Distributional reinforcement learning with linear function approximation
This work addresses a theoretical gap for researchers in reinforcement learning, offering the first proof of convergence for a distributional algorithm with function approximation, though it is incremental in nature.
The paper tackles the limited theoretical understanding of practical distributional reinforcement learning methods by adapting the Cramér distance to arbitrary vectors, deriving a new algorithm with formal convergence guarantees for policy evaluation when combined with linear function approximation, and providing evidence that Cramér-based methods may perform worse than directly approximating the value function.
Despite many algorithmic advances, our theoretical understanding of practical distributional reinforcement learning methods remains limited. One exception is Rowland et al. (2018)'s analysis of the C51 algorithm in terms of the Cramér distance, but their results only apply to the tabular setting and ignore C51's use of a softmax to produce normalized distributions. In this paper we adapt the Cramér distance to deal with arbitrary vectors. From it we derive a new distributional algorithm which is fully Cramér-based and can be combined to linear function approximation, with formal guarantees in the context of policy evaluation. In allowing the model's prediction to be any real vector, we lose the probabilistic interpretation behind the method, but otherwise maintain the appealing properties of distributional approaches. To the best of our knowledge, ours is the first proof of convergence of a distributional algorithm combined with function approximation. Perhaps surprisingly, our results provide evidence that Cramér-based distributional methods may perform worse than directly approximating the value function.