WEIRDO：弱残差正则化的Doob h-变换扩散对齐

中文摘要 AI 辅助

我们研究在推理时估计引导项的问题，该引导项将扩散生成模型所学分布导向倾斜目标$q_0 \propto w\\,p_0$。基于随机最优控制方法，我们观察到精确的漂移修正等于Doob $h$-函数对数的梯度，并研究从样本中估计该梯度的问题。在本文中，我们假设预训练模型的得分函数可用，倾斜权重有界且为正，参考分布具有有界支撑，且不需要权重的光滑性。我们引入一个带惩罚的最小二乘风险，其中惩罚项为$h$-函数满足的空间-时间调和方程残差，以对偶Sobolev范数度量，并推导出所得引导估计的平方误差的高概率界。由于惩罚在目标处消失，估计器无正则化偏差，且在有利情形下其收敛速度快于估计光滑回归函数一阶导数的极小极大速率。假设$w$有界且为正，且对某个$\mathrm{s} \in (0,\infty]$有$\mathbb{E}_{p_0}[w^{-\mathrm{s}}] < \infty$，参考数据具有紧支撑，我们证明引导项在平方$L^2$意义下以速率$\varepsilon_n^{\mathrm{s}/(\mathrm{s}+4)}$可估计，其中$\varepsilon_n = n^{-2(\beta-1)/(2(\beta-1)+d)}$。我们还将所得界转移到估计引导采样器与精确引导采样器边缘分布之间的全变差距离上，并通过数值实验说明所提方法的性能。

英文摘要

We study the problem of estimating the guidance that steers the distribution learned by a diffusion generative model toward a tilted target $q_0 \propto w\,p_0$ 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 $\mathbb{E}_{p_0}[w^{-\mathrm{s}}] < \infty$ for some $\mathrm{s} \in (0,\infty]$, and that the reference data are compactly supported, we prove that the guidance is estimable in squared $L^2$ at rate $\varepsilon_n^{\mathrm{s}/(\mathrm{s}+4)}$, where $\varepsilon_n = n^{-2(β-1)/(2(β-1)+d)}.$ 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.

发表机构

• HSE University（高等经济大学）

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