# Revisiting Landscape Analysis in Deep Neural Networks: Eliminating Decreasing Paths to Infinity

arXiv: https://arxiv.org/abs/1912.13472 · HTML: https://mlsanity.com/paper/1912.13472/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Dec 2019.

## Built on by (most cited first, 6 of 6)

- [NTK-SAP: Improving neural network pruning by aligning training dynamics](https://mlsanity.com/paper/2304.02840/) (2023)
- [When Expressivity Meets Trainability: Fewer than n Neurons Can Work](https://mlsanity.com/paper/2210.12001/) (2022)
- [Achieving Small Test Error in Mildly Overparameterized Neural Networks](https://mlsanity.com/paper/2104.11895/) (2021)
- [Flat Channels to Infinity in Neural Loss Landscapes](https://mlsanity.com/paper/2506.14951/) (2025)
- [Global Convergence Analysis of Deep Linear Networks with A One-neuron Layer](https://mlsanity.com/paper/2201.02761/) (2022)
- [Noise Regularizes Over-parameterized Rank One Matrix Recovery, Provably](https://mlsanity.com/paper/2202.03535/) (2022)

## Abstract (arXiv)

Traditional landscape analysis of deep neural networks aims to show that no sub-optimal local minima exist in some appropriate sense. From this, one may be tempted to conclude that descent algorithms which escape saddle points will reach a good local minimum. However, basic optimization theory tell us that it is also possible for a descent algorithm to diverge to infinity if there are paths leading to infinity, along which the loss function decreases. It is not clear whether for non-linear neural networks there exists one setting that no bad local-min and no decreasing paths to infinity can be simultaneously achieved. In this paper, we give the first positive answer to this question. More specifically, for a large class of over-parameterized deep neural networks with appropriate regularizers, the loss function has no bad local minima and no decreasing paths to infinity. The key mathematical trick is to show that the set of regularizers which may be undesirable can be viewed as the image of a Lipschitz continuous mapping from a lower-dimensional Euclidean space to a higher-dimensional Euclidean space, and thus has zero measure.