To this end, high-probability guarantees have been considered in the literature [35, 64, 20, 32, 22]. These results allow to control the risk associated with the worst-case tail events as theyspecify howmanyiterations would be sufficient toensureG(xk,yk) issufficiently small foranygivenfailure probability q (0,1).

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The problem(1) with µy > 0 is called a weakly convex-strongly concave(WCSC) saddle-point problem, whereas forµy =0,itiscalledaweakly convex-merely concave(WCMC) saddle-point problem.

Inference-time alignment for diffusion models aims to adapt a pre-trained diffusion model toward a target distribution without retraining the base score network, thereby preserving the generative capacity of the base model while enforcing desired properties at the inference time. A central mechanism for achieving such alignment is guidance, which modifies the sampling dynamics through an additional drift term. In this work, we introduce Doob's matching, a novel framework for guidance estimation grounded in Doob's $h$-transform. Our approach formulates guidance as the gradient of logarithm of an underlying Doob's $h$-function and employs gradient-penalized regression to simultaneously estimate both the $h$-function and its gradient, resulting in a consistent estimator of the guidance. Theoretically, we establish non-asymptotic convergence rates for the estimated guidance. Moreover, we analyze the resulting controllable diffusion processes and prove non-asymptotic convergence guarantees for the generated distributions in the 2-Wasserstein distance.