Isoperimetry is All We Need: Langevin Posterior Sampling for RL with Sublinear Regret (2412.20824v2)

Abstract: Common assumptions, like linear or RKHS models, and Gaussian or log-concave posteriors over the models, do not explain practical success of RL across a wider range of distributions and models. Thus, we study how to design RL algorithms with sublinear regret for isoperimetric distributions, specifically the ones satisfying the Log-Sobolev Inequality (LSI). LSI distributions include the standard setups of RL theory, and others, such as many non-log-concave and perturbed distributions. First, we show that the Posterior Sampling-based RL (PSRL) algorithm yields sublinear regret if the data distributions satisfy LSI and some mild additional assumptions. Also, when we cannot compute or sample from an exact posterior, we propose a Langevin sampling-based algorithm design: LaPSRL. We show that LaPSRL achieves order-optimal regret and subquadratic complexity per episode. Finally, we deploy LaPSRL with a Langevin sampler -- SARAH-LD, and test it for different bandit and MDP environments. Experimental results validate the generality of LaPSRL across environments and its competitive performance with respect to the baselines.

• The paper demonstrates that Posterior Sampling-based RL using isoperimetric distributions can achieve sublinear regret under Log-Sobolev Inequality conditions.
• It introduces LaPSRL, a Langevin sampling-based adaptation that efficiently approximates intractable posteriors in high-dimensional or non-convex settings.
• Empirical results in bandit and MDP environments confirm that LaPSRL attains order-optimal performance with subquadratic computational complexity per episode.

Isoperimetry in Reinforcement Learning: Sublinear Regret through Langevin Sampling

The paper "Isoperimetry is All We Need: Langevin Posterior Sampling for RL with Sublinear Regret" by Emilio Jorge et al. presents an in-depth exploration of designing Reinforcement Learning (RL) algorithms with sublinear regret, leveraging isoperimetric properties. The focus lies on distributions that adhere to the Log-Sobolev Inequality (LSI).

Key Contributions

1. Isoperimetric Distributions in RL: The authors propose moving beyond conventional assumptions such as linear models or posteriors that are Gaussian/log-concave, which do not entirely elucidate the practical success of RL. They target isoperimetric distributions, including those conforming to LSI, for algorithm design.
2. Theoretical Justification: The paper establishes that Posterior Sampling-based RL (PSRL) can yield sublinear regret when the data distributions comply with LSI, under some mild assumptions. This insight broadens the applicability of PSRL to a wider scope of distribution models beyond traditional assumptions.
3. Practical Algorithm - LaPSRL: Recognizing the impracticality of exact posterior computation or sampling in many scenarios, the authors propose LaPSRL, a Langevin sampling-based adaptation to PSRL. They demonstrate that LaPSRL achieves order-optimal regret with subquadratic complexity per episode.
4. Empirical Validation: LaPSRL, implemented with the SARAH-LD Langevin sampler, was experimentally validated in different bandit and Markov Decision Process (MDP) environments. It showed competent performance against existing benchmarks, underlining its versatility and efficiency.

Methodological Insights

Isoperimetry and LSI: Central to the paper is the utilization of isoperimetric properties, specifically LSI, to facilitate efficient sampling in RL contexts. This approach ensures proper concentration of empirical statistics which can mitigate common issues such as model error and sampling bias.

Langevin Sampling Techniques: The paper underscores the use of Langevin-based methods for approximate sampling, crucial for handling scenarios where exact posterior computation is impractical due to high-dimensionality or non-convexity.

Implications and Future Directions

The implications of this research span both theoretical and practical realms:

• Theoretical Expansion: This work extends the theoretical framework of RL by integrating isoperimetry, enhancing the robustness of PSRL across a broader array of distributions.
• Practical Application: By providing a practicable approach to approximate sampling in RL, this research holds promise for improving performance in real-world applications where data distributions are complex and non-conventional.
• Future Exploration: The paper opens avenues for further research on adapting these principles to more complex neural architectures like Bayesian neural networks and explores connections with areas like mean-field neural networks.

By advancing a comprehensive framework that accommodates LSI distributions, this paper contributes significantly to closing the theory-to-practice gap in RL, laying the groundwork for adaptive RL systems that are more generalizable and efficient in diverse settings.