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

arXiv: https://arxiv.org/abs/2609.05727 · HTML: https://mlsanity.com/paper/2609.05727/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 59 references that are in the map.
- First posted on arXiv 4 Sep 2026.

## Builds on (most cited first, 3 of 3)

- [DanceGRPO: Unleashing GRPO on Visual Generation](https://mlsanity.com/paper/2505.07818/) (2025)
- [Fine-Tuning of Continuous-Time Diffusion Models as Entropy-Regularized Control](https://mlsanity.com/paper/2402.15194/) (2024)
- [Reinforce Adjoint Matching: Scaling RL Post-Training of Diffusion and Flow-Matching Models](https://mlsanity.com/paper/2605.10759/) (2026)

## Abstract (arXiv)

We develop Newton Matching, a unified framework for fine-tuning and sampling in generative modeling. The target is πproptoμ eτ r, where r is the reward, τ>0 the inverse temperature, and μ 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 ρ, each stage takes a tangential step generated by the regularized reward r-frac1τlog(ρ/μ), 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 < η ≤ τ, global convergence under mild conditions, and local quadratic convergence for full steps (η=τ). 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 ρ=π. 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.