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Quantum Speedups in Regret Analysis of Infinite Horizon Average-Reward Markov Decision Processes

Bhargav Ganguly, Yang Xu, Vaneet Aggarwal

TL;DR

This work introduces Quantum-UCRL, a quantum-enhanced model-based RL algorithm for infinite-horizon average-reward MDPs. It combines optimistic policy optimization with quantum mean estimation (via QBounded) to accelerate transition-probability estimation and tighten Bellman-error based regret analysis. Theoretical results establish a martingale-free regret bound that scales with the MDP's size and mixing time, demonstrating potential quantum speedups for long-horizon RL. The paper also discusses practical aspects of quantum measurements and suggests directions such as parameterized quantum RL to extend these gains further.

Abstract

This paper investigates the potential of quantum acceleration in addressing infinite horizon Markov Decision Processes (MDPs) to enhance average reward outcomes. We introduce an innovative quantum framework for the agent's engagement with an unknown MDP, extending the conventional interaction paradigm. Our approach involves the design of an optimism-driven tabular Reinforcement Learning algorithm that harnesses quantum signals acquired by the agent through efficient quantum mean estimation techniques. Through thorough theoretical analysis, we demonstrate that the quantum advantage in mean estimation leads to exponential advancements in regret guarantees for infinite horizon Reinforcement Learning. Specifically, the proposed Quantum algorithm achieves a regret bound of $\tilde{\mathcal{O}}(1)$, a significant improvement over the $\tilde{\mathcal{O}}(\sqrt{T})$ bound exhibited by classical counterparts.

Quantum Speedups in Regret Analysis of Infinite Horizon Average-Reward Markov Decision Processes

TL;DR

This work introduces Quantum-UCRL, a quantum-enhanced model-based RL algorithm for infinite-horizon average-reward MDPs. It combines optimistic policy optimization with quantum mean estimation (via QBounded) to accelerate transition-probability estimation and tighten Bellman-error based regret analysis. Theoretical results establish a martingale-free regret bound that scales with the MDP's size and mixing time, demonstrating potential quantum speedups for long-horizon RL. The paper also discusses practical aspects of quantum measurements and suggests directions such as parameterized quantum RL to extend these gains further.

Abstract

This paper investigates the potential of quantum acceleration in addressing infinite horizon Markov Decision Processes (MDPs) to enhance average reward outcomes. We introduce an innovative quantum framework for the agent's engagement with an unknown MDP, extending the conventional interaction paradigm. Our approach involves the design of an optimism-driven tabular Reinforcement Learning algorithm that harnesses quantum signals acquired by the agent through efficient quantum mean estimation techniques. Through thorough theoretical analysis, we demonstrate that the quantum advantage in mean estimation leads to exponential advancements in regret guarantees for infinite horizon Reinforcement Learning. Specifically, the proposed Quantum algorithm achieves a regret bound of , a significant improvement over the bound exhibited by classical counterparts.
Paper Structure (13 sections, 13 theorems, 59 equations, 1 figure, 2 algorithms)

This paper contains 13 sections, 13 theorems, 59 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work and Preliminaries
  3. Problem Formulation
  4. Algorithmic Framework
  5. Theoretical Results
  6. Conclusions
  7. Appendix A: Convexity of Optimization Problem
  8. Appendix B: A brief overview of Quantum Amplitude Amplification
  9. Appendix C: Proof of Lemma 3
  10. Appendix D: Proof of Lemma 4
  11. Appendix E: Proof of Lemma 5
  12. Appendix F: Some Auxiliary Lemmas used in this Paper
  13. Appendix G: Proof of Theorem 1

Key Result

Lemma 1

Let $X$ be a d-dimensional bounded random variable such that $||X||_2 \leq 1$. Given three reals $L_2 \in (0,1]$, $\delta \in (0,1)$ and $n \geq 1$ such that $\mathbb{E}[||X||_2] \leq L_2$, the multivariate bounded estimator $\texttt{QBounded}_d(X,L_2,n,\delta)$ obtained by Algorithm algo: QBounded

Figures (1)

  • Figure 1: Agent's interaction at round $t$ with the MDP Environment and accessible quantum transition oracle

Theorems & Definitions (27)

  • Definition 1: Random Variable, Definition 2.2 of cornelissen2022near
  • Definition 2: Quantum Experiment
  • Definition 3: Quantum Evaluation Oracle
  • Lemma 1: Quantum multivariate bounded estimator, Theorem 3.3 of cornelissen2022near
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 17 more