We study the problem of estimating the guidance that steers the distribution learned by a diffusion generative model toward a tilted target
q0∝wp0 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
Ep0[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
εns/(s+4), where
εn=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.