Non-attracting Regions of Local Minima in Deep and Wide Neural Networks

TL;DR

This paper investigates the loss surface of deep neural networks, showing that suboptimal local minima exist but are connected and can be escaped, explaining why wide networks often perform well despite local minima.

Contribution

It constructs examples of suboptimal local minima in neural networks and analyzes their connectedness and conditions leading to saddle points, advancing understanding of loss landscapes.

Findings

Suboptimal local minima exist in neural networks with sigmoid activation.

These minima form connected sets that can be escaped via non-increasing paths.

Conditions are characterized under which minima become saddle points.

Abstract

Understanding the loss surface of neural networks is essential for the design of models with predictable performance and their success in applications. Experimental results suggest that sufficiently deep and wide neural networks are not negatively impacted by suboptimal local minima. Despite recent progress, the reason for this outcome is not fully understood. Could deep networks have very few, if at all, suboptimal local optima? or could all of them be equally good? We provide a construction to show that suboptimal local minima (i.e., non-global ones), even though degenerate, exist for fully connected neural networks with sigmoid activation functions. The local minima obtained by our construction belong to a connected set of local solutions that can be escaped from via a non-increasing path on the loss curve. For extremely wide neural networks of decreasing width after the wide layer, we prove that every suboptimal local minimum belongs to such a connected set. This provides a partial explanation for the successful application of deep neural networks. In addition, we also characterize under what conditions the same construction leads to saddle points instead of local minima for deep neural networks.