# Adjoint Matching through the Lens of the Stochastic Maximum Principle in Optimal Control

arXiv: https://arxiv.org/abs/2604.08580 · HTML: https://mlsanity.com/paper/2604.08580/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 63 references that are in the map.
- First posted on arXiv 28 Mar 2026.

## Builds on (most cited first, 30 of 49)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Make-A-Video: Text-to-Video Generation without Text-Video Data](https://mlsanity.com/paper/2209.14792/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Training Diffusion Models with Reinforcement Learning](https://mlsanity.com/paper/2305.13301/) (2023)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [DPOK: Reinforcement Learning for Fine-tuning Text-to-Image Diffusion Models](https://mlsanity.com/paper/2305.16381/) (2023)
- [Directly Fine-Tuning Diffusion Models on Differentiable Rewards](https://mlsanity.com/paper/2309.17400/) (2023)
- [Voicebox: Text-Guided Multilingual Universal Speech Generation at Scale](https://mlsanity.com/paper/2306.15687/) (2023)
- [Aligning Text-to-Image Diffusion Models with Reward Backpropagation](https://mlsanity.com/paper/2310.03739/) (2023)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Fine-Tuning of Continuous-Time Diffusion Models as Entropy-Regularized Control](https://mlsanity.com/paper/2402.15194/) (2024)
- [Audiobox: Unified Audio Generation with Natural Language Prompts](https://mlsanity.com/paper/2312.15821/) (2023)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
- [Theoretical guarantees for sampling and inference in generative models with latent diffusions](https://mlsanity.com/paper/1903.01608/) (2019)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)

## Built on by (most cited first, 6 of 6)

- [Non-Asymptotic Error Bounds for SMC with Biased Proposals: Application to Conditional Diffusion Sampling](https://mlsanity.com/paper/2607.04780/) (2026)
- [Lifted Schr"odinger Bridges for Gaussian Mixture Endpoints: Projection Gaps and Path-Space Obstructions](https://mlsanity.com/paper/2605.24795/) (2026)
- [Ensemble Controlled-Flow Filtering for Implicit Data Assimilation](https://mlsanity.com/paper/2607.12975/) (2026)
- [Unified Optimality Conditions for Stochastic Optimal Control in the Rough Path and Ito Frameworks](https://mlsanity.com/paper/2609.38395/) (2026)
- [Reinforcement Learning via Fixed-point Flow Matching](https://mlsanity.com/paper/2610.12848/) (2026)
- [Weighted Adjoint Matching: Finetuning Flow Matching Models with Time-Dependent Control Cost](https://mlsanity.com/paper/2610.20035/) (2026)

## Abstract (arXiv)

Reward fine-tuning of diffusion and flow models and sampling from tilted or Boltzmann distributions can both be formulated as stochastic optimal control (SOC) problems, where learning an optimal generative dynamics corresponds to optimizing a control under SDE constraints. In this work, we revisit and generalize Adjoint Matching, a recently proposed SOC-based method for learning optimal controls, and place it on a rigorous footing by deriving it from the Stochastic Maximum Principle (SMP). We formulate a general Hamiltonian adjoint matching objective for SOC problems with control-dependent drift and diffusion and convex running costs, and show that its expected value has the same first variation as the original SOC objective. As a consequence, critical points satisfy the Hamilton--Jacobi--Bellman (HJB) stationarity conditions. In the important practical case of state- and control-independent diffusion, we recover the lean adjoint matching loss previously introduced, which avoids second-order terms and whose critical points coincide with the optimal control under mild uniqueness assumptions. Numerical experiments confirm that the extra terms it discards become necessary once the diffusion is state-dependent. Finally, we show that adjoint matching can be precisely interpreted as a continuous-time method of successive approximations induced by the SMP, yielding a practical and implementable alternative to classical SMP-based algorithms, which are obstructed by intractable martingale terms in the stochastic setting. These results are also of independent interest to the stochastic control community, providing new implementable objectives and a viable pathway for SMP-based iterations in stochastic problems.