Stability and Generalisation in Batch Reinforcement Learning

Overfitting has been recently acknowledged as a key limiting factor in the capabilities of reinforcement learning algorithms, despite little theoretical characterisation. We provide a theoretical examination of overfitting in the context of batch reinforcement learning, through the fundamental relationship between algorithmic stability (Bousquet & Elisseeff, 2002)–which characterises the effect of a change at a single data point–and the generalisation gap–which quantifies overfitting. Examining a popular fitted policy evaluation method with linear value function approximation, we characterise the dynamics of overfitting in the RL context. We provide finite sample, finite time, polynomial bounds on the generalisation gap in RL. In addition, our approach applies to a class of algorithms which only partially fit to temporal difference errors, as is common in deep RL, rather than perfectly optimising at each step. As such, our results characterise an unexplored bias-variance trade-off in the frequency of target network updates. To do so, our work extends the stochastic gradient-based approach of Hardt et al. (2016) to the iterative methods more common in RL. We find that under regimes where learning requires few iterations, the expected temporal difference error over the dataset is representative of the true performance on the MDP, indicating that, as is the case in supervised learning, good generalisation in RL can be ensured through the use of algorithms that learn quickly.