Learning Against Nature: Minimax Regret and the Price of Robustness

2026-02-16

AI Summary

This study addresses how a decision maker forms robust posterior beliefs to minimize worst-case regret when data quality is uncertain. The authors propose a robust learning framework grounded in the minimax regret criterion, modeling belief updating as a zero-sum game between the decision maker and an adversarial nature. By endogenously deriving the local perturbation scale under signal ambiguity—specifically, of order \(1/\sqrt{n}\)—they uncover the asymptotic cost of robustness and a bias attributable to “under-inference,” establishing a duality with the local asymptotic alternatives paradigm in statistical theory. Key findings include that adversarial data generation induces strictly positive prior regret even with infinite data; however, when the true data-generating process is informative, the underlying state can be consistently learned, albeit at a sub-exponential convergence rate.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Machine Learning: Adversarial Learning & Robustness
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study how a decision-maker (DM) learns from data of unknown quality to form robust, ''general-purpose'' posterior beliefs. We develop a framework for robust learning and belief formation under a minimax-regret criterion, cast as a zero-sum game: the DM chooses posterior beliefs to minimize ex-ante regret, while an adversarial Nature selects the data-generating process (DGP). We show that, in large samples of $n$ signal draws, Nature optimally induces ambiguity by choosing a process whose precision converges to the uninformative signals at the rate $1/\sqrt{n}$. As a result, learning against the adversarial DGP is nontrivial as well as incomplete: the DM's ex-ante regret remains strictly positive even with an infinite amount of data. However, when the true DGP is fixed and informative (even if only slightly), our DM with a robust updating rule eventually learns the state with enough data. Still, learning occurs at a sub-exponential rate -- quantifying the asymptotic price of robustness -- and it exhibits ''under-inference'' bias. Our framework provides a decision-theoretic dual to the local alternatives method in asymptotic statistics, deriving the characteristic $1/\sqrt{n}$-scaling endogenously from the signal ambiguity.

Problem

• minimax regret
• robust learning
• data-generating process
• adversarial nature
• asymptotic price of robustness

Innovation

• minimax regret
• robust learning
• adversarial data-generating process
• signal ambiguity
• under-inference bias