Discrepancy and Fisher information

TL;DR

This paper introduces an online algorithm for maintaining a symmetric random walk within a convex body, controlling discarded steps via Fisher information, with a dimension-free bound for the cube.

Contribution

It presents a novel online method that uses Fisher information to bound step discard rates, ensuring efficient random walk confinement in convex bodies.

Findings

Expected discarded steps are controlled by Fisher-information-type quantity.

For the cube, the algorithm achieves a dimension-free bound.

A walk with unit Euclidean steps can be kept bounded with minimal discarded steps.

Abstract

We give an online algorithm that keeps a symmetric random walk inside a convex body by discarding some of its steps. The expected number of discarded steps is controlled by a Fisher-information-type quantity associated with the body. For the cube, this gives a dimension-free bound: a walk with unit Euclidean steps can be kept bounded in all coordinates while discarding only a small constant fraction of the steps on average.