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Chasing Convex Bodies and Functions with Black-Box Advice

Abstract

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\textit{consistency}, while also ensuring worst-case robustness\textit{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 (2+ϵ)(\sqrt{2}+\epsilon)-consistency and O(Cϵ2)\mathcal{O}(\frac{C}{\epsilon^2})-robustness for any ϵ>0\epsilon > 0, where CC is the competitive ratio of an algorithm for convex function chasing or a subclass thereof. The second, BDINTERP, achieves (1+ϵ)(1+\epsilon)-consistency and O(CDϵ)\mathcal{O}(\frac{CD}{\epsilon})-robustness when the problem has bounded diameter DD. Further, we show that BDINTERP achieves near-optimal consistency-robustness trade-off for the special case where cost functions are α\alpha-polyhedral.

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