Efficient Learning of Stationary Diffusions with Stein-type Discrepancies

Fabian Bleile, Sarah Lumpp, Mathias Drton

TL;DR: We propose SKDS, a Stein-type objective that learns stationary diffusions with theoretical guarantees and reduced computational cost.

Abstract: Learning a stationary diffusion amounts to estimating the parameters of a stochastic differential equation whose stationary distribution matches a target distribution. We build on the recently introduced kernel deviation from stationarity (KDS), which enforces stationarity by evaluating expectations of the diffusion's generator in a reproducing kernel Hilbert space. Leveraging the connection between KDS and Stein discrepancies, we introduce the Stein-type KDS (SKDS) as an alternative formulation. We prove that a vanishing SKDS guarantees alignment of the learned diffusion’s stationary distribution with the target. Furthermore, under broad parametrizations, SKDS is convex with an empirical version that is $\epsilon$-quasiconvex with high probability. Empirically, learning with SKDS attains comparable accuracy to KDS while substantially reducing computational cost, and yields improvements over the majority of competitive baselines.

Submission Number: 553