Logarithmic Regret for Reinforcement Learning with Linear Function Approximation

Reinforcement learning (RL) with linear function approximation has received increasing attention recently. However, existing work has focused on obtaining √ 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., 2020) , the LSVI-UCB algorithm can achieve O(d 3 H 5 /gap min • log(T )) regret; and under the linear mixture MDP assumption (Ayoub et al., 2020) , the UCRL-VTR algorithm can achieve O(d 2 H 5 /gap min • log 3 (T )) regret, where d is the dimension of feature mapping, H is the length of episode, gap min is the minimal sub-optimality gap, and 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.