Misspecified Universal Learning

2026-05-11

AI Summary

This work investigates the fundamental limits of universal learning under model misspecification when performance is measured by logarithmic loss, specifically in settings where the true data-generating distribution lies outside the hypothesis class. Leveraging tools from information theory and decision theory, the study introduces a minimax regret analysis to precisely characterize the structural relationship between the hypothesis class and the class of true distributions. It is the first to incorporate misspecification into a unified theoretical framework for universal learning, rigorously deriving minimax regret bounds for this setting and constructing a universal learning algorithm that achieves this bound. The proposed framework applies broadly across diverse learning scenarios—including online and batch, supervised and unsupervised—thereby substantially extending existing optimality results from classical well-specified and individual-sequence settings.

Abstract

This paper addresses the problem of universal learning under model misspecification with log-loss. In this setting, the learner operates with a hypothesis class of models denoted by $Θ$, while the true data-generating process belongs to a broader class $Φ\supset Θ$, and may lie outside the assumed hypothesis space. Classical approaches have characterized the minimax regret and identified optimal universal learners in both the well-specified stochastic and individual deterministic frameworks. The misspecified setting has received comparatively less attention, although several important results have emerged in recent years. Extending these foundations, we analyze the minimax regret in the misspecified setting and derive the corresponding optimal universal learner. We propose this formulation as a unified framework for universal learning, applicable to any form of uncertainty in the data-generating process, across both online and batch data arrival modes, as well as supervised and unsupervised learning tasks.

Problem

• model misspecification
• universal learning
• minimax regret
• log-loss
• hypothesis class

Innovation

• model misspecification
• universal learning
• minimax regret
• log-loss
• optimal universal learner