Blackwell's Approachability with Approximation Algorithms
This work addresses the challenge of applying Blackwell's approachability in computationally hard settings, making it relevant for scenarios involving complex optimization problems, though it is incremental in extending existing theory.
The paper tackles the problem of efficiently guaranteeing approachability in repeated vector-valued games when action sets are NP-hard to optimize over, by showing that with approximation algorithms, the player can efficiently approach a scaled target set with optimal rate.
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 $α_{\mX} \geq 1$ and $ 1 \geq α_{\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 $α_{\mX}α_{\mY}^{-1}S$ 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.