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

Vikram Krishnamurthy

TL;DR

This work identifies a minimal concentration-based framework that underpins many optimism-based bandit algorithms, reducing the regret analysis to a single high-probability estimator concentration condition plus two deterministic lemmas (radius collapse and optimism-forced deviations). It provides unified, near-minimal proofs for UCB, UCB-V, linear UCB, and GP-UCB in finite-action settings and extends the approach to a broad set of variants, including heteroskedastic, heavy-tailed, misspecified, surrogate-reward, and contextual-structure cases via sketch arguments. The main contribution is a clear, general template for achieving logarithmic regret under broad conditions, clarifying when optimism-based indices yield near-optimal performance. The framework also outlines extensions to randomized index policies and discusses the boundaries where the approach does not apply, such as adversarial or fully general contextual bandits.

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.

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

TL;DR

This work identifies a minimal concentration-based framework that underpins many optimism-based bandit algorithms, reducing the regret analysis to a single high-probability estimator concentration condition plus two deterministic lemmas (radius collapse and optimism-forced deviations). It provides unified, near-minimal proofs for UCB, UCB-V, linear UCB, and GP-UCB in finite-action settings and extends the approach to a broad set of variants, including heteroskedastic, heavy-tailed, misspecified, surrogate-reward, and contextual-structure cases via sketch arguments. The main contribution is a clear, general template for achieving logarithmic regret under broad conditions, clarifying when optimism-based indices yield near-optimal performance. The framework also outlines extensions to randomized index policies and discusses the boundaries where the approach does not apply, such as adversarial or fully general contextual bandits.

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.

Paper Structure

This paper contains 16 sections, 4 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Overview and Scope
  3. Main Condition: Confidence and Concentration
  4. Meta-Result. Logarithmic Regret
  5. Examples
  6. Example 1: UCB (Bounded Rewards)
  7. Example 2: UCB-V (Bounded Variance)
  8. Example 3: Linear UCB (Sketch)
  9. Example 4: GP-UCB (Finite-Arm Sketch)
  10. Example 5: Bandits with Heteroskedastic Noise (Sketch).
  11. Example 6: PAC (Anytime) Confidence Bounds (Sketch).
  12. Example 7: Heavy-Tailed Stochastic Bandits (Sketch).
  13. Example 8: Misspecified Linear Bandits with Gap-Adjusted Assumptions (Sketch).
  14. Example 9: Bandits with Machine Learning-Assisted Surrogate Rewards (Sketch).
  15. Example 10: Stochastic Contextual Linear Bandits with Diverse Contexts (Sketch)
  16. ...and 1 more sections

Key Result

Lemma 1

Under Condition cond:concentration and confidence level $\delta$, for each suboptimal arm $i$ with gap $\Delta_i$, there exists an integer such that $r_i(m)\le \Delta_i/4$ for all $m\ge m_0$.

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1: Radius collapse
  • proof
  • Lemma 2: Optimism forces a deviation
  • proof
  • Theorem 1: Logarithmic regret
  • proof
  • ...and 2 more