Related papers: Exponential tractability of $L_2$-approximation wi…
A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions $1, 2, \ldots$ grows at a moderate rate, i.e., the sequence of problems is tractable. Here, we focus…
Value function approximation has demonstrated phenomenal empirical success in reinforcement learning (RL). Nevertheless, despite a handful of recent progress on developing theory for RL with linear function approximation, the understanding…
We study the integration and approximation problems for monotone and convex bounded functions that depend on $d$ variables, where $d$ can be arbitrarily large. We consider the worst case error for algorithms that use finitely many function…
We study multivariate problems like function approximation, numerical integration, global optimization and dispersion. We obtain new results on the information complexity $n(\varepsilon,d)$ of these problems. The information complexity is…
We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to approximate, rather then exactly represent, a given hypothesis class. We show that…
In this paper we consider integration and $L_2$-approximation for functions over $\RR^s$ from weighted Hermite spaces. The first part of the paper is devoted to a comparison of several weighted Hermite spaces that appear in literature,…
We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous…
We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and…
It is well-known that modern neural networks are vulnerable to adversarial examples. To mitigate this problem, a series of robust learning algorithms have been proposed. However, although the robust training error can be near zero via some…
We consider an \eps-approximation by n-term partial sums of the Karhunen-Lo\`eve expansion to d-parametric random fields of tensor product-type in the average case setting. We investigate the behavior, as d tends to infinity, of the…
We consider approximation or recovery of functions based on a finite number of function evaluations. This is a well-studied problem in optimal recovery, machine learning, and numerical analysis in general, but many fundamental insights were…
We study multivariate $\boldsymbol{L}_{\infty}$-approximation for a weighted Korobov space of periodic functions for which the Fourier coefficients decay exponentially fast. The weights are defined, in particular, in terms of two sequences…
We study d-variate approximation problems in the average case setting with respect to a zero-mean Gaussian measure. Our interest is focused on measures having a structure of non-homogeneous linear tensor product, where covariance kernel is…
We study loss functions that measure the accuracy of a prediction based on multiple data points simultaneously. To our knowledge, such loss functions have not been studied before in the area of property elicitation or in machine learning…
Let us assume that $f$ is a continuous function defined on the unit ball of $\mathbb R^d$, of the form $f(x) = g (A x)$, where $A$ is a $k \times d$ matrix and $g$ is a function of $k$ variables for $k \ll d$. We are given a budget $m \in…
Function approximation has been an indispensable component in modern reinforcement learning algorithms designed to tackle problems with large state spaces in high dimensions. This paper reviews recent results on error analysis for these…
In this work a general approach to compute a compressed representation of the exponential $\exp(h)$ of a high-dimensional function $h$ is presented. Such exponential functions play an important role in several problems in Uncertainty…
In this paper, we investigate the sample complexity of policy evaluation in infinite-horizon offline reinforcement learning (also known as the off-policy evaluation problem) with linear function approximation. We identify a hard regime…
We study the average case complexity of a linear multivariate problem $(\lmp)$ defined on functions of $d$ variables. We consider two classes of information. The first $\lstd$ consists of function values and the second $\lall$ of all…
We study how much a linear program (LP) can be compressed when solved repeatedly, given prior knowledge about its objective function. Existing data-driven projection methods learn low-dimensional surrogate LPs with approximate…