Nested Convex Bodies are Chaseable

Abstract

In the Convex Body Chasing problem, we are given an initial point v 0 ∈ R d and an online sequence of n convex bodies F 1 , . . . , F n . When we receive F i , we are required to move inside F i . Our goal is to minimize the total distance traveled. This fundamental online problem was first studied by Friedman and Linial (DCG 1993). They proved an Ω( √ d) lower bound on the competitive ratio, and conjectured that a competitive ratio depending only on d is possible. However, despite much interest in the problem, the conjecture remains wide open.We consider the setting in which the convex bodies are nested: F 1 ⊃ . . . ⊃ F n . The nested setting is closely related to extending the online LP framework of Buchbinder and Naor (ESA 2005) to arbitrary linear constraints. Moreover, this setting retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial's conjecture. In this work, we give a f (d)-competitive algorithm for chasing nested convex bodies in R d .

Better Bounds for Online Line Chasing

“…The Friedman-Linial conjecture has remained open for over two decades. In the last several years this topic has experienced a sudden increase in research activity, partly motivated by connections to machine learning (see [3,7]), resulting in rapid progress. In 2016, Antoniadis et al [1] gave a 2 O(d) -competitive algorithm for chasing affine spaces of any dimension.…”

“…In 2016, Antoniadis et al [1] gave a 2 O(d) -competitive algorithm for chasing affine spaces of any dimension. In 2018, Bansal et al [3] gave an algorithm with competitive ratio 2 O(d log d) for nested families of convex sets, where the input set sequence satisfies X 1 ⊇ X 2 ⊇ ... ⊇ X m . Soon later their bound was improved to O(d log d) by Argue et al [2], and then to O( √ d log d) by Bubeck et al [6].…”