Cross-Time Directional Selection in Diffusion Sampling

Recent work has popularized a practical recipe in diffusion and flow matching: predict the clean signal $x$, convert it to a velocity, and train through a velocity-space loss. The conversion contains a singular endpoint amplification and therefore appears prone to unstable optimization, yet recent systems obtain strong empirical results with this recipe. We investigate this tension through the integrability of the pre-optimizer stochastic-gradient second moment. Under stated initialization conditions, the moment diverges under Uniform sampling; boundary-suppressing sampling can restore integrability under an additional upper-growth condition. We then show that prediction--loss alignment eliminates this conversion-induced source of non-integrability. Under a uniform moment bound, alignment yields a finite second moment for every timestep density, including Uniform sampling. Controlled experiments across continuous and binary settings reproduce the predicted sampler-dependent instability and show that aligned objectives remain trainable across the tested samplers. These results reconcile pointwise amplification with sampler-dependent empirical success and support alignment as a principled route to more robust flow-matching training.

We propose D2PO (Dynamic Direct Preference Optimization), a principled framework for optimizing diffusion sampling policies with respect to timestep schedules and classifier-free guidance (CFG) weights. Our work is motivated by a fundamental limitation of existing student-teacher regression frameworks; low-NFE student samplers are trained to mimic high-NFEteachers, often sacrificing high-frequency texture fidelity while preserving coarse global structures, thereby misaligning the sampler with perceptual quality. D2PO addresses this challenge by reformulating sampler optimization as a preference-based alignment problem, leveraging the Direct Preference Optimization (DPO) framework. To make DPO applicable to diffusion samplers, we model the sampling policy as an energy-based model (EBM), transforming preference comparisons into tractable energy differences. We further introduce a novel energy formulation derived directly from the pretrained score network, enabling preference evaluation in perturbed spaces that jointly capture structural consistency and fine-grained details. Moreover, we introduce dynamic preferences, where the preferred samples used for alignment progressively improve as the sampling policies are learned. This self-improving mechanism replaces rigid static teacher supervision with an iterative, preference-guided refinement process, providing progressively stronger alignment signals. Extensive experiments demonstrate that D2PO aligns diffusion samplers with perceptual quality more faithfully, unlocking the full potential of high-quality teachers and consistently outperforming conventional regression-based schedulers under low-NFE constraints.

Sampling latency compounds in diffusion workflows, where users generate and discard many candidates before keeping one. Surprisingly, the poor outputs of standard few-step samplers do not reflect a lack of reconstruction capacity: by optimizing only the initial noise, frozen 3-4-step samplers can closely reproduce their corresponding full-step outputs. Building on this finding, we learn corrections to the initial noise and denoising updates using endpoint supervision, improving correspondence with full-step outputs generated from the same noise and prompt. The resulting previews allow users to screen candidates cheaply and reserve full-step generation for promising ones. Input correction also transfers across sampling budgets without retraining. Experiments show substantial improvements in reference fidelity, including 53-78% lower reconstruction MSE than retrained LD3 on unconditional benchmarks, alongside improved ranking preservation and candidate selection on SD1.5, SDXL, and FLUX.1-dev.