Distributional reinforcement learning with linear function approximation

TL;DR

This paper introduces a new distributional reinforcement learning algorithm based on the Cramér distance that works with linear function approximation and provides the first convergence proof for such methods, highlighting potential performance drawbacks.

Contribution

It adapts the Cramér distance for arbitrary vectors and develops a distributional algorithm with formal guarantees in policy evaluation, extending theoretical understanding.

Findings

First convergence proof of distributional RL with function approximation.

Cramér-based methods may underperform compared to direct value function approximation.

The new method generalizes distributional RL to arbitrary real vectors, losing probabilistic interpretation.

Abstract

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 Cram\'er 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 Cram\'er distance to deal with arbitrary vectors. From it we derive a new distributional algorithm which is fully Cram\'er-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 Cram\'er-based distributional methods may perform worse than directly approximating the value function.