Logarithmic Regret for Reinforcement Learning with Linear Function Approximation
This work addresses the challenge of improving regret bounds from √T to logarithmic for RL researchers and practitioners, representing a significant theoretical advance rather than an incremental improvement.
The paper tackles the problem of achieving logarithmic regret in reinforcement learning with linear function approximation, showing that under linear MDP assumptions, algorithms like LSVI-UCB and UCRL-VTR can achieve regret bounds of Õ(d³H⁵/gap_min·log(T)) and Õ(d²H⁵/gap_min·log³(T)), respectively, which are the first such logarithmic bounds in this context.
Reinforcement learning (RL) with linear function approximation has received increasing attention recently. However, existing work has focused on obtaining $\sqrt{T}$-type regret bound, where $T$ is the number of interactions with the MDP. In this paper, we show that logarithmic regret is attainable under two recently proposed linear MDP assumptions provided that there exists a positive sub-optimality gap for the optimal action-value function. More specifically, under the linear MDP assumption (Jin et al. 2019), the LSVI-UCB algorithm can achieve $\tilde{O}(d^{3}H^5/\text{gap}_{\text{min}}\cdot \log(T))$ regret; and under the linear mixture MDP assumption (Ayoub et al. 2020), the UCRL-VTR algorithm can achieve $\tilde{O}(d^{2}H^5/\text{gap}_{\text{min}}\cdot \log^3(T))$ regret, where $d$ is the dimension of feature mapping, $H$ is the length of episode, $\text{gap}_{\text{min}}$ is the minimal sub-optimality gap, and $\tilde O$ hides all logarithmic terms except $\log(T)$. To the best of our knowledge, these are the first logarithmic regret bounds for RL with linear function approximation. We also establish gap-dependent lower bounds for the two linear MDP models.