Local minima in training of neural networks

TL;DR

This paper investigates the existence of bad local minima in neural network training, showing that under certain data and initialization conditions, networks can indeed get trapped in suboptimal minima, challenging previous assumptions.

Contribution

It provides counter-examples demonstrating that local minima can affect neural network training under realistic conditions, contrary to common beliefs.

Findings

Counter-examples exist where networks encounter bad local minima.

Assumptions on data and initialization are crucial for theoretical guarantees.

Realistic scenarios can lead to susceptibility to local minima.

Abstract

There has been a lot of recent interest in trying to characterize the error surface of deep models. This stems from a long standing question. Given that deep networks are highly nonlinear systems optimized by local gradient methods, why do they not seem to be affected by bad local minima? It is widely believed that training of deep models using gradient methods works so well because the error surface either has no local minima, or if they exist they need to be close in value to the global minimum. It is known that such results hold under very strong assumptions which are not satisfied by real models. In this paper we present examples showing that for such theorem to be true additional assumptions on the data, initialization schemes and/or the model classes have to be made. We look at the particular case of finite size datasets. We demonstrate that in this scenario one can construct counter-examples (datasets or initialization schemes) when the network does become susceptible to bad local minima over the weight space.