The sample complexity of level set approximation

TL;DR

This paper investigates the sample complexity of approximating level sets of unknown functions using sequential queries, introducing algorithms that achieve optimal rates under various smoothness assumptions.

Contribution

It introduces the Bisect and Approximate algorithms that reduce level set approximation to local function approximation, providing rate-optimal guarantees.

Findings

Achieves rate-optimal sample complexity for Hölder functions.

Shows improved rates under additional smoothness assumptions.

Provides a framework for efficient level set approximation.

Abstract

We study the problem of approximating the level set of an unknown function by sequentially querying its values. We introduce a family of algorithms called Bisect and Approximate through which we reduce the level set approximation problem to a local function approximation problem. We then show how this approach leads to rate-optimal sample complexity guarantees for H{\"o}lder functions, and we investigate how such rates improve when additional smoothness or other structural assumptions hold true.