Bellman Contraction under MMD: A Unified Framework

We introduce a mathematical framework for studying contraction of the distributional Bellman operator under Maximum Mean Discrepancy (MMD). We show that contraction reduces to conditional positive definiteness of an associated kernel constructed from the base kernel, which we call the Bellman difference kernel. Focusing on radial kernels, we use Schoenberg's theorem to connect this condition to Bernstein functions, yielding contraction criteria, valid in arbitrary dimension, that can be checked analytically or numerically. Our framework unifies known results for the Gaussian RBF, power-distance, and multiquadric kernels; using numerical search, we further identify 12 new contraction-inducing kernels. We also propose chaining, a general construction that produces contraction-inducing kernels at any prescribed rate. To reconcile theory with practice, we introduce conditional positive definiteness on a set, which explains why the Gaussian RBF kernel can remain a valid choice on bounded supports despite its lack of global contraction. Finally, we identify a fundamental tradeoff between contraction rate and discriminative power, where kernels that contract faster tend to discriminate less between distributions.