CoSCoDiff: Consensus Splitting for Coordinating Objectives and Constraints in Diffusion Models

Controllable diffusion models are used for task-specific generation, which often optimize objectives while satisfying feasibility constraints. Existing methods commonly optimize these requirements jointly, where conflicting updates can degrade sample quality and weaken constraint satisfaction. To address this issue, we explicitly separate constraints from objectives and formulate controllable diffusion sampling as a constrained stochastic optimal control problem amenable to variable splitting. Building on this formulation, we propose CoSCoDiff, a training-free framework that coordinates the two components through terminal-state consensus splitting. To approximate the intractable trajectory-level optimization, CoSCoDiff combines sequential rollout with Tweedie prediction to construct tractable local control updates. These updates propagate objective and consensus signals to intermediate noisy states, while a constraint operator enforces feasibility on the clean sample. The framework accommodates existing objective-guided methods and supports both primal-only and primal–dual coordination, allowing historical dual feedback to be retained or omitted in the presence of approximation errors. Comprehensive experiments demonstrate that CoSCoDiff effectively coordinates objective optimization and hard constraint satisfaction, and achieves the best overall performance among the baselines across heterogeneous control requirements.