# Landscape Connectivity and Dropout Stability of SGD Solutions for Over-parameterized Neural Networks

arXiv: https://arxiv.org/abs/1912.10095 · HTML: https://mlsanity.com/paper/1912.10095/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Dec 2019.

## Built on by (most cited first, 30 of 35)

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- [On Quantum Speedups for Nonconvex Optimization via Quantum Tunneling Walks](https://mlsanity.com/paper/2209.14501/) (2022)
- [Exploring Neural Network Landscapes: Star-Shaped and Geodesic Connectivity](https://mlsanity.com/paper/2404.06391/) (2024)
- [Stable Recovery of Entangled Weights: Towards Robust Identification of Deep Neural Networks from Minimal Samples](https://mlsanity.com/paper/2101.07150/) (2021)
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- [Is Grokking a Computational Glass Relaxation?](https://mlsanity.com/paper/2505.11411/) (2025)
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- [Limiting fluctuation and trajectorial stability of multilayer neural networks with mean field training](https://mlsanity.com/paper/2110.15954/) (2021)
- [Deep Architecture Connectivity Matters for Its Convergence: A Fine-Grained Analysis](https://mlsanity.com/paper/2205.05662/) (2022)
- [Mean-field analysis for heavy ball methods: Dropout-stability, connectivity, and global convergence](https://mlsanity.com/paper/2210.06819/) (2022)
- [Mode Connectivity in Auction Design](https://mlsanity.com/paper/2305.11005/) (2023)
- [Mode Connectivity and Data Heterogeneity of Federated Learning](https://mlsanity.com/paper/2309.16923/) (2023)
- [Do We Really Need Permutations? Impact of Model Width on Linear Mode Connectivity](https://mlsanity.com/paper/2510.08023/) (2025)
- [Overparametrization bends the landscape: BBP transitions at initialization in simple Neural Networks](https://mlsanity.com/paper/2510.18435/) (2025)
- [WGAN with an Infinitely Wide Generator Has No Spurious Stationary Points](https://mlsanity.com/paper/2102.07541/) (2021)

## Abstract (arXiv)

The optimization of multilayer neural networks typically leads to a solution with zero training error, yet the landscape can exhibit spurious local minima and the minima can be disconnected. In this paper, we shed light on this phenomenon: we show that the combination of stochastic gradient descent (SGD) and over-parameterization makes the landscape of multilayer neural networks approximately connected and thus more favorable to optimization. More specifically, we prove that SGD solutions are connected via a piecewise linear path, and the increase in loss along this path vanishes as the number of neurons grows large. This result is a consequence of the fact that the parameters found by SGD are increasingly dropout stable as the network becomes wider. We show that, if we remove part of the neurons (and suitably rescale the remaining ones), the change in loss is independent of the total number of neurons, and it depends only on how many neurons are left. Our results exhibit a mild dependence on the input dimension: they are dimension-free for two-layer networks and depend linearly on the dimension for multilayer networks. We validate our theoretical findings with numerical experiments for different architectures and classification tasks.