Primal-Dual Inference Enables Constrained Diffusion Models

Researchers have developed a new method called Constrained Diffusion Models with Primal-Dual Inference (PDI) to address the challenge of sampling from optimal distributions in entropy-regularized optimization problems, particularly those involving average constraints. This approach formalizes constrained sampling within the Lagrangian dual domain, where the desired optimal distribution is a Gibbs distribution parameterized by an optimal dual variable. Unlike methods that pre-estimate and fix the dual multiplier, PDI dynamically infers both the optimal primal distribution and its corresponding dual variable simultaneously. In each step of the reverse diffusion process, the model denoises using a score field conditioned on the current multiplier, then updates this multiplier via dual ascent based on the estimated constraint violation of the denoised samples. To facilitate this, a single dual-conditioned score network is trained across the family of Gibbs distributions induced by the dual variables encountered during inference. The paper provides theoretical guarantees, proving that the time average of the generated dual variables converges to a neighborhood of the dual optimum and bounding the impact of any residual dual mismatch on the final distribution. PDI's effectiveness is demonstrated across various applications, including constrained Gaussian mixture sampling, wireless resource allocation, and portfolio management.