Title:Newton Matching for Generative Modeling: A Unified Framework for Fine-Tuning and Sampling

Abstract:We develop Newton Matching, a unified framework for fine-tuning and sampling in generative modeling. The target is $\pi\propto\mu e^{\tau r}$, where $r$ is the reward, $\tau>0$ the inverse temperature, and $\mu$ denotes the pretrained model's terminal density for fine-tuning or the constant $1$ for sampling. We shift the paradigm from isolated losses to iterative optimization over canonical models: population minimizers of standard conditional matching for terminal densities. Under compatible smooth-realization assumptions, canonical velocities form a manifold diffeomorphic to the density manifold. Transporting the Fisher-Rao metric and mixture connection to this manifold, we show that the reverse-KL Hessian equals the metric, so the Newton direction coincides with the negative Fisher-Rao gradient. At terminal density $\rho$, each stage takes a tangential step generated by the regularized reward $r-\frac1\tau\log(\rho/\mu)$, followed by terminal-density-preserving canonicalization. This canonical retraction yields an exact finite-stepsize density characterization. For the ideal iteration, we prove strict reverse-KL descent away from the target for $0 < \eta \le \tau$, global convergence under mild conditions, and local quadratic convergence for full steps ($\eta=\tau$). Covariance and gradient forms, each with forward or reverse regression-pair constructions, yield sample-wise tangential-update losses with the same population minimizer, without importance sampling or full-trajectory backpropagation. We develop approximate updates and define critical-point consistency as vanishing tangential displacement if and only if $\rho=\pi$. We recover representative methods as exact realizations, critical-point-consistent approximations, or objective-altering variants, enabling modular algorithm design. Our work advances the theory and algorithms of reinforcement learning for generative models.

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