Generating DDPM-based Samples from Tilted Distributions

Reward guidance methods sample an exponential tilt of a base density using a diffusion model trained with only samples from the base distribution. Existing methods use what we call external guidance: additive terms in the drift or multiplicative terms on the transition probabilities. We propose internal guidance which leaves the output of the base denoiser unmodified and instead changes the point at which it is evaluated. For discrete distributions embedded in the hypercube $\{-1,1\}^{d}$, the denoiser of any linearly tilted density is exactly the base denoiser evaluated at a shifted and rescaled input. In practice the base denoiser is a neural network trained on noised samples of the base distribution. The required evaluation points for sampling the tilted density would be out-of-distribution for basic noising of the base distribution. We modify the training algorithm with random diagonal rescalings of the data, so that the denoiser network learns both in- and out-of-distribution evaluations of the denoiser relative to the base distribution. A single denoiser network samples binary reward tilts, class labels, and infilling with no fine-tuning and no external guidance term.

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.

Diffusion models are typically sampled independently, even when the downstream objective is to obtain a diverse set of candidates. We introduce a variance-weighted batch distribution that favours collections of samples with large empirical spread after a prescribed linear feature map. The target is specified explicitly, and the sampler is derived as the corresponding Doob $h$-transform of independent diffusion dynamics. The resulting correction has a compact form: an interaction term that repels posterior denoised means, together with a curvature term that moves particles to the region of higher feature variance. This yields an interacting-particle sampler with a transparent probabilistic target rather than a heuristic repulsive drift.