Expressivity and Statistical Trade-offs in Diffusion Policy Learning

Diffusion-based policies have recently emerged as powerful policy parameterizations for reinforcement learning, representing state-conditioned action distributions as terminal laws of diffusion processes with parameterized drifts. This terminal-law representation has shown substantial expressive flexibility in practice, enabling diffusion policies to model complex, multimodal, and highly non-Gaussian action distributions; however, it remains unclear what mathematically drives this expressivity and how to fully exploit it when the policy is learned from finite data. In this paper, we identify the drift Lipschitz budget K as a central quantity governing the expressivity and statistical behavior of diffusion policies. We quantify expressivity through approximation: diffusion policies with K-Lipschitz drifts can concentrate near optimal deterministic policies and achieve value approximation error of order 1/K; moreover, we prove a matching lower bound under nondegenerate diffusion noise. This increased expressivity comes with a statistical cost. When the drift is parameterized by neural networks, increasing K improves approximation but increases statistical complexity. Balancing these two terms yields a finite-sample performance gap of order tildeO(n-2/(m+6)) for generic neural-network drifts, and a sharper rate tildeO(n-2/(m+4)) for one-sided dissipative drift classes, where n is the sample size and m is the dimension of the state space. Numerical experiments provide empirical evidence for the sample-dependent trade-off in K, supporting both theoretical regimes. Our framework also suggests a practical implementation principle: choose the diffusion budget K according to the available sample size, and then select a neural-network architecture with the corresponding fixed Lipschitz coefficient.