WEIRDO: WEak resIdual Regularized DOob's h-transform diffusion alignment

Abstract

We study the problem of estimating the guidance that steers the distribution learned by a diffusion generative model toward a tilted target q₀ ∝ w p₀ at inference time. Relying on the stochastic optimal control approach, we observe that the exact drift correction is the gradient of the logarithm of Doob's h-function, and we study the problem of estimating it from a sample. In the present paper, we assume that the score of the pretrained model is available, that the tilting weight is bounded and positive, and that the reference distribution has a bounded support, no smoothness of the weight is required. Introducing a penalized least-squares risk in which the penalty is the residual of the space-time harmonicity equation satisfied by the h-function, measured in a dual Sobolev norm, we derive high-probability bounds on the squared error of the resulting guidance estimate. Since the penalty vanishes at the target, the estimator is free of regularization bias, and in favourable scenarios its rate of convergence is faster than the minimax rate of estimating first-order derivatives of a smooth regression function. Assuming that w is bounded and positive with 𝔼_p₀[w^−s] < ∞ for some s ∈ (0,∞], and that the reference data are compactly supported, we prove that the guidance is estimable in squared L² at rate ε_n^s/(s+4), where ε_n = n^{−2(β−1)/(2(β−1)+d)}. We also transfer the obtained bounds to the total variation distance between the marginals of the estimated and the exactly guided samplers, and illustrate the performance of the suggested approach with numerical experiments.

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Cite

BibTeX RIS

BibTeX

@misc{suchkov2026weirdo,
  title = {{WEIRDO: WEak resIdual Regularized DOob's h-transform diffusion alignment}},
  author = {Denis Suchkov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39531},
  eprint = {2609.39531},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39531},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - WEIRDO: WEak resIdual Regularized DOob's h-transform diffusion alignment
AU  - Denis Suchkov
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39531
UR  - https://arxiv.org/abs/2609.39531
AB  - We study the problem of estimating the guidance that steers the distribution learned by a diffusion generative model toward a tilted target q_0 ∝ w p_0 at inference time. Relying on the stochastic optimal control approach, we observe that the exact drift correction is the gradient of the logarithm of Doob's h-function, and we study the problem of estimating it from a sample. In the present paper, we assume that the score of the pretrained model is available, that the tilting weight is bounded and positive, and that the reference distribution has a bounded support, no smoothness of the weight is required. Introducing a penalized least-squares risk in which the penalty is the residual of the space-time harmonicity equation satisfied by the h-function, measured in a dual Sobolev norm, we derive high-probability bounds on the squared error of the resulting guidance estimate. Since the penalty vanishes at the target, the estimator is free of regularization bias, and in favourable scenarios its rate of convergence is faster than the minimax rate of estimating first-order derivatives of a smooth regression function. Assuming that w is bounded and positive with 𝔼_p_0[w^−s] < ∞ for some s ∈ (0,∞], and that the reference data are compactly supported, we prove that the guidance is estimable in squared L^2 at rate ε_n^s/(s+4), where ε_n = n^−2(β−1)/(2(β−1)+d). We also transfer the obtained bounds to the total variation distance between the marginals of the estimated and the exactly guided samplers, and illustrate the performance of the suggested approach with numerical experiments.
ER  -

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