Approximate Function Evaluation via Multi-Armed Bandits

TL;DR

This paper introduces an adaptive sampling algorithm for efficiently estimating a smooth function's value at an unknown point, leveraging noisy oracle samples and learning the importance of each coordinate.

Contribution

The paper presents a novel instance-adaptive algorithm that learns optimal sampling strategies for function evaluation under noise, with proven asymptotic optimality for linear functions.

Findings

The adaptive algorithm achieves high-probability $ ext{epsilon}$-accurate estimates.

Numerical experiments demonstrate significant improvements over non-adaptive methods.

The method generalizes to heteroskedastic noise scenarios.

Abstract

We study the problem of estimating the value of a known smooth function $f$ at an unknown point $\boldsymbol{\mu} \in \mathbb{R}^n$, where each component $\mu_i$ can be sampled via a noisy oracle. Sampling more frequently components of $\boldsymbol{\mu}$ corresponding to directions of the function with larger directional derivatives is more sample-efficient. However, as $\boldsymbol{\mu}$ is unknown, the optimal sampling frequencies are also unknown. We design an instance-adaptive algorithm that learns to sample according to the importance of each coordinate, and with probability at least $1-\delta$ returns an $\epsilon$ accurate estimate of $f(\boldsymbol{\mu})$. We generalize our algorithm to adapt to heteroskedastic noise, and prove asymptotic optimality when $f$ is linear. We corroborate our theoretical results with numerical experiments, showing the dramatic gains afforded by adaptivity.