Adaptive Approximation of the Minimum of Brownian Motion

TL;DR

This paper introduces an adaptive algorithm for approximating the minimum of a Brownian motion on [0,1], achieving faster convergence rates than traditional nonadaptive methods by adaptively selecting evaluation points.

Contribution

The paper presents a novel adaptive algorithm that significantly improves the convergence rate for estimating the minimum of Brownian motion compared to existing nonadaptive approaches.

Findings

Adaptive algorithm converges at an arbitrarily high polynomial rate.

Outperforms the standard 1/2 convergence rate of nonadaptive methods.

Demonstrates the effectiveness of adaptive sampling in stochastic optimization.

Abstract

We study the error in approximating the minimum of a Brownian motion on the unit interval based on finitely many point evaluations. We construct an algorithm that adaptively chooses the points at which to evaluate the Brownian path. In contrast to the $1/2$ convergence rate of optimal nonadaptive algorithms, the proposed adaptive algorithm converges at an arbitrarily high polynomial rate.