Nearly Minimax Optimal Reinforcement Learning for Linear Markov Decision Processes
This provides a complete solution for optimal reinforcement learning in linear MDPs, which is foundational for advancing RL theory and applications.
The paper tackles the problem of reinforcement learning with linear function approximation in episodic time-inhomogeneous linear Markov decision processes, achieving a nearly minimax optimal regret of $ ilde O(d\sqrt{H^3K})$ with the first computationally efficient algorithm.
We study reinforcement learning (RL) with linear function approximation. For episodic time-inhomogeneous linear Markov decision processes (linear MDPs) whose transition probability can be parameterized as a linear function of a given feature mapping, we propose the first computationally efficient algorithm that achieves the nearly minimax optimal regret $\tilde O(d\sqrt{H^3K})$, where $d$ is the dimension of the feature mapping, $H$ is the planning horizon, and $K$ is the number of episodes. Our algorithm is based on a weighted linear regression scheme with a carefully designed weight, which depends on a new variance estimator that (1) directly estimates the variance of the optimal value function, (2) monotonically decreases with respect to the number of episodes to ensure a better estimation accuracy, and (3) uses a rare-switching policy to update the value function estimator to control the complexity of the estimated value function class. Our work provides a complete answer to optimal RL with linear MDPs, and the developed algorithm and theoretical tools may be of independent interest.