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We study the reinforcement learning problem for discounted Markov Decision Processes (MDPs) under the tabular setting. We propose a model-based algorithm named UCBVI-$\gamma$, which is based on the \emph{optimism in the face of uncertainty…

Deep metric learning (DML) aims to minimize empirical expected loss of the pairwise intra-/inter- class proximity violations in the embedding space. We relate DML to feasibility problem of finite chance constraints. We show that minimizer…

Choosing appropriate step sizes is critical for reducing the computational cost of training large-scale neural network models. Mini-batch sub-sampling (MBSS) is often employed for computational tractability. However, MBSS introduces a…

We consider the problem of learning optimal solutions of a partially known linear optimization problem and recovering its underlying cost function where a set of past decisions and the feasible set are known. We develop a new framework,…

The predictive normalized maximum likelihood (pNML) approach has recently been proposed as the min-max optimal solution to the batch learning problem where both the training set and the test data feature are individuals, known sequences.…

Statistical inverse learning aims at recovering an unknown function $f$ from randomly scattered and possibly noisy point evaluations of another function $g$, connected to $f$ via an ill-posed mathematical model. In this paper we blend…

Min-max problems have broad applications in machine learning, including learning with non-decomposable loss and learning with robustness to data distribution. Convex-concave min-max problem is an active topic of research with efficient…

We construct minimax optimal non-asymptotic confidence sets for low rank matrix recovery algorithms such as the Matrix Lasso or Dantzig selector. These are employed to devise adaptive sequential sampling procedures that guarantee recovery…

We study problem-dependent rates, i.e., generalization errors that scale near-optimally with the variance, the effective loss, or the gradient norms evaluated at the "best hypothesis." We introduce a principled framework dubbed "uniform…

The Minimum Description Length (MDL) principle is solidly based on a provably ideal method of inference using Kolmogorov complexity. We test how the theory behaves in practice on a general problem in model selection: that of learning the…

Matrix product operators allow efficient descriptions (or realizations) of states on a 1D lattice. We consider the task of learning a realization of minimal dimension from copies of an unknown state, such that the resulting operator is…

This article focuses on minimax programming problems subject to addition-{\L}ukasiewicz fuzzy relational inequalities. We first establish two necessary and sufficient conditions that a solution of the fuzzy relational inequalities is a…

Consider the Maximum Weight Independent Set problem for rectangles: given a family of weighted axis-parallel rectangles in the plane, find a maximum-weight subset of non-overlapping rectangles. The problem is notoriously hard both in the…

The theory of Chebyshev (uniform) approximation for univariate polynomial and piecewise polynomial functions has been studied for decades. The optimality conditions are based on the notion of alternating sequence. However, the extension the…

We consider the problem of estimating how well a model class is capable of fitting a distribution of labeled data. We show that it is often possible to accurately estimate this "learnability" even when given an amount of data that is too…

Systems of fuzzy relation equations and inequalities in which an unknown fuzzy relation is on the one side of the equation or inequality are linear systems. They are the most studied ones, and a vast literature on linear systems focuses on…

Federated learning is a communication-efficient training process that alternates between local training at the edge devices and averaging the updated local model at the central server. Nevertheless, it is impractical to achieve a perfect…

The solution, $x$, of the linear system of equations $A x\approx b$ arising from the discretization of an ill-posed integral equation with a square integrable kernel $H(s,t)$ is considered. The Tikhonov regularized solution $ x(\lambda)$ is…

An emerging line of work has shown that machine-learned predictions are useful to warm-start algorithms for discrete optimization problems, such as bipartite matching. Previous studies have shown time complexity bounds proportional to some…

We study the problem of multivariate $L_2$-approximation of functions in a weighted Korobov space using a median lattice-based algorithm recently proposed by the authors. In the original work, the algorithm requires knowledge of the…