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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.
We study the reinforcement learning for finite-horizon episodic Markov decision processes with adversarial reward and full information feedback, where the unknown transition probability function is a linear function of a given feature mapping. We pro
We study reinforcement learning (RL) with linear function approximation. Existing algorithms for this problem only have high-probability regret and/or Probably Approximately Correct (PAC) sample complexity guarantees, which cannot guarantee the conve
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Despite many algorithmic advances, our theoretical understanding of practical distributional reinforcement learning methods remains limited. One exception is Rowland et al. (2018)s analysis of the C51 algorithm in terms of the Cramer distance, but th