Computational Intractability of Strategizing against Online Learners

2025-03-06

AI Summary

This work studies how an optimizer can compute an optimal response against opponents employing no-regret online learning algorithms—such as Multiplicative Weights Update (MWU)—in multi-agent games. The authors prove that, unless P = NP, no polynomial-time algorithm can compute an approximately optimal response to a standard MWU learner, thereby establishing the NP-hardness of this problem. Furthermore, via a refined reduction, they derive an Ω(T) strong hardness lower bound—the first such result surpassing prior Θ(1) weak intractability bounds, which applied only to fictitious play. This is the first work to reveal a fundamental computational barrier to strategy optimization against no-regret learners in general normal-form games, ruling out the existence of universal efficient algorithms. The results provide rigorous complexity-theoretic boundaries for adversarial learning, mechanism design, and equilibrium computation.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Learning to Search
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Online learning algorithms are widely used in strategic multi-agent settings, including repeated auctions, contract design, and pricing competitions, where agents adapt their strategies over time. A key question in such environments is how an optimizing agent can best respond to a learning agent to improve its own long-term outcomes. While prior work has developed efficient algorithms for the optimizer in special cases - such as structured auction settings or contract design - no general efficient algorithm is known. In this paper, we establish a strong computational hardness result: unless $mathsf{P} = mathsf{NP}$, no polynomial-time optimizer can compute a near-optimal strategy against a learner using a standard no-regret algorithm, specifically Multiplicative Weights Update (MWU). Our result proves an $Omega(T)$ hardness bound, significantly strengthening previous work that only showed an additive $Theta(1)$ impossibility result. Furthermore, while the prior hardness result focused on learners using fictitious play - an algorithm that is not no-regret - we prove intractability for a widely used no-regret learning algorithm. This establishes a fundamental computational barrier to finding optimal strategies in general game-theoretic settings.

Problem

• Computational hardness of strategizing against online learners
• No polynomial-time optimizer for near-optimal strategies
• Intractability with no-regret learning algorithms like MWU

Innovation

• Proves computational hardness for polynomial-time optimizers
• Focuses on no-regret learning algorithm MWU
• Establishes Ω(T) hardness bound in game theory