Random weights of DNNs and emergence of fixed points

TL;DR

This paper investigates how the distribution of random weights in deep neural networks affects the number and stability of fixed points, revealing different behaviors for light and heavy-tail distributions and their dependence on network architecture.

Contribution

It provides a novel analysis of fixed points in randomly initialized DNNs, highlighting the impact of weight distribution tails and network depth on fixed point properties.

Findings

Light tails lead to a single stable fixed point across architectures.

Heavy tails result in multiple stable fixed points.

Number of fixed points varies non-monotonically with network depth.

Abstract

This paper is concerned with a special class of deep neural networks (DNNs) where the input and the output vectors have the same dimension. Such DNNs are widely used in applications, e.g., autoencoders. The training of such networks can be characterized by their fixed points (FPs). We are concerned with the dependence of the FPs number and their stability on the distribution of randomly initialized DNNs' weight matrices. Specifically, we consider the i.i.d. random weights with heavy and light-tail distributions. Our objectives are twofold. First, the dependence of FPs number and stability of FPs on the type of the distribution tail. Second, the dependence of the number of FPs on the DNNs' architecture. We perform extensive simulations and show that for light tails (e.g., Gaussian), which are typically used for initialization, a single stable FP exists for broad types of architectures. In contrast, for heavy tail distributions (e.g., Cauchy), which typically appear in trained DNNs, a number of FPs emerge. We further observe that these FPs are stable attractors and their basins of attraction partition the domain of input vectors. Finally, we observe an intriguing non-monotone dependence of the number of fixed points $Q(L)$ on the DNNs' depth $L$. The above results were first obtained for untrained DNNs with two types of distributions at initialization and then verified by considering DNNs in which the heavy tail distributions arise in training.