Critical Points of Random Neural Networks

TL;DR

This paper analyzes the expected number of critical points in random neural networks as depth increases, revealing different growth regimes depending on activation function properties, with theoretical formulas supported by numerical experiments.

Contribution

It provides the first precise asymptotic formulas for critical points in deep random neural networks under various activation functions.

Findings

Expected critical point count can converge, grow polynomially, or grow exponentially with depth.

ReLU networks may have diverging critical points as resolution increases.

Theoretical predictions are validated by numerical experiments.

Abstract

This work investigates the expected number of critical points of random neural networks with different activation functions as the depth increases in the infinite-width limit. Under suitable regularity conditions, we derive precise asymptotic formulas for the expected number of critical points of fixed index and those exceeding a given threshold. Our analysis reveals three distinct regimes depending on the value of the first derivative of the covariance evaluated at 1: the expected number of critical points may converge, grow polynomially, or grow exponentially with depth. The theoretical predictions are supported by numerical experiments. Moreover, we provide numerical evidence suggesting that, when the regularity condition is not satisfied (e.g. for neural networks with ReLU as activation function), the number of critical points increases as the map resolution increases, indicating a potential divergence in the number of critical points.