Blackwell's Approachability with Approximation Algorithms

2025-02-06

AI Summary

This paper investigates the theoretical limits of Blackwell approachability under computational constraints: specifically, when players’ and adversaries’ action sets correspond to solution spaces of NP-hard problems, accessible only via α_X- and α_Y-approximation algorithms. The central question is: which target sets remain efficiently approachable in this setting? The authors introduce the first Blackwell approachability framework compatible with approximation algorithms, proving that the scaled downward-closed set (α_X/α_Y)·S is polynomial-time approachable. For the one-sided approximation case, they design a simpler, optimal online algorithm. Their analysis provides tight convergence-rate guarantees, thereby relaxing the classical Blackwell theory’s requirement of exact optimization. This advancement extends approachability to realistic computationally bounded settings—such as adversarial online learning and games involving NP-hard decision problems—while preserving rigorous theoretical foundations.

Technology Category

• Search and Optimization: Adversarial Search
• Game Theory and Economic Paradigms: Adversarial Learning
• Multiagent Systems: Adversarial Agents

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 extit{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 extit{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.

Problem

• Blackwell's approachability problem
• approximation algorithms
• NP-Hard optimization

Innovation

• Approximation algorithms for complex action sets
• Efficient approachability with optimal rate
• Simplified algorithms for partial approximation access