Learning Against Nature: Minimax Regret and the Price of Robustness

TL;DR

This paper develops a robust learning framework using minimax regret, showing that ambiguity persists even with large data, leading to incomplete learning and an inherent bias, and quantifies the asymptotic cost of robustness.

Contribution

It introduces a decision-theoretic model for robust learning under ambiguity, linking it to asymptotic statistics and characterizing the price of robustness.

Findings

Nature induces ambiguity with precision converging to uninformative signals at rate 1/√n.

Ex-ante regret remains positive even with infinite data under worst-case DGP.

Learning is asymptotically slow and biased when the true DGP is informative.

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.