# How I learned to stop worrying and love StopGrads: Stationarity, Convergence, and a case study on Flow Map Learning

arXiv: https://arxiv.org/abs/2609.16222 · HTML: https://mlsanity.com/paper/2609.16222/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 48 references that are in the map.
- First posted on arXiv 14 Sep 2026.

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

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion](https://mlsanity.com/paper/2310.02279/) (2023)
- [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Simplifying, Stabilizing and Scaling Continuous-Time Consistency Models](https://mlsanity.com/paper/2410.11081/) (2024)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation](https://mlsanity.com/paper/1806.02450/) (2018)
- [The Mechanics of n-Player Differentiable Games](https://mlsanity.com/paper/1802.05642/) (2018)
- [Improved Mean Flows: On the Challenges of Fastforward Generative Models](https://mlsanity.com/paper/2512.02012/) (2025)
- [How to build a consistency model: Learning flow maps via self-distillation](https://mlsanity.com/paper/2505.18825/) (2025)
- [Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference](https://mlsanity.com/paper/1703.09194/) (2017)
- [PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2306.08827/) (2023)
- [Align Your Flow: Scaling Continuous-Time Flow Map Distillation](https://mlsanity.com/paper/2506.14603/) (2025)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)
- [Differentiable Game Mechanics](https://mlsanity.com/paper/1905.04926/) (2019)
- [Discrete Flow Maps](https://mlsanity.com/paper/2604.09784/) (2026)
- [Terminal Velocity Matching](https://mlsanity.com/paper/2511.19797/) (2025)
- [Geometric Insights into the Convergence of Nonlinear TD Learning](https://mlsanity.com/paper/1905.12185/) (2019)
- [Why Target Networks Stabilise Temporal Difference Methods](https://mlsanity.com/paper/2302.12537/) (2023)
- [Approximate Temporal Difference Learning is a Gradient Descent for Reversible Policies](https://mlsanity.com/paper/1805.00869/) (2018)
- [Bridging the Gap Between Target Networks and Functional Regularization](https://mlsanity.com/paper/2106.02613/) (2021)
- [Fast and Unified Path Gradient Estimators for Normalizing Flows](https://mlsanity.com/paper/2403.15881/) (2024)
- [Dual Perspectives on Non-Contrastive Self-Supervised Learning](https://mlsanity.com/paper/2507.01028/) (2025)
- [Stabilizing Consistency Training: A Flow Map Analysis and Self-Distillation](https://mlsanity.com/paper/2601.22679/) (2026)
- [Flow Map Learning via Nongradient Vector Flow](https://mlsanity.com/paper/2607.26398/) (2026)

## Abstract (arXiv)

Stopgrads are widely used in training machine learning models, but stopgrads can alter the gradient, stationary points and convergence guarantees of the original objective, which can make stopgrad training theoretically ungrounded. We introduce a stopgrad regression principle, which identifies a general template for stopgrad objectives with a closed-form characterization of stationary points and their uniqueness, unifying stopgrad objectives for flow maps, reinforcement learning, and diffusion samplers. We provide theoretical grounding for optimizing stopgrad flow map objectives by showing their unique stationary point is the true flow map, and showing positive convergence results for Eulerian and Lagrangian objectives, including MeanFlow and improved MeanFlow. Remarkably, we show that under functional semi-gradient flow, the learned flow map has a closed-form expression composing the initial flow map and the true flow map. We additionally use our stopgrad regression principle to propose modified stopgrad placements for flow map objectives which reduce training memory by 2x.