# Global Convergence of SGD On Two Layer Neural Nets

arXiv: https://arxiv.org/abs/2210.11452 · HTML: https://mlsanity.com/paper/2210.11452/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 57 of its 71 references that are in the map.
- First posted on arXiv 20 Oct 2022.

## Builds on (most cited first, 30 of 57)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [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)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks](https://mlsanity.com/paper/1811.08888/) (2018)
- [Regularization Matters: Generalization and Optimization of Neural Nets v.s. their Induced Kernel](https://mlsanity.com/paper/1810.05369/) (2018)
- [On the linearity of large non-linear models: when and why the tangent kernel is constant](https://mlsanity.com/paper/2010.01092/) (2020)
- [An Improved Analysis of Training Over-parameterized Deep Neural Networks](https://mlsanity.com/paper/1906.04688/) (2019)
- [Polylogarithmic width suffices for gradient descent to achieve arbitrarily small test error with shallow ReLU networks](https://mlsanity.com/paper/1909.12292/) (2019)
- [Finite Depth and Width Corrections to the Neural Tangent Kernel](https://mlsanity.com/paper/1909.05989/) (2019)
- [Optimal Regularization Can Mitigate Double Descent](https://mlsanity.com/paper/2003.01897/) (2020)
- [Mean Field Analysis of Neural Networks: A Central Limit Theorem](https://mlsanity.com/paper/1808.09372/) (2018)
- [On the Power of Over-parametrization in Neural Networks with Quadratic Activation](https://mlsanity.com/paper/1803.01206/) (2018)
- [Dynamics of Deep Neural Networks and Neural Tangent Hierarchy](https://mlsanity.com/paper/1909.08156/) (2019)
- [Recovery Guarantees for One-hidden-layer Neural Networks](https://mlsanity.com/paper/1706.03175/) (2017)
- [What Can ResNet Learn Efficiently, Going Beyond Kernels?](https://mlsanity.com/paper/1905.10337/) (2019)
- [Harnessing the Power of Infinitely Wide Deep Nets on Small-data Tasks](https://mlsanity.com/paper/1910.01663/) (2019)
- [How Much Over-parameterization Is Sufficient to Learn Deep ReLU Networks?](https://mlsanity.com/paper/1911.12360/) (2019)
- [On the Convergence Rate of Training Recurrent Neural Networks](https://mlsanity.com/paper/1810.12065/) (2018)

## Built on by (most cited first, 5 of 5)

- [Training a Two Layer ReLU Network Analytically](https://mlsanity.com/paper/2304.02972/) (2023)
- [Langevin Monte-Carlo Provably Learns Depth Two Neural Nets at Any Size and Data](https://mlsanity.com/paper/2503.10428/) (2025)
- [Convergent Stochastic Training of Multi-Headed Attention and Understanding LoRA](https://mlsanity.com/paper/2605.07959/) (2026)
- [Gradient Flow Convergence Guarantee for General Neural Network Architectures](https://mlsanity.com/paper/2509.23887/) (2025)
- [Emergent Low-Rank Training Dynamics in MLPs with Smooth Activations](https://mlsanity.com/paper/2602.06208/) (2026)

## Abstract (arXiv)

In this note, we consider appropriately regularized ℓ2-empirical risk of depth 2 nets with any number of gates and show bounds on how the empirical loss evolves for SGD iterates on it -- for arbitrary data and if the activation is adequately smooth and bounded like sigmoid and tanh. This in turn leads to a proof of global convergence of SGD for a special class of initializations. We also prove an exponentially fast convergence rate for continuous time SGD that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show the existence of Frobenius norm regularized loss functions on constant-sized neural nets which are "Villani functions" and thus be able to build on recent progress with analyzing SGD on such objectives. Most critically the amount of regularization required for our analysis is independent of the size of the net.