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Itô maps for any-step SDEs

NeurIPS 2026 · Image generation › Fast diffusion sampling · approach: Diffusion theory and learned samplers · 0 citations (OpenAlex)

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Proposes Itô maps, Brownian-path-conditioned any-step flow maps for generative SDEs, trained via Lagrangian/progressive self-distillation, and uses them to build gradient-based and gradient-free optimal-control estimators for reward steering.

Problem

One-step generative models distill deterministic ODE flows and cannot represent the stochastic endpoint conditional law p_{1|t}(x1|xt) needed for posterior sampling and inference-time steering with SDE dynamics.

Approach

Condition a flow map on the realized Brownian path (compressed via KL or dyadic Haar features) so the SDE trajectory becomes deterministic, enabling one-pass any-step prediction on the same path; derive It\^o-G, It\^o-GF and BEL estimators of the optimal control from it.

Key results

• On ImageNet-256 steering, Itô-G achieves average ImageReward 1.88 vs 1.83 for MFM-G, 0.55 for DPS and 1.36 for Best-of-100.
• On ImageNet-256 steering with HPSv2, Itô-G reaches 0.38 vs 0.26 for MFM-G and 0.28 for Best-of-10.
• On 2D GMM posterior steering with 128 MC samples, Itô-G attains S-W2 0.16 and MMD 0.024, improving over MFM-G on both.

Surprising findings

• Eulerian self-distillation and MeanFlow-type objectives are shown to be unable to train Itô maps, since they do not constrain the Brownian displacement.
• Gradient-free BEL estimators give only small gains on ImageNet steering, with BEL-I sometimes below the unsteered SDE on ImageNet reward (e.g., ImageReward 0.49 vs 0.42 SDE, and worse than SDE on some prompts).

Limitations

• Number of KL modes and dyadic depth chosen empirically; systematic study left for future work.
• Strongest ImageNet steering relies on a pretrained Decoupled MeanFlow teacher; self-distilled model lags strong deterministic flow maps.
• Gradient-free estimators show mixed strength depending on setting.

Experimental setup

Datasets: MNIST ImageNet-256

Benchmarks: ImageNet-256 steering MNIST class-mixture steering 2D GMM posterior sampling

Metrics: average ImageReward average HPSv2 S-W2 MMD L2 class-frequency error pixel MSE

Baselines: DPS MFM-G MFM-GF Best-of-N SMC FreeDoM MPGD

Builds on: stochastic interpolants flow maps Lagrangian self-distillation Mean Flow Doob h-transform

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