Does DQN Learn?
This addresses a foundational gap in reinforcement learning theory for researchers, revealing critical limitations in widely used methods.
The paper tackles the problem of Deep Q-Network (DQN) failing to improve upon initial policies, showing numerically that DQN often returns worse policies and providing a theoretical explanation for sub-optimal behaviors in linear DQN, including scenarios where it identifies the worst policy.
A primary requirement for any reinforcement learning method is that it should produce policies that improve upon the initial guess. In this work, we show that the widely used Deep Q-Network (DQN) fails to satisfy this minimal criterion -- even when it gets to see all possible states and actions infinitely often (a condition under which tabular Q-learning is guaranteed to converge to the optimal Q-value function). Our specific contributions are twofold. First, we numerically show that DQN often returns a policy that performs worse than the initial one. Second, we offer a theoretical explanation for this phenomenon in linear DQN, a simplified version of DQN that uses linear function approximation in place of neural networks while retaining the other key components such as $ε$-greedy exploration, experience replay, and target network. Using tools from differential inclusion theory, we prove that the limit points of linear DQN correspond to fixed points of projected Bellman operators. Crucially, we show that these fixed points need not relate to optimal -- or even near-optimal -- policies, thus explaining linear DQN's sub-optimal behaviors. We also give a scenario where linear DQN always identifies the worst policy. Our work fills a longstanding gap in understanding the convergence behaviors of Q-learning with function approximation and $ε$-greedy exploration.