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Researcher profile

Alessandro Rudi

Alessandro Rudi contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Machine LearningArtificial Intelligencemath.OCmath.STStatistics TheoryComputer VisionData Structures and Algorithmsmath.FAquant-ph

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Published work

15 published item(s)

preprint2026arXiv

Minimax Optimal Variance-Aware Regret Bounds for Multinomial Logistic MDPs

We study reinforcement learning for episodic Markov Decision Processes (MDPs) whose transitions are modelled by a multinomial logistic (MNL) model. Existing algorithms for MNL mixture MDPs yield a regret of $\smash{\tilde{O}(dH^2\sqrt{T})}$ (Li et al., 2024), where $d$ is the feature dimension, $H$ the episode length, and $T$ the number of episodes. Inspired by the logistic bandit literature (Abeille et al., 2021; Faury et al., 2022; Boudart et al., 2026), we introduce a problem-dependent constant $\barσ\_T \leq 1/2$, measuring the normalised average variance of the optimal downstream value function along the learner's trajectory. We propose an algorithm achieving a regret of $\smash{\tilde{O}(dH^2\barσ\_T\sqrt{T})}$, which recovers the existing bound in the worst case and improves upon it for structured MDPs. For instance, for KL-constrained robust MDPs, $\barσ\_T = O(H^{-1})$, reducing the horizon dependence by a factor $H$. We further establish a matching $\smash{Ω(dH^2\barσ\_T\sqrt{T})}$ lower bound, proving minimax optimality (up to logarithmic factors) and fully characterising the regret complexity of MNL mixture MDPs for the first time.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Learning PSD-valued functions using kernel sums-of-squares

Shape constraints such as positive semi-definiteness (PSD) for matrices or convexity for functions play a central role in many applications in machine learning and sciences, including metric learning, optimal transport, and economics. Yet, very few function models exist that enforce PSD-ness or convexity with good empirical performance and theoretical guarantees. In this paper, we introduce a kernel sum-of-squares model for functions that take values in the PSD cone, which extends kernel sums-of-squares models that were recently proposed to encode non-negative scalar functions. We provide a representer theorem for this class of PSD functions, show that it constitutes a universal approximator of PSD functions, and derive eigenvalue bounds in the case of subsampled equality constraints. We then apply our results to modeling convex functions, by enforcing a kernel sum-of-squares representation of their Hessian, and show that any smooth and strongly convex function may be thus represented. Finally, we illustrate our methods on a PSD matrix-valued regression task, and on scalar-valued convex regression.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Measuring dissimilarity with diffeomorphism invariance

Measures of similarity (or dissimilarity) are a key ingredient to many machine learning algorithms. We introduce DID, a pairwise dissimilarity measure applicable to a wide range of data spaces, which leverages the data's internal structure to be invariant to diffeomorphisms. We prove that DID enjoys properties which make it relevant for theoretical study and practical use. By representing each datum as a function, DID is defined as the solution to an optimization problem in a Reproducing Kernel Hilbert Space and can be expressed in closed-form. In practice, it can be efficiently approximated via Nyström sampling. Empirical experiments support the merits of DID.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Non-Convex Optimization with Certificates and Fast Rates Through Kernel Sums of Squares

We consider potentially non-convex optimization problems, for which optimal rates of approximation depend on the dimension of the parameter space and the smoothness of the function to be optimized. In this paper, we propose an algorithm that achieves close to optimal a priori computational guarantees, while also providing a posteriori certificates of optimality. Our general formulation builds on infinite-dimensional sums-of-squares and Fourier analysis, and is instantiated on the minimization of multivariate periodic functions.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Nyström Kernel Mean Embeddings

Kernel mean embeddings are a powerful tool to represent probability distributions over arbitrary spaces as single points in a Hilbert space. Yet, the cost of computing and storing such embeddings prohibits their direct use in large-scale settings. We propose an efficient approximation procedure based on the Nyström method, which exploits a small random subset of the dataset. Our main result is an upper bound on the approximation error of this procedure. It yields sufficient conditions on the subsample size to obtain the standard $n^{-1/2}$ rate while reducing computational costs. We discuss applications of this result for the approximation of the maximum mean discrepancy and quadrature rules, and illustrate our theoretical findings with numerical experiments.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

On the Benefits of Large Learning Rates for Kernel Methods

This paper studies an intriguing phenomenon related to the good generalization performance of estimators obtained by using large learning rates within gradient descent algorithms. First observed in the deep learning literature, we show that a phenomenon can be precisely characterized in the context of kernel methods, even though the resulting optimization problem is convex. Specifically, we consider the minimization of a quadratic objective in a separable Hilbert space, and show that with early stopping, the choice of learning rate influences the spectral decomposition of the obtained solution on the Hessian's eigenvectors. This extends an intuition described by Nakkiran (2020) on a two-dimensional toy problem to realistic learning scenarios such as kernel ridge regression. While large learning rates may be proven beneficial as soon as there is a mismatch between the train and test objectives, we further explain why it already occurs in classification tasks without assuming any particular mismatch between train and test data distributions.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

On the Consistency of Max-Margin Losses

The foundational concept of Max-Margin in machine learning is ill-posed for output spaces with more than two labels such as in structured prediction. In this paper, we show that the Max-Margin loss can only be consistent to the classification task under highly restrictive assumptions on the discrete loss measuring the error between outputs. These conditions are satisfied by distances defined in tree graphs, for which we prove consistency, thus being the first losses shown to be consistent for Max-Margin beyond the binary setting. We finally address these limitations by correcting the concept of Max-Margin and introducing the Restricted-Max-Margin, where the maximization of the loss-augmented scores is maintained, but performed over a subset of the original domain. The resulting loss is also a generalization of the binary support vector machine and it is consistent under milder conditions on the discrete loss.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spaces

In this paper, we study regression problems over a separable Hilbert space with the square loss, covering non-parametric regression over a reproducing kernel Hilbert space. We investigate a class of spectral/regularized algorithms, including ridge regression, principal component regression, and gradient methods. We prove optimal, high-probability convergence results in terms of variants of norms for the studied algorithms, considering a capacity assumption on the hypothesis space and a general source condition on the target function. Consequently, we obtain almost sure convergence results with optimal rates. Our results improve and generalize previous results, filling a theoretical gap for the non-attainable cases.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Second order conditions to decompose smooth functions as sums of squares

We consider the problem of decomposing a regular non-negative function as a sum of squares of functions which preserve some form of regularity. In the same way as decomposing non-negative polynomials as sum of squares of polynomials allows to derive methods in order to solve global optimization problems on polynomials, decomposing a regular function as a sum of squares allows to derive methods to solve global optimization problems on more general functions. As the regularity of the functions in the sum of squares decomposition is a key indicator in analyzing the convergence and speed of convergence of optimization methods, it is important to have theoretical results guaranteeing such a regularity. In this work, we show second order sufficient conditions in order for a $p$ times continuously differentiable non-negative function to be a sum of squares of $p-2$ differentiable functions. The main hypothesis is that, locally, the function grows quadratically in directions which are orthogonal to its set of zeros. The novelty of this result, compared to previous works is that it allows sets of zeros which are continuous as opposed to discrete, and also applies to manifolds as opposed to open sets of $\R^d$. This has applications in problems where manifolds of minimizers or zeros typically appear, such as in optimal transport, and for minimizing functions defined on manifolds.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A General Framework for Consistent Structured Prediction with Implicit Loss Embeddings

We propose and analyze a novel theoretical and algorithmic framework for structured prediction. While so far the term has referred to discrete output spaces, here we consider more general settings, such as manifolds or spaces of probability measures. We define structured prediction as a problem where the output space lacks a vectorial structure. We identify and study a large class of loss functions that implicitly defines a suitable geometry on the problem. The latter is the key to develop an algorithmic framework amenable to a sharp statistical analysis and yielding efficient computations. When dealing with output spaces with infinite cardinality, a suitable implicit formulation of the estimator is shown to be crucial.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Approximating Hamiltonian dynamics with the Nyström method

Simulating the time-evolution of quantum mechanical systems is BQP-hard and expected to be one of the foremost applications of quantum computers. We consider classical algorithms for the approximation of Hamiltonian dynamics using subsampling methods from randomized numerical linear algebra. We derive a simulation technique whose runtime scales polynomially in the number of qubits and the Frobenius norm of the Hamiltonian. As an immediate application, we show that sample based quantum simulation, a type of evolution where the Hamiltonian is a density matrix, can be efficiently classically simulated under specific structural conditions. Our main technical contribution is a randomized algorithm for approximating Hermitian matrix exponentials. The proof leverages a low-rank, symmetric approximation via the Nyström method. Our results suggest that under strong sampling assumptions there exist classical poly-logarithmic time simulations of quantum computations.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Consistent Structured Prediction with Max-Min Margin Markov Networks

Max-margin methods for binary classification such as the support vector machine (SVM) have been extended to the structured prediction setting under the name of max-margin Markov networks ($M^3N$), or more generally structural SVMs. Unfortunately, these methods are statistically inconsistent when the relationship between inputs and labels is far from deterministic. We overcome such limitations by defining the learning problem in terms of a "max-min" margin formulation, naming the resulting method max-min margin Markov networks ($M^4N$). We prove consistency and finite sample generalization bounds for $M^4N$ and provide an explicit algorithm to compute the estimator. The algorithm achieves a generalization error of $O(1/\sqrt{n})$ for a total cost of $O(n)$ projection-oracle calls (which have at most the same cost as the max-oracle from $M^3N$). Experiments on multi-class classification, ordinal regression, sequence prediction and ranking demonstrate the effectiveness of the proposed method.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Non-parametric Models for Non-negative Functions

Linear models have shown great effectiveness and flexibility in many fields such as machine learning, signal processing and statistics. They can represent rich spaces of functions while preserving the convexity of the optimization problems where they are used, and are simple to evaluate, differentiate and integrate. However, for modeling non-negative functions, which are crucial for unsupervised learning, density estimation, or non-parametric Bayesian methods, linear models are not applicable directly. Moreover, current state-of-the-art models like generalized linear models either lead to non-convex optimization problems, or cannot be easily integrated. In this paper we provide the first model for non-negative functions which benefits from the same good properties of linear models. In particular, we prove that it admits a representer theorem and provide an efficient dual formulation for convex problems. We study its representation power, showing that the resulting space of functions is strictly richer than that of generalized linear models. Finally we extend the model and the theoretical results to functions with outputs in convex cones. The paper is complemented by an experimental evaluation of the model showing its effectiveness in terms of formulation, algorithmic derivation and practical results on the problems of density estimation, regression with heteroscedastic errors, and multiple quantile regression.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Structured and Localized Image Restoration

We present a novel approach to image restoration that leverages ideas from localized structured prediction and non-linear multi-task learning. We optimize a penalized energy function regularized by a sum of terms measuring the distance between patches to be restored and clean patches from an external database gathered beforehand. The resulting estimator comes with strong statistical guarantees leveraging local dependency properties of overlapping patches. We derive the corresponding algorithms for energies based on the mean-squared and Euclidean norm errors. Finally, we demonstrate the practical effectiveness of our model on different image restoration problems using standard benchmarks.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Structured Prediction with Partial Labelling through the Infimum Loss

Annotating datasets is one of the main costs in nowadays supervised learning. The goal of weak supervision is to enable models to learn using only forms of labelling which are cheaper to collect, as partial labelling. This is a type of incomplete annotation where, for each datapoint, supervision is cast as a set of labels containing the real one. The problem of supervised learning with partial labelling has been studied for specific instances such as classification, multi-label, ranking or segmentation, but a general framework is still missing. This paper provides a unified framework based on structured prediction and on the concept of infimum loss to deal with partial labelling over a wide family of learning problems and loss functions. The framework leads naturally to explicit algorithms that can be easily implemented and for which proved statistical consistency and learning rates. Experiments confirm the superiority of the proposed approach over commonly used baselines.

0 citations0 reviewsNo code linkedOpen access