A Kernel Perspective on Behavioural Metrics for Markov Decision Processes

Behavioural metrics have been shown to be an effective mechanism for constructing representations in reinforcement learning. We present a novel perspective on behavioural metrics for Markov decision processes via the use of positive definite kernels. We leverage this new perspective to define a new metric that is provably equivalent to the recently introduced MICo distance (Castro et al., 2021). The kernel perspective further enables us to provide new theoretical results, which has so far eluded prior work. These include bounding value function differences by means of our metric, and the demonstration that our metric can be provably embedded into a finite-dimensional Euclidean space with low distortion error. These are two crucial properties when using behavioural metrics for reinforcement learning representations. We complement our theory with strong empirical results that demonstrate the effectiveness of these methods in practice.