Sample-Efficient Reinforcement Learning for Linearly-Parameterized MDPs with a Generative Model

TL;DR

This paper demonstrates that for large-scale MDPs with linear features, model-based RL and Q-learning can learn near-optimal policies efficiently, with sample complexities depending on feature dimension rather than state-action space size.

Contribution

The paper provides tight sample complexity bounds for model-based RL and Q-learning in linearly-parameterized MDPs, matching minimax lower bounds and showing efficiency when feature dimension is small.

Findings

Sample complexity scales with feature dimension K, not state-action space size.

Model-based approach matches minimax lower bounds in sample efficiency.

Q-learning also achieves near-optimal sample complexity under the linear feature assumption.

Abstract

The curse of dimensionality is a widely known issue in reinforcement learning (RL). In the tabular setting where the state space $\mathcal{S}$ and the action space $\mathcal{A}$ are both finite, to obtain a nearly optimal policy with sampling access to a generative model, the minimax optimal sample complexity scales linearly with $|\mathcal{S}|\times|\mathcal{A}|$, which can be prohibitively large when $\mathcal{S}$ or $\mathcal{A}$ is large. This paper considers a Markov decision process (MDP) that admits a set of state-action features, which can linearly express (or approximate) its probability transition kernel. We show that a model-based approach (resp.$~$Q-learning) provably learns an $\varepsilon$-optimal policy (resp.$~$Q-function) with high probability as soon as the sample size exceeds the order of $\frac{K}{(1-\gamma)^{3}\varepsilon^{2}}$ (resp.$~$$\frac{K}{(1-\gamma)^{4}\varepsilon^{2}}$), up to some logarithmic factor. Here $K$ is the feature dimension and $\gamma\in(0,1)$ is the discount factor of the MDP. Both sample complexity bounds are provably tight, and our result for the model-based approach matches the minimax lower bound. Our results show that for arbitrarily large-scale MDP, both the model-based approach and Q-learning are sample-efficient when $K$ is relatively small, and hence the title of this paper.