Reinforcement Learning with Function Approximation: From Linear to Nonlinear

Function approximation has been an indispensable component in modern reinforcement learning algorithms designed to tackle problems with large state space in high dimensions. This paper reviews the recent results on the error analysis of those reinforcement learning algorithms in the settings of linear or nonlinear approximation, with an emphasis on the approximation error and the estimation error/sample complexity. We discuss different properties related to the approximation error and concrete conditions on the transition probability and reward function under which these properties hold true. The sample complexity in reinforcement learning is more complicated for analysis compared to supervised learning, mainly due to the distribution mismatch phenomenon. With assumptions on the linear structure of the problem, there are various algorithms in the literature that can achieve polynomial sample complexity with respect to the number of features, episode length, and accuracy, although the minimax rate has not been achieved yet. These results rely on the $L^\infty$ and UCB estimation of estimation error, which can handle the distribution mismatch phenomenon. The problem and analysis become much more challenging in the setting of nonlinear function approximation since both $L^\infty$ and UCB estimation are inadequate to help bound the error with a good rate in high dimensions. We discuss different additional assumptions needed to handle the distribution mismatch and derive meaningful results for nonlinear RL problems.