Optimal Horizon-Free Reward-Free Exploration for Linear Mixture MDPs
This work addresses the challenge of sample-efficient exploration in reinforcement learning for long-horizon problems, which is incremental as it builds on prior reward-free RL methods but introduces horizon-free guarantees.
The paper tackles the problem of reward-free reinforcement learning with linear function approximation, where existing algorithms have sample complexities that depend polynomially on the planning horizon, making them intractable for long-horizon problems. The result is a new algorithm for linear mixture MDPs that achieves a sample complexity of Õ(d²ε⁻²) episodes to find an ε-optimal policy, with only polylogarithmic dependence on the horizon, and provides a matching lower bound up to logarithmic factors.
We study reward-free reinforcement learning (RL) with linear function approximation, where the agent works in two phases: (1) in the exploration phase, the agent interacts with the environment but cannot access the reward; and (2) in the planning phase, the agent is given a reward function and is expected to find a near-optimal policy based on samples collected in the exploration phase. The sample complexities of existing reward-free algorithms have a polynomial dependence on the planning horizon, which makes them intractable for long planning horizon RL problems. In this paper, we propose a new reward-free algorithm for learning linear mixture Markov decision processes (MDPs), where the transition probability can be parameterized as a linear combination of known feature mappings. At the core of our algorithm is uncertainty-weighted value-targeted regression with exploration-driven pseudo-reward and a high-order moment estimator for the aleatoric and epistemic uncertainties. When the total reward is bounded by $1$, we show that our algorithm only needs to explore $\tilde O( d^2\varepsilon^{-2})$ episodes to find an $\varepsilon$-optimal policy, where $d$ is the dimension of the feature mapping. The sample complexity of our algorithm only has a polylogarithmic dependence on the planning horizon and therefore is "horizon-free". In addition, we provide an $Ω(d^2\varepsilon^{-2})$ sample complexity lower bound, which matches the sample complexity of our algorithm up to logarithmic factors, suggesting that our algorithm is optimal.