Prior-dependent analysis of posterior sampling reinforcement learning with function approximation
(2403.11175)Abstract
This work advances randomized exploration in reinforcement learning (RL) with function approximation modeled by linear mixture MDPs. We establish the first prior-dependent Bayesian regret bound for RL with function approximation; and refine the Bayesian regret analysis for posterior sampling reinforcement learning (PSRL), presenting an upper bound of ${\mathcal{O}}(d\sqrt{H3 T \log T})$, where $d$ represents the dimensionality of the transition kernel, $H$ the planning horizon, and $T$ the total number of interactions. This signifies a methodological enhancement by optimizing the $\mathcal{O}(\sqrt{\log T})$ factor over the previous benchmark (Osband and Van Roy, 2014) specified to linear mixture MDPs. Our approach, leveraging a value-targeted model learning perspective, introduces a decoupling argument and a variance reduction technique, moving beyond traditional analyses reliant on confidence sets and concentration inequalities to formalize Bayesian regret bounds more effectively.
Overview
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The paper analyzes how the variance of prior distributions affects learning efficiency in reinforcement learning (RL) with function approximation, focusing on Posterior Sampling for Reinforcement Learning (PSRL) under linear mixture Markov Decision Processes (MDPs).
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It introduces a prior-dependent Bayesian regret bound to understand the impact of prior distribution's variance on learning and presents an improved prior-free Bayesian regret bound for PSRL.
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A novel posterior variance reduction theorem is discussed, showing how posterior variance of model parameters decreases predictably based on prior distribution variance and environmental dynamics variance.
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The study concludes with implications for the design of prior distributions in RL and suggests directions for future research, indicating the importance of prior knowledge in effective learning strategies.
Prior-dependent Analysis of Posterior Sampling Reinforcement Learning with Function Approximation
Introduction
Reinforcement Learning (RL) with function approximation is a significant aspect of building sophisticated artificial intelligence systems capable of learning and making decisions from complex and high-dimensional data. The incorporation of priors, reflecting pre-existing knowledge or assumptions about the environment's dynamics, plays a crucial role in accelerating learning in RL. This paper presents a formal analysis of Posterior Sampling for Reinforcement Learning (PSRL) under the framework of linear mixture Markov Decision Processes (MDPs). Our focus is to elucidate how the variance of prior distributions influences learning efficacy, leading to a nuanced understanding of Bayesian regret in RL with function approximation.
Key Contributions
Our study introduces several novel contributions to the domain of RL with function approximation, particularly within the scope of linear mixture MDPs:
- We establish a prior-dependent Bayesian regret bound, providing insights into how the prior distribution's variance impacts learning efficiency.
- An improved prior-free Bayesian regret bound is presented for PSRL, enhancing understanding of the relation between regret and the choice of prior.
- A methodological advancement is achieved through a decoupling argument and a variance reduction theorem, circumventing traditional dependence on confidence bounds for regret analysis.
Technical Novelty
Posterior Variance Reduction
At the heart of our analysis is a novel posterior variance reduction theorem. This theorem elucidates a reduction in the posterior variance of the true model parameters in a predictable manner, contingent upon the variance of the prior distribution and the variance induced by environmental dynamics. This reduction is captured as:
- $\mathbb{E}[\rmGamma{\ell+1, h} \given \hist{\ell, h}] \preceq \rmGamma{\ell, h} - \frac{ \rmGamma{\ell, h} X{\ell, h} X{\ell, h}{\top} \rmGamma{\ell, h} }{ \bar{\sigma}{\ell, h}2 + X{\ell, h}{\top} \rmGamma{\ell, h} X{\ell, h}}$.
Decoupling Argument
Through a decoupling lemma, we bridge the regrets to the posterior variance over models, allowing for a prior-dependent analysis. This lemma signifies an escalation from classical analyses, providing a stronger foundation for regret bound estimations:
- $\mathbb{E}[\sum{\ell=1}L \abs{ \Delta{\ell, h}(s_{\ell, h}) }] \leq \sqrt{d \mathbb{E}[...]}$, linking regret directly to cumulative posterior variance.
Regret Analysis
The crux of our analytical contributions lies in aptly quantifying the impact of prior knowledge on learning efficacy. Through meticulous decomposition of regret into cumulative variance and potential, we highlight how prior knowledge innately modulates exploration-exploitation dynamics. By bounding the cumulative variance and cumulative potential, we derive both prior-dependent and improved prior-free Bayesian regret bounds for PSRL, thereby offering a comprehensive perspective on the role of priors in RL.
Implications and Future Directions
This paper's insights into the prior-dependent analysis foster a deeper understanding of the Bayesian methods in RL, encouraging more informed selection and design of prior distributions. Furthermore, the methodological novelties introduced pave the way for future investigations into RL's broader aspects, including exploration strategies and model mis-specification. Speculatively, our analysis hints at a more refined conjecture regarding the lower bounds of Bayesian regret in RL with function approximation, underscoring the significance of prior knowledge in achieving optimal learning trajectories.
In conclusion, our paper enhances the theoretical underpinnings of RL with function approximation, particularly for linear mixture MDPs, by introducing a prior-dependent Bayesian regret bound and refining the understanding of PSRL through a novel decoupling argument and posterior variance reduction technique. This work not only extends our comprehension of Bayesian regret in RL but also sets the stage for future explorations into efficient learning strategies leveraging prior knowledge.
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