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Information-Theoretic Minimax Regret Bounds for Reinforcement Learning based on Duality

Raghav Bongole, Amaury Gouverneur, Borja Rodríguez-Gálvez, Tobias J. Oechtering, Mikael Skoglund

TL;DR

This research establishes minimax theorems and uses bounds on the Bayesian regret to derive minimax regret bounds, and establishes minimax theorems and uses bounds on the Bayesian regret to perform minimax regret analysis using these minimax theorems.

Abstract

We study agents acting in an unknown environment where the agent's goal is to find a robust policy. We consider robust policies as policies that achieve high cumulative rewards for all possible environments. To this end, we consider agents minimizing the maximum regret over different environment parameters, leading to the study of minimax regret. This research focuses on deriving information-theoretic bounds for minimax regret in Markov Decision Processes (MDPs) with a finite time horizon. Building on concepts from supervised learning, such as minimum excess risk (MER) and minimax excess risk, we use recent bounds on the Bayesian regret to derive minimax regret bounds. Specifically, we establish minimax theorems and use bounds on the Bayesian regret to perform minimax regret analysis using these minimax theorems. Our contributions include defining a suitable minimax regret in the context of MDPs, finding information-theoretic bounds for it, and applying these bounds in various scenarios.

Information-Theoretic Minimax Regret Bounds for Reinforcement Learning based on Duality

TL;DR

This research establishes minimax theorems and uses bounds on the Bayesian regret to derive minimax regret bounds, and establishes minimax theorems and uses bounds on the Bayesian regret to perform minimax regret analysis using these minimax theorems.

Abstract

We study agents acting in an unknown environment where the agent's goal is to find a robust policy. We consider robust policies as policies that achieve high cumulative rewards for all possible environments. To this end, we consider agents minimizing the maximum regret over different environment parameters, leading to the study of minimax regret. This research focuses on deriving information-theoretic bounds for minimax regret in Markov Decision Processes (MDPs) with a finite time horizon. Building on concepts from supervised learning, such as minimum excess risk (MER) and minimax excess risk, we use recent bounds on the Bayesian regret to derive minimax regret bounds. Specifically, we establish minimax theorems and use bounds on the Bayesian regret to perform minimax regret analysis using these minimax theorems. Our contributions include defining a suitable minimax regret in the context of MDPs, finding information-theoretic bounds for it, and applying these bounds in various scenarios.

Paper Structure

This paper contains 10 sections, 4 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Model and Definitions
  4. Minimax Theorems
  5. Minimax theorem with continuity conditions
  6. Upper bounds on minimax regret
  7. Finite Multi-arm Bandit Problem with bounded reward
  8. Linear Bandits
  9. Contextual Bandits
  10. Conclusion

Key Result

Proposition 1

Let $\mathbb{P}_\Theta$ be absolutely continuous with respect to $\mu$ with density $p_\Theta$. Then, the Bayesian regret can be expressed as where the supremum is taken over the function space ${f: \mathcal{S} \times \mathcal{O} \rightarrow \mathcal{A}}$.

Theorems & Definitions (14)

  • Definition 1: Utility
  • Definition 2: Optimal Utility
  • Definition 3: Regret
  • Definition 4: Minimax Regret
  • Definition 5: Bayesian Regret
  • Proposition 1
  • proof
  • Definition 6: Minimum Bayesian Regret
  • Definition 7: Worst-case MBR
  • Theorem 1
  • ...and 4 more