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

TL;DR

This paper reveals that many optimism-based stochastic bandit algorithms share a common regret analysis structure, providing a unifying theorem that simplifies understanding and extends to new variants.

Contribution

It introduces a simple unifying theorem that captures the core analysis of various optimism-based bandit algorithms, simplifying proofs and extending to new variants.

Findings

Unified regret analysis framework for optimism bandit algorithms

Simplified proofs for classical algorithms like UCB and GP-UCB

Extension of the framework to contemporary bandit variants

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.