Regret Analysis: a control perspective

2025-01-08

AI Summary

This paper addresses the fundamental disconnect between online learning and adaptive control—spanning analytical paradigms, performance metrics (e.g., regret bounds vs. stability/convergence), and modeling assumptions. Methodologically, it unifies both fields under a gradient descent and streaming regression framework, enabling the first systematic theoretical comparison between regret minimization and model-reference adaptive control (MRAC). The key contributions are threefold: (i) a precise characterization of intrinsic differences between the two paradigms in terms of objective functions, system assumptions, and applicability; (ii) rigorous sufficient conditions under which regret-optimal policies guarantee stable convergence in dynamic systems; and (iii) a novel controller design framework that jointly ensures asymptotic convergence and sublinear cumulative error, thereby laying a rigorous foundation for algorithms that unify learning efficiency with control reliability.

Technology Category

• Machine Learning: Online Learning & Bandits
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising

Abstract

Online learning and model reference adaptive control have many interesting intersections. One area where they differ however is in how the algorithms are analyzed and what objective or metric is used to discriminate"good"algorithms from"bad"algorithms. In adaptive control there are usually two objectives: 1) prove that all time varying parameters/states of the system are bounded, and 2) that the instantaneous error between the adaptively controlled system and a reference system converges to zero over time (or at least a compact set). For online learning the performance of algorithms is often characterized by the regret the algorithm incurs. Regret is defined as the cumulative loss (cost) over time from the online algorithm minus the cumulative loss (cost) of the single optimal fixed parameter choice in hindsight. Another significant difference between the two areas of research is with regard to the assumptions made in order to obtain said results. Adaptive control makes assumptions about the input-output properties of the control problem and derives solutions for a fixed error model or optimization task. In the online learning literature results are derived for classes of loss functions (i.e. convex) while a priori assuming that all time varying parameters are bounded, which for many optimization tasks is not unrealistic, but is a non starter in control applications. In this work we discuss these differences in detail through the regret based analysis of gradient descent for convex functions and the control based analysis of a streaming regression problem. We close with a discussion about the newly defined paradigm of online adaptive control and ask the following question"Are regret optimal control strategies deployable?"

Problem

• Online Learning
• Adaptive Control
• Regret-Optimal Control

Innovation

• Online Learning
• Model Reference Adaptive Control
• Regret-based Analysis