Chasing Convex Bodies and Functions with Black-Box Advice
Nicolas Christianson , Tinashe Handina , and Adam Wierman
In Proceedings of Thirty Fifth Conference on Learning Theory , 2022
We consider the problem of convex function chasing with black-box advice, where an online decision-maker aims to minimize the total cost of making and switching between decisions in a normed vector space, aided by black-box advice such as the decisions of a machine-learned algorithm. The decision-maker seeks cost comparable to the advice when it performs well, known as \emphconsistency, while also ensuring worst-case \emphrobustness even when the advice is adversarial. We first consider the common paradigm of algorithms that switch between the decisions of the advice and a competitive algorithm, showing that no algorithm in this class can improve upon 3-consistency while staying robust. We then propose two novel algorithms that bypass this limitation by exploiting the problem’s convexity. The first, \textscInterp, achieves (\sqrt2+ε)-consistency and \mathcalO(\fracCε^2)-robustness for any ε> 0, where C is the competitive ratio of an algorithm for convex function chasing or a subclass thereof. The second, \textscBdInterp, achieves (1+ε)-consistency and \mathcalO(\fracCDε)-robustness when the problem has bounded diameter D. Further, we show that \textscBdInterp achieves near-optimal consistency-robustness trade-off for the special case where cost functions are α-polyhedral.