Tackling Heavy-Tailed Rewards in Reinforcement Learning with Function Approximation: Minimax Optimal and Instance-Dependent Regret Bounds

While numerous works have focused on devising efficient algorithms for reinforcement learning (RL) with uniformly bounded rewards, it remains an open question whether sample or time-efficient algorithms for RL with large state-action space exist when the rewards are heavy-tailed, i.e., with only finite (1+ϵ)-th moments for some ϵ∈(0,1]. In this work, we address the challenge of such rewards in RL with linear function approximation. We first design an algorithm, Heavy-OFUL, for heavy-tailed linear bandits, achieving an instance-dependent T-round regret of Õ(d T^1-ϵ/2(1+ϵ)√(∑_t=1^T ν_t^2) + d T^1-ϵ/2(1+ϵ)), the first of this kind. Here, d is the feature dimension, and ν_t^1+ϵ is the (1+ϵ)-th central moment of the reward at the t-th round. We further show the above bound is minimax optimal when applied to the worst-case instances in stochastic and deterministic linear bandits. We then extend this algorithm to the RL settings with linear function approximation. Our algorithm, termed as Heavy-LSVI-UCB, achieves the first computationally efficient instance-dependent K-episode regret of Õ(d √(H 𝒰^*) K^1/1+ϵ + d √(H 𝒱^* K)). Here, H is length of the episode, and 𝒰^*, 𝒱^* are instance-dependent quantities scaling with the central moment of reward and value functions, respectively. We also provide a matching minimax lower bound Ω(d H K^1/1+ϵ + d √(H^3 K)) to demonstrate the optimality of our algorithm in the worst case. Our result is achieved via a novel robust self-normalized concentration inequality that may be of independent interest in handling heavy-tailed noise in general online regression problems.