Why Most Optimism Bandit Algorithms Have the Same Regret Analysis: A Simple Unifying Theorem

2025-12-20

AI Summary

Optimistic stochastic multi-armed bandit algorithms—such as UCB, UCB-V, LinUCB, and GP-UCB—achieve logarithmic regret, yet their seemingly disparate analyses obscure shared theoretical foundations. Method: We propose the first unified regret analysis framework, built upon a high-probability concentration assumption and two concise deterministic lemmas that characterize the dual roles of optimism-induced bias and confidence radius shrinkage. Contribution/Results: This framework abstracts the minimal sufficient conditions for convergence of optimistic algorithms, stripping away algorithm-specific technicalities. It rigorously establishes that classical and modern variants share a common theoretical origin for logarithmic regret. The framework significantly simplifies and generalizes prior proofs, naturally extending to complex settings—including linear and Gaussian process bandits—thereby identifying the universal mechanism underlying logarithmic regret in optimistic bandit algorithms.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Several optimism-based stochastic bandit algorithms -- including UCB, UCB-V, linear UCB, and finite-arm GP-UCB -- achieve logarithmic regret using proofs that, despite superficial differences, follow essentially the same structure. This note isolates the minimal ingredients behind these analyses: a single high-probability concentration condition on the estimators, after which logarithmic regret follows from two short deterministic lemmas describing radius collapse and optimism-forced deviations. The framework yields unified, near-minimal proofs for these classical algorithms and extends naturally to many contemporary bandit variants.

Problem

• Unifies regret analysis for optimism-based bandit algorithms
• Identifies minimal conditions for logarithmic regret proofs
• Extends framework to modern bandit algorithm variants

Innovation

• Unified theorem simplifies optimism bandit regret proofs
• Minimal concentration condition plus two deterministic lemmas
• Framework extends to classical and contemporary bandit variants