A New Interpretation of the Certainty-Equivalence Approach for PAC Reinforcement Learning with a Generative Model

2025-01-05

AI Summary

This paper addresses the low sample efficiency and strong environmental assumptions of the Certainty Equivalence Method (CEM) in PAC reinforcement learning under unknown environments. We propose the Trajectory Tree Method (TTM), offering a novel perspective and establishing, for the first time, theoretical equivalence between CEM and TTM. Methodologically, we leverage generative models, relax the conventional bounded-reward assumption, and construct policies via maximum likelihood estimation. Theoretically, we derive a tight sample complexity upper bound under a small failure probability δ and prove, for the first time, minimax optimality of CEM/TTM in non-stationary finite-horizon MDPs. Key contributions include: (i) significantly improved sample complexity for CEM under small δ—surpassing existing bounds; (ii) a matching lower bound, thereby establishing PAC optimality for finite-horizon MDPs; and (iii) a simpler, more general analytical framework that unifies CEM and TTM, applicable to both stationary and non-stationary settings.

Technology Category

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

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Reinforcement learning (RL) enables an agent interacting with an unknown MDP $M$ to optimise its behaviour by observing transitions sampled from $M$. A natural entity that emerges in the agent's reasoning is $widehat{M}$, the maximum likelihood estimate of $M$ based on the observed transitions. The well-known extit{certainty-equivalence} method (CEM) dictates that the agent update its behaviour to $widehat{pi}$, which is an optimal policy for $widehat{M}$. Not only is CEM intuitive, it has been shown to enjoy minimax-optimal sample complexity in some regions of the parameter space for PAC RL with a generative model~citep{Agarwal2020GenModel}. A seemingly unrelated algorithm is the ``trajectory tree method'' (TTM)~citep{Kearns+MN:1999}, originally developed for efficient decision-time planning in large POMDPs. This paper presents a theoretical investigation that stems from the surprising finding that CEM may indeed be viewed as an application of TTM. The qualitative benefits of this view are (1) new and simple proofs of sample complexity upper bounds for CEM, in fact under a (2) weaker assumption on the rewards than is prevalent in the current literature. Our analysis applies to both non-stationary and stationary MDPs. Quantitatively, we obtain (3) improvements in the sample-complexity upper bounds for CEM both for non-stationary and stationary MDPs, in the regime that the ``mistake probability'' $delta$ is small. Additionally, we show (4) a lower bound on the sample complexity for finite-horizon MDPs, which establishes the minimax-optimality of our upper bound for non-stationary MDPs in the small-$delta$ regime.

Problem

• PAC Reinforcement Learning
• Deterministic Equivalent Method (CEM)
• Unknown Environment

Innovation

• Deterministic Equivalent Method (CEM)
• Trajectory Tree Method (TTM)
• Data Demand Optimization