On the power of adaption and randomization

TL;DR

This paper establishes bounds on the advantages of adaptive and randomized algorithms over deterministic ones in linear operator approximation, with optimal results for symmetric sets and insights into non-symmetric cases.

Contribution

It provides theoretical bounds on adaptive and randomized algorithms' performance, unifies concepts of n-widths and s-numbers, and explores extensions to non-linear approximation.

Findings

Optimal bounds for symmetric convex sets

Unified framework connecting n-widths and s-numbers

Extensions to non-linear widths and function value approximation

Abstract

We present bounds on the maximal gain of adaptive and randomized algorithms over non-adaptive, deterministic ones for approximating linear operators on convex sets. If the sets are additionally symmetric, then our results are optimal. For non-symmetric sets, we unify some notions of $n$-widths and s-numbers, and show their connection to minimal errors. We also discuss extensions to non-linear widths and approximation based on function values, and conclude with a list of open problems.