# Global Exponential Convergence of Two-Layer Linear Network Training

arXiv: https://arxiv.org/abs/2610.09356 · HTML: https://mlsanity.com/paper/2610.09356/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 34 references that are in the map.
- First posted on arXiv 7 Oct 2026.

## Builds on (most cited first, 19 of 19)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Deep Learning without Poor Local Minima](https://mlsanity.com/paper/1605.07110/) (2016)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced](https://mlsanity.com/paper/1806.00900/) (2018)
- [Deep linear neural networks with arbitrary loss: All local minima are global](https://mlsanity.com/paper/1712.01473/) (2017)
- [Global Convergence of Gradient Descent for Asymmetric Low-Rank Matrix Factorization](https://mlsanity.com/paper/2106.14289/) (2021)
- [Abide by the Law and Follow the Flow: Conservation Laws for Gradient Flows](https://mlsanity.com/paper/2307.00144/) (2023)
- [On the Global Convergence of Gradient Descent for multi-layer ResNets in the mean-field regime](https://mlsanity.com/paper/2110.02926/) (2021)
- [On the Convergence Rate of LoRA Gradient Descent](https://mlsanity.com/paper/2512.18248/) (2025)
- [Understanding the training of infinitely deep and wide ResNets with Conditional Optimal Transport](https://mlsanity.com/paper/2403.12887/) (2024)
- [A convergence result of a continuous model of deep learning via a Lojasiewicz--Simon inequality](https://mlsanity.com/paper/2311.15365/) (2023)
- [A Convergence Analysis of Nesterov's Accelerated Gradient Method in Training Deep Linear Neural Networks](https://mlsanity.com/paper/2204.08306/) (2022)
- [Intrinsic training dynamics of deep neural networks](https://mlsanity.com/paper/2508.07370/) (2025)
- [Positive semidefinite interval of matrix pencil and its applications for the generalized trust region subproblems](https://mlsanity.com/paper/2302.14352/) (2023)

## Abstract (arXiv)

We prove global exponential (linear) convergence with an explicit rate in the rich scaling for wide two-layer linear networks trained with smooth Polyak-Lojasiewicz predictor losses. Gradient flow in the factors closes exactly in terms of a finite-dimensional Bures flow of the neuron law covariance, in which the predictor dynamics are preconditioned by hidden covariance blocks. Mean-field conservation laws provide uniform spectral lower bounds on the hidden preconditioning blocks when the initial covariance satisfies a spectral support gap condition. This condition encompasses positive definiteness while still allowing for singular initializations. For an initial covariance Sigma0 = σ2 Id, the loss converges to the global minimum with linear rate at least 4σ2kappa, where kappa is the PL constant. We establish stability of this rate under finite-width sampling, as well as global convergence of factor gradient descent for an explicit stepsize interval depending on smoothness, the initial loss, and conserved spectral margins. Our argument extends layerwise to deep linear ResNets, subject to a residual-path bound. In the case of heavy-ball momentum, training dynamics close instead over positions and velocities in terms of a lifted phase covariance. Linear convergence holds under an explicit condition on the energy and damping, specifying a window of admissible dampings. For two-scale white initializations, this interval is nonempty for sufficiently large position scales, with a fixed initial loss gap and velocity covariance. Numerical experiments illustrate the covariance geometry and compare the predicted and observed rates.