Chasing Convex Bodies and Functions with Black-Box Advice

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 consistency, while also ensuring worst-case robustness 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, INTERP, achieves (√2̅ + ϵ)-consistency and 2̅(C/ϵ²)-robustness for any ϵ > 0, where C is the competitive ratio of an algorithm for convex function chasing or a subclass thereof. The second, BDINTERP, achieves (1 + ϵ)-consistency and O(CD/ϵ)-robustness when the problem has bounded diameter D. Further, we show that BDINTERP achieves near-optimal consistency-robustness trade-off for the special case where cost functions are α-polyhedral.