Distributional Reinforcement Learning with Unconstrained Monotonic Neural Networks

TL;DR

This paper introduces a novel distributional RL algorithm using unconstrained monotonic neural networks that can learn multiple distribution representations and compares different probability metrics.

Contribution

It presents the first distributional RL method supporting learning of PDF, CDF, and QF representations using UMNNs, decoupling distribution representation from the metric.

Findings

UMDQN supports learning three distribution representations.

Comparison of KL, Cramer, and Wasserstein metrics reveals their strengths and weaknesses.

Identifies limitations of the Wasserstein distance in distributional RL.

Abstract

The distributional reinforcement learning (RL) approach advocates for representing the complete probability distribution of the random return instead of only modelling its expectation. A distributional RL algorithm may be characterised by two main components, namely the representation of the distribution together with its parameterisation and the probability metric defining the loss. The present research work considers the unconstrained monotonic neural network (UMNN) architecture, a universal approximator of continuous monotonic functions which is particularly well suited for modelling different representations of a distribution. This property enables the efficient decoupling of the effect of the function approximator class from that of the probability metric. The research paper firstly introduces a methodology for learning different representations of the random return distribution (PDF, CDF and QF). Secondly, a novel distributional RL algorithm named unconstrained monotonic deep Q-network (UMDQN) is presented. To the authors' knowledge, it is the first distributional RL method supporting the learning of three, valid and continuous representations of the random return distribution. Lastly, in light of this new algorithm, an empirical comparison is performed between three probability quasi-metrics, namely the Kullback-Leibler divergence, Cramer distance, and Wasserstein distance. The results highlight the main strengths and weaknesses associated with each probability metric together with an important limitation of the Wasserstein distance.