of quantum mechanical theory without requiring hyperparameter tuning or architectural modifications. On a benchmark dataset of 27 transition metals, franken outperforms optimized kernel-based methods in both training time and accuracy, reducing model training from tens of hours to minutes on a single GPU. We further demonstrate the framework's strong data-efficiency by training stable and accurate potentials for bulk water and the Pt(111)/water interface using just tens of training structures. Our open-source implementation (https://franken.readthedocs.io) offers a fast and practical solution for training potentials and deploying them for molecular dynamics simulations across diverse systems.

physics.comp-ph↗

arXiv2025

Learning convolution operators on compact Abelian groups

We consider the problem of learning convolution operators associated to compact Abelian groups. We study a regularization-based approach and provide corresponding learning guarantees under natural regularity conditions on the convolution kernel. More precisely, we assume the convolution kernel is a function in a translation invariant Hilbert space and analyze a natural ridge regression (RR) estimator. Building on existing results for RR, we characterize the accuracy of the estimator in terms of finite sample bounds. Interestingly, regularity assumptions which are classical in the analysis of RR, have a novel and natural interpretation in terms of space/frequency localization. Theoretical results are illustrated by numerical simulations.

cs.LG↗

arXiv2025

The Nyström method for convex loss functions

We investigate an extension of classical empirical risk minimization, where the hypothesis space consists of a random subspace within a given Hilbert space. Specifically, we examine the Nyström method where the subspaces are defined by a random subset of the data. This approach recovers Nyström approximations used in kernel methods as a specific case. Using random subspaces naturally leads to computational advantages, but a key question is whether it compromises the learning accuracy. Recently, the tradeoffs between statistics and computation have been explored for the square loss and self-concordant losses, such as the logistic loss. In this paper, we extend these analyses to general convex Lipschitz losses, which may lack smoothness, such as the hinge loss used in support vector machines. Our main results show the existence of various scenarios where computational gains can be achieved without sacrificing learning performance. When specialized to smooth loss functions, our analysis recovers most previous results. Moreover, it allows to consider classification problems and translate the surrogate risk bounds into classification error bounds. Indeed, this gives the opportunity to compare the effect of Nyström approximations when combined with different loss functions such as the hinge or the square loss.

stat.ML↗

arXiv2025

Linear quadratic control of nonlinear systems with Koopman operator learning and the Nyström 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 (Nyström 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 Nyström 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.

math.OC↗

arXiv2025

Towards a Learning Theory of Representation Alignment

It has recently been argued that AI models' representations are becoming aligned as their scale and performance increase. Empirical analyses have been designed to support this idea and conjecture the possible alignment of different representations toward a shared statistical model of reality. In this paper, we propose a learning-theoretic perspective to representation alignment. First, we review and connect different notions of alignment based on metric, probabilistic, and spectral ideas. Then, we focus on stitching, a particular approach to understanding the interplay between different representations in the context of a task. Our main contribution here is relating properties of stitching to the kernel alignment of the underlying representation. Our results can be seen as a first step toward casting representation alignment as a learning-theoretic problem.

cs.LG↗

arXiv2025

Q-learning with temporal memory to navigate turbulence

We consider the problem of olfactory searches in a turbulent environment. We focus on agents that respond solely to odor stimuli, with no access to spatial perception nor prior information about the odor. We ask whether navigation to a target can be learned robustly within a sequential decision making framework. We develop a reinforcement learning algorithm using a small set of interpretable olfactory states and train it with realistic turbulent odor cues. By introducing a temporal memory, we demonstrate that two salient features of odor traces, discretized in few olfactory states, are sufficient to learn navigation in a realistic odor plume. Performance is dictated by the sparse nature of turbulent odors. An optimal memory exists which ignores blanks within the plume and activates a recovery strategy outside the plume. We obtain the best performance by letting agents learn their recovery strategy and show that it is mostly casting cross wind, similar to behavior observed in flying insects. The optimal strategy is robust to substantial changes in the odor plumes, suggesting minor parameter tuning may be sufficient to adapt to different environments.

physics.bio-ph↗

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

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

Assumption violations in causal discovery and the robustness of score matching

When domain knowledge is limited and experimentation is restricted by ethical, financial, or time constraints, practitioners turn to observational causal discovery methods to recover the causal structure, exploiting the statistical properties of their data. Because causal discovery without further assumptions is an ill-posed problem, each algorithm comes with its own set of usually untestable assumptions, some of which are hard to meet in real datasets. Motivated by these considerations, this paper extensively benchmarks the empirical performance of recent causal discovery methods on observational i.i.d. data generated under different background conditions, allowing for violations of the critical assumptions required by each selected approach. Our experimental findings show that score matching-based methods demonstrate surprising performance in the false positive and false negative rate of the inferred graph in these challenging scenarios, and we provide theoretical insights into their performance. This work is also the first effort to benchmark the stability of causal discovery algorithms with respect to the values of their hyperparameters. Finally, we hope this paper will set a new standard for the evaluation of causal discovery methods and can serve as an accessible entry point for practitioners interested in the field, highlighting the empirical implications of different algorithm choices.

stat.ME↗

arXiv2024

Trust And Balance: Few Trusted Samples Pseudo-Labeling and Temperature Scaled Loss for Effective Source-Free Unsupervised Domain Adaptation

Deep Neural Networks have significantly impacted many computer vision tasks. However, their effectiveness diminishes when test data distribution (target domain) deviates from the one of training data (source domain). In situations where target labels are unavailable and the access to the labeled source domain is restricted due to data privacy or memory constraints, Source-Free Unsupervised Domain Adaptation (SF-UDA) has emerged as a valuable tool. Recognizing the key role of SF-UDA under these constraints, we introduce a novel approach marked by two key contributions: Few Trusted Samples Pseudo-labeling (FTSP) and Temperature Scaled Adaptive Loss (TSAL). FTSP employs a limited subset of trusted samples from the target data to construct a classifier to infer pseudo-labels for the entire domain, showing simplicity and improved accuracy. Simultaneously, TSAL, designed with a unique dual temperature scheduling, adeptly balance diversity, discriminability, and the incorporation of pseudo-labels in the unsupervised adaptation objective. Our methodology, that we name Trust And Balance (TAB) adaptation, is rigorously evaluated on standard datasets like Office31 and Office-Home, and on less common benchmarks such as ImageCLEF-DA and Adaptiope, employing both ResNet50 and ViT-Large architectures. Our results compare favorably with, and in most cases surpass, contemporary state-of-the-art techniques, underscoring the effectiveness of our methodology in the SF-UDA landscape.

cs.CV↗

arXiv2024

Sim2Real Bilevel Adaptation for Object Surface Classification using Vision-Based Tactile Sensors

In this paper, we address the Sim2Real gap in the field of vision-based tactile sensors for classifying object surfaces. We train a Diffusion Model to bridge this gap using a relatively small dataset of real-world images randomly collected from unlabeled everyday objects via the DIGIT sensor. Subsequently, we employ a simulator to generate images by uniformly sampling the surface of objects from the YCB Model Set. These simulated images are then translated into the real domain using the Diffusion Model and automatically labeled to train a classifier. During this training, we further align features of the two domains using an adversarial procedure. Our evaluation is conducted on a dataset of tactile images obtained from a set of ten 3D printed YCB objects. The results reveal a total accuracy of 81.9%, a significant improvement compared to the 34.7% achieved by the classifier trained solely on simulated images. This demonstrates the effectiveness of our approach. We further validate our approach using the classifier on a 6D object pose estimation task from tactile data.

cs.RO↗

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↗