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We develop a minimax theory for operator learning, where the goal is to estimate an unknown operator between separable Hilbert spaces from finitely many noisy input-output samples. For uniformly bounded Lipschitz operators, we prove…

In clinical trials and other applications, we often see regions of the feature space that appear to exhibit interesting behaviour, but it is unclear whether these observed phenomena are reflected at the population level. Focusing on a…

Policy learning in modern operations environments faces a fundamental tension between limited operational data and the large, often continuous, state and action spaces over which good decisions must be identified and deployed. We study…

Empirical Risk Minimization (ERM) algorithms are widely used in a variety of estimation and prediction tasks in signal-processing and machine learning applications. Despite their popularity, a theory that explains their statistical…

We prove risk bounds for binary classification in high-dimensional settings when the sample size is allowed to be smaller than the dimensionality of the training set observations. In particular, we prove upper bounds for both 'compressive…

Stochastic gradient descent (SGD) is a widely used algorithm in machine learning, particularly for neural network training. Recent studies on SGD for canonical quadratic optimization or linear regression show it attains well generalization…

Stochastic approximation (SA) is a classical approach for stochastic convex optimization. Previous studies have demonstrated that the convergence rate of SA can be improved by introducing either smoothness or strong convexity condition. In…

We study square loss in a realizable time-series framework with martingale difference noise. Our main result is a fast rate excess risk bound which shows that whenever a trajectory hypercontractivity condition holds, the risk of the…

We build a theoretical framework for designing and understanding practical meta-learning methods that integrates sophisticated formalizations of task-similarity with the extensive literature on online convex optimization and sequential…

In this paper we consider learning in passive setting but with a slight modification. We assume that the target expected loss, also referred to as target risk, is provided in advance for learner as prior knowledge. Unlike most studies in…

We study theoretical properties of a broad class of regularized algorithms with vector-valued output. These spectral algorithms include kernel ridge regression, kernel principal component regression, various implementations of gradient…

This paper proves, in very general settings, that convex risk minimization is a procedure to select a unique conditional probability model determined by the classification problem. Unlike most previous work, we give results that are general…

We develop and analyze $M$-estimation methods for divergence functionals and the likelihood ratios of two probability distributions. Our method is based on a non-asymptotic variational characterization of $f$-divergences, which allows the…

High-dimensional data is common in multiple areas, such as health care and genomics, where the number of features can be tens of thousands. In such scenarios, the large number of features often leads to inefficient learning. Constraint…

We study the approximation capacity of deep ReLU recurrent neural networks (RNNs) and explore the convergence properties of nonparametric least squares regression using RNNs. We derive upper bounds on the approximation error of RNNs for…

We develop an approach for estimating models described via conditional moment restrictions, with a prototypical application being non-parametric instrumental variable regression. We introduce a min-max criterion function, under which the…

Convergence bounds are one of the main tools to obtain information on the performance of a distributed machine learning task, before running the task itself. In this work, we perform a set of experiments to assess to which extent, and in…

This paper presents a systematic study on gap-dependent sample complexity in offline reinforcement learning. Prior work showed when the density ratio between an optimal policy and the behavior policy is upper bounded (the optimal policy…

We investigate the stochastic optimization problem of minimizing population risk, where the loss defining the risk is assumed to be weakly convex. Compositions of Lipschitz convex functions with smooth maps are the primary examples of such…

Often in the analysis of first-order methods, assuming the existence of a quadratic growth bound (a generalization of strong convexity) facilitates much stronger convergence analysis. Hence the analysis is done twice, once for the general…