Nearly Minimax Optimal Reinforcement Learning for Linear Mixture Markov Decision Processes

• URL: http://arxiv.org/abs/2012.08507v2
• Date: Thu, 7 Jan 2021 18:57:11 GMT
• Title: Nearly Minimax Optimal Reinforcement Learning for Linear Mixture Markov Decision Processes
• Authors: Dongruo Zhou and Quanquan Gu and Csaba Szepesvari
• Abstract summary: We study reinforcement learning with linear function approximation where the underlying transition probability kernel of the Markov decision process (MDP) is a linear mixture model. We propose a new, computationally efficient algorithm with linear function approximation named $textUCRL-VTR+$ for the aforementioned linear mixture MDPs. To the best of our knowledge, these are the first computationally efficient, nearly minimax optimal algorithms for RL with linear function approximation.
• Score: 91.38793800392108
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study reinforcement learning (RL) with linear function approximation where the underlying transition probability kernel of the Markov decision process (MDP) is a linear mixture model (Jia et al., 2020; Ayoub et al., 2020; Zhou et al., 2020) and the learning agent has access to either an integration or a sampling oracle of the individual basis kernels. We propose a new Bernstein-type concentration inequality for self-normalized martingales for linear bandit problems with bounded noise. Based on the new inequality, we propose a new, computationally efficient algorithm with linear function approximation named $\text{UCRL-VTR}^{+}$ for the aforementioned linear mixture MDPs in the episodic undiscounted setting. We show that $\text{UCRL-VTR}^{+}$ attains an $\tilde O(dH\sqrt{T})$ regret where $d$ is the dimension of feature mapping, $H$ is the length of the episode and $T$ is the number of interactions with the MDP. We also prove a matching lower bound $\Omega(dH\sqrt{T})$ for this setting, which shows that $\text{UCRL-VTR}^{+}$ is minimax optimal up to logarithmic factors. In addition, we propose the $\text{UCLK}^{+}$ algorithm for the same family of MDPs under discounting and show that it attains an $\tilde O(d\sqrt{T}/(1-\gamma)^{1.5})$ regret, where $\gamma\in [0,1)$ is the discount factor. Our upper bound matches the lower bound $\Omega(d\sqrt{T}/(1-\gamma)^{1.5})$ proved by Zhou et al. (2020) up to logarithmic factors, suggesting that $\text{UCLK}^{+}$ is nearly minimax optimal. To the best of our knowledge, these are the first computationally efficient, nearly minimax optimal algorithms for RL with linear function approximation.

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