Ctrl-Z Sampling: Scaling Diffusion Sampling with Controlled Random Zigzag Explorations

Diffusion models have achieved unprecedented success in text-aligned generation, largely driven by Classifier-Free Guidance (CFG). However, standard CFG operates strictly on instantaneous gradients, omitting the intrinsic curvature of the data manifold. Recent methods like Zigzag-sampling (Z-Sampling) explicitly traverse multi-step forward-backward trajectories to probe this curvature, significantly improving semantic alignment. Yet, these explicit traversals triple the Neural Function Evaluation (NFE) cost and introduce unconstrained truncation errors from off-manifold evaluations, causing cumulative drift from the true marginal distribution. In this paper, we theoretically demonstrate that the explicit zigzag sequence is topologically reducible. We propose Implicit Z-Sampling, rigorously proving that intermediate states can be algebraically annihilated via operator dualities, physically eliminating off-manifold approximation errors. To push sampling efficiency to its theoretical lower bound, we introduce $Z^2$-Sampling (Zero-cost Zigzag Sampling). Exploiting the Probability Flow ODE's temporal coherence, $Z^2$-Sampling couples implicit algebraic collapse with a dynamically cached Temporal Semantic Surrogate. This restores the standard 2-NFE baseline without sacrificing semantic exploration. We formally prove via Backward Error Analysis that this discrete collapse inherently synthesizes a directional derivative curvature penalty. Finally, extensive evaluations demonstrate that $Z^2$-Sampling structurally shatters the performance-efficiency Pareto frontier. We validate its universal applicability across diverse architectures (U-Nets, DiTs) and modalities (image/video), establishing seamless orthogonality with advanced alignment frameworks (AYS, Diffusion-DPO).

We develop a principled framework for analyzing and designing noise schedules in diffusion models. We show that one can recast this design problem as an optimal control problem, whose state is the Fisher information of the diffusion process which evolves according to an ODE and the control input is the noise schedule. The objective of the optimal control problem is a functional involving the Fisher information, which is shown to be an upper bound on the Kullback-Leibler sampling error. By solving this optimal control problem, we obtain sufficient conditions on noise schedules under which state-of-the-art $\tilde{\mathcal{O}} (d/n)$ sampling error is achievable, where $d$ is the data dimension and $n$ is the number of discretization steps. While existing theoretical work also prove that $\tilde{\mathcal{O}}(d/n)$ sampling error bounds are achievable, these results hold for specific noise schedules, which do not include the schedules used in practice. Under a further parametric assumption on the data distribution, we show that one can obtain closed-form expressions for the noise schedules. These noise schedules generalize standard empirical schedules such as exponential and sigmoid schedules by allowing additional parameters that can be tuned. Systematically tuning the parameters of these schedules yields new schedules that achieve superior FID scores on image generation benchmarks.

Adapting pretrained diffusion models to downstream objectives such as inverse problems often requires expensive test-time guidance or optimization. We propose a principled framework for generating high-quality reward-aligned samples at substantially reduced inference cost. Our approach formulates test-time adaptation as a hierarchical variational model, where control is amortized into a lightweight yet expressive stochastic policy. This formulation naturally supports few-step diffusion sampling: large step sizes enable fast inference, while the learned policy maintains sample quality by providing structured per-step control. The resulting fully amortized sampler achieves a strong quality--speed tradeoff, matching or exceeding recent test-time scaling baselines while requiring significantly less compute. For example, on 4x super-resolution, our method achieves better perceptual quality with more than 5x faster inference compared to the best-performing baseline. We further extend our approach to a semi-amortized regime that combines cheap amortized proposals with limited test-time optimization, achieving state-of-the-art perceptual quality across several challenging inverse problems.