  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.
