Blackwell's Approachability with Approximation Algorithms

TL;DR

This paper extends Blackwell's approachability theory to settings where players can only access their action sets via approximation algorithms, establishing conditions for efficient approachability with scaled target sets.

Contribution

It introduces a framework for approachability with approximation algorithms, providing new algorithms and guarantees when action sets are computationally hard to optimize.

Findings

Downward closure of scaled target set is efficiently approachable.

Approachability guarantees hold under monotone preferences.

Simpler algorithms are available when only one side has approximation access.

Abstract

We revisit Blackwell's celebrated approachability problem which considers a repeated vector-valued game between a player and an adversary. Motivated by settings in which the action set of the player or adversary (or both) is difficult to optimize over, for instance when it corresponds to the set of all possible solutions to some NP-Hard optimization problem, we ask what can the player guarantee \textit{efficiently}, when only having access to these sets via approximation algorithms with ratios $\alpha_{\mX} \geq 1$ and $ 1 \geq \alpha_{\mY} > 0$, respectively. Assuming the player has monotone preferences, in the sense that he does not prefer a vector-valued loss $\ell_1$ over $\ell_2$ if $\ell_2 \leq \ell_1$, we establish that given a Blackwell instance with an approachable target set $S$, the downward closure of the appropriately-scaled set $\alpha_{\mX}\alpha_{\mY}^{-1}S$ is \textit{efficiently} approachable with optimal rate. In case only the player's or adversary's set is equipped with an approximation algorithm, we give simpler and more efficient algorithms.