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.

翻译：本文提出了一种基于原始-对偶推论（PDI）的约束扩散模型，用于从具有平均约束的熵正则化优化问题的最优分布中采样。我们在拉格朗日对偶域中形式化约束采样，其中最优分布呈现为以最优对偶变量为索引的吉布斯分布形式。PDI并非在采样前估计该对偶乘子并在生成过程中将其固定，而是联合推断最优原始分布及其参数化对偶变量。每个反向扩散步骤利用与当前乘子相关的分数场进行去噪，随后通过基于去噪样本估计的约束违反度执行对偶上升来更新乘子。为实现这种条件分数场，我们针对推论过程中遇到的对偶变量所诱导的吉布斯分布族，训练单一的双重条件分数网络。我们证明了推论轨迹中生成的对偶变量的时间平均收敛于对偶最优值的邻域，并通过与调度相关的稳定因子约束残余对偶失配对终端分布的影响。我们在高斯混合模型的约束采样、无线资源分配及投资组合管理任务上评估了PDI的性能。