[edit]

# Chasing Convex Bodies and Functions with Black-Box Advice

*Proceedings of Thirty Fifth Conference on Learning Theory*, PMLR 178:867-908, 2022.

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