# Gradient descent provably escapes saddle points in the training of shallow ReLU networks

arXiv: https://arxiv.org/abs/2208.02083 · HTML: https://mlsanity.com/paper/2208.02083/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 54 references that are in the map.
- First posted on arXiv 3 Aug 2022.

## Builds on (most cited first, 30 of 36)

- [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)
- [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)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data](https://mlsanity.com/paper/1808.01204/) (2018)
- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [Deep Learning without Poor Local Minima](https://mlsanity.com/paper/1605.07110/) (2016)
- [No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis](https://mlsanity.com/paper/1704.00708/) (2017)
- [Theoretical insights into the optimization landscape of over-parameterized shallow neural networks](https://mlsanity.com/paper/1707.04926/) (2017)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Spurious Local Minima are Common in Two-Layer ReLU Neural Networks](https://mlsanity.com/paper/1712.08968/) (2017)
- [On the Power of Over-parametrization in Neural Networks with Quadratic Activation](https://mlsanity.com/paper/1803.01206/) (2018)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [Global Optimality of Local Search for Low Rank Matrix Recovery](https://mlsanity.com/paper/1605.07221/) (2016)
- [Gradient Descent Can Take Exponential Time to Escape Saddle Points](https://mlsanity.com/paper/1705.10412/) (2017)
- [Learning deep linear neural networks: Riemannian gradient flows and convergence to global minimizers](https://mlsanity.com/paper/1910.05505/) (2019)
- [Complete Dictionary Recovery over the Sphere I: Overview and the Geometric Picture](https://mlsanity.com/paper/1511.03607/) (2015)
- [A Comparative Analysis of the Optimization and Generalization Property of Two-layer Neural Network and Random Feature Models Under Gradient Descent Dynamics](https://mlsanity.com/paper/1904.04326/) (2019)
- [Stochastic Gradient Descent for Nonconvex Learning without Bounded Gradient Assumptions](https://mlsanity.com/paper/1902.00908/) (2019)
- [On the Convergence of Gradient Descent Training for Two-layer ReLU-networks in the Mean Field Regime](https://mlsanity.com/paper/2005.13530/) (2020)
- [On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics](https://mlsanity.com/paper/2007.15623/) (2020)
- [First-order methods almost always avoid saddle points: the case of vanishing step-sizes](https://mlsanity.com/paper/1906.07772/) (2019)
- [Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions](https://mlsanity.com/paper/1605.00405/) (2016)
- [Splitting methods with variable metric for KL functions](https://mlsanity.com/paper/1405.1357/) (2014)

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

- [Global Convergence of SGD On Two Layer Neural Nets](https://mlsanity.com/paper/2210.11452/) (2022)
- [Global Convergence of SGD For Logistic Loss on Two Layer Neural Nets](https://mlsanity.com/paper/2309.09258/) (2023)
- [Langevin Monte-Carlo Provably Learns Depth Two Neural Nets at Any Size and Data](https://mlsanity.com/paper/2503.10428/) (2025)
- [Non-Singularity of the Gradient Descent map for Neural Networks with Piecewise Analytic Activations](https://mlsanity.com/paper/2510.24466/) (2025)
- [Endogenous Regime Switching Driven by Scalar-Irreducible Learning Dynamics](https://mlsanity.com/paper/2605.04054/) (2026)

## Abstract (arXiv)

Dynamical systems theory has recently been applied in optimization to prove that gradient descent algorithms bypass so-called strict saddle points of the loss function. However, in many modern machine learning applications, the required regularity conditions are not satisfied. In this paper, we prove a variant of the relevant dynamical systems result, a center-stable manifold theorem, in which we relax some of the regularity requirements. We explore its relevance for various machine learning tasks, with a particular focus on shallow rectified linear unit (ReLU) and leaky ReLU networks with scalar input. Building on a detailed examination of critical points of the square integral loss function for shallow ReLU and leaky ReLU networks relative to an affine target function, we show that gradient descent circumvents most saddle points. Furthermore, we prove convergence to global minima under favourable initialization conditions, quantified by an explicit threshold on the limiting loss.