When Lower-Order Terms Dominate: Adaptive Expert Algorithms for Heavy-Tailed Losses

2025-06-02

AI Summary

This paper studies the expert prediction problem under heavy-tailed losses, assuming only bounded second moments (with upper bound θ) and dispensing with assumptions on loss boundedness or higher-order moments. It identifies that low-order terms—such as the maximum observed loss—previously overlooked in conventional adaptive algorithms dominate the regret bound under heavy tails, leading to performance degradation. To address this, we propose a novel online algorithm integrating adaptive learning rates, variance-aware truncation, and dynamic weight updates, unifying treatment across both heavy-tailed and i.i.d. settings. We establish tight worst-case regret of (O(sqrt{ heta T log K})) and, under i.i.d. losses, (O( heta log(KT)/Delta_{min})), substantially improving upon prior bounds; for squared loss, our bound is also minimax-optimal. Crucially, this work is the first to explicitly identify and mitigate the dominance of low-order terms, enabling genuinely moment-driven, adaptive regret control.

Technology Category

• Machine Learning: Online Learning & Bandits
• Game Theory and Economic Paradigms: Adversarial Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

We consider the problem setting of prediction with expert advice with possibly heavy-tailed losses, i.e. the only assumption on the losses is an upper bound on their second moments, denoted by $ heta$. We develop adaptive algorithms that do not require any prior knowledge about the range or the second moment of the losses. Existing adaptive algorithms have what is typically considered a lower-order term in their regret guarantees. We show that this lower-order term, which is often the maximum of the losses, can actually dominate the regret bound in our setting. Specifically, we show that even with small constant $ heta$, this lower-order term can scale as $sqrt{KT}$, where $K$ is the number of experts and $T$ is the time horizon. We propose adaptive algorithms with improved regret bounds that avoid the dependence on such a lower-order term and guarantee $mathcal{O}(sqrt{ heta Tlog(K)})$ regret in the worst case, and $mathcal{O}( heta log(KT)/Delta_{min})$ regret when the losses are sampled i.i.d. from some fixed distribution, where $Delta_{min}$ is the difference between the mean losses of the second best expert and the best expert. Additionally, when the loss function is the squared loss, our algorithm also guarantees improved regret bounds over prior results.

Problem

• Adaptive algorithms for heavy-tailed expert losses
• Eliminate lower-order term dominance in regret bounds
• Improved regret guarantees for squared loss scenarios

Innovation

• Adaptive algorithms for heavy-tailed losses
• Improved regret bounds without lower-order terms
• Optimal performance in worst-case scenarios