Value Function Approximations via Kernel Embeddings for No-Regret Reinforcement Learning

TL;DR

This paper introduces a kernel embedding-based RL algorithm that learns transition representations in a reproducing kernel Hilbert space, achieving regret bounds without estimating transition probabilities, suitable for large or continuous state-action spaces.

Contribution

The paper proposes CME-RL, an online model-based RL algorithm using kernel embeddings for transition distributions, providing regret guarantees and bypassing transition probability estimation.

Findings

Achieves a regret bound of order O(Hmma_N\u221a{N})

Applicable to domains with kernels, large or continuous spaces

Provides new insights into kernel methods for RL

Abstract

We consider the regret minimization problem in reinforcement learning (RL) in the episodic setting. In many real-world RL environments, the state and action spaces are continuous or very large. Existing approaches establish regret guarantees by either a low-dimensional representation of the stochastic transition model or an approximation of the $Q$-functions. However, the understanding of function approximation schemes for state-value functions largely remains missing. In this paper, we propose an online model-based RL algorithm, namely the CME-RL, that learns representations of transition distributions as embeddings in a reproducing kernel Hilbert space while carefully balancing the exploitation-exploration tradeoff. We demonstrate the efficiency of our algorithm by proving a frequentist (worst-case) regret bound that is of order $\tilde{O}\big(H\gamma_N\sqrt{N}\big)$\footnote{ $\tilde{O}(\cdot)$ hides only absolute constant and poly-logarithmic factors.}, where $H$ is the episode length, $N$ is the total number of time steps and $\gamma_N$ is an information theoretic quantity relating the effective dimension of the state-action feature space. Our method bypasses the need for estimating transition probabilities and applies to any domain on which kernels can be defined. It also brings new insights into the general theory of kernel methods for approximate inference and RL regret minimization.