Lipschitz Selectors may not Yield Competitive Algorithms for Convex Body Chasing

TL;DR

This paper demonstrates that Lipschitz selectors, despite their mathematical appeal, do not necessarily produce competitive algorithms for convex body chasing, challenging assumptions about their effectiveness.

Contribution

It provides a counterexample of a Lipschitz selector that is non-competitive and characterizes conditions under which Lipschitz selectors guarantee competitiveness.

Findings

A Lipschitz selector can be non-competitive for convex body chasing.

Lipschitz continuity with respect to Hausdorff distance does not imply competitiveness.

Lipschitz selectors with respect to an Lp-type distance are competitive only when p=1.

Abstract

The current best algorithms for convex body chasing problem in online algorithms use the notion of the Steiner point of a convex set. In particular, the algorithm which always moves to the Steiner point of the request set is $O(d)$ competitive for nested convex body chasing, and this is optimal among memoryless algorithms [Bubeck et al. 2020]. A memoryless algorithm coincides with the notion of a selector in functional analysis. The Steiner point is noted for being Lipschitz with respect to the Hausdorff metric, and for achieving the minimal Lipschitz constant possible. It is natural to ask whether every selector with this Lipschitz property yields a competitive algorithm for nested convex body chasing. We answer this question in the negative by exhibiting a selector which yields a non-competitive algorithm for nested convex body chasing but is Lipschitz with respect to Hausdorff distance. Furthermore, we show that being Lipschitz with respect to an $L_p$-type analog to the Hausdorff distance is sufficient to guarantee competitiveness if and only if $p=1$.