Constrained Diffusion Models with Primal-Dual Inference
For practitioners needing to sample from distributions with average constraints (e.g., resource allocation, finance), PDI provides a principled and effective method that avoids separate dual estimation, though it is an incremental extension of diffusion models.
This paper introduces primal-dual inference (PDI) for constrained diffusion models, enabling joint inference of the optimal primal distribution and dual variable during reverse diffusion. PDI achieves strong performance on constrained sampling tasks, including wireless resource allocation and portfolio management, with theoretical convergence guarantees.
This paper develops constrained diffusion models with primal-dual inference (PDI) to sample from optimal distributions of entropy-regularized optimization problems with \emph{average} constraints. We formalize constrained sampling in the Lagrangian dual domain, where the optimal distribution takes the form of a Gibbs distribution indexed by the optimal dual variable. Rather than estimating this dual multiplier before sampling and freezing it throughout generation, PDI jointly infers the optimal primal distribution and its parametrizing dual variable. Each reverse diffusion step denoises using the score field associated with the current multiplier and then updates the multiplier through dual ascent using the estimated constraint violation of the denoised samples. To enable this conditional score field, we train a single dual-conditioned score network over the family of Gibbs distributions induced by the dual variables encountered during inference. We prove that the time average of the dual variables generated along the inference trajectory converges to a neighborhood of the dual optimum and bound the effect of residual dual mismatch on the terminal distribution through schedule-dependent stability factors. We evaluate PDI on constrained sampling from a mixture of Gaussians, wireless resource allocation, and portfolio management.