An Analytical Formula of Population Gradient for two-layered ReLU network and its Applications in Convergence and Critical Point Analysis

TL;DR

This paper derives an analytical formula for the population gradient of a two-layer ReLU network, enabling theoretical analysis of critical points and convergence, with implications for training dynamics and initialization strategies.

Contribution

It provides the first analytical formula for the population gradient of two-layer ReLU networks and analyzes critical points, convergence, and symmetry-breaking phenomena.

Findings

Critical points outside the teacher hyperplane form manifolds.

Convergence to the teacher parameters is probable under specific initialization.

Spontaneous symmetry-breaking occurs with multiple ReLU nodes upon small perturbations.

Abstract

In this paper, we explore theoretical properties of training a two-layered ReLU network $g(\mathbf{x}; \mathbf{w}) = \sum_{j=1}^K \sigma(\mathbf{w}_j^T\mathbf{x})$ with centered $d$-dimensional spherical Gaussian input $\mathbf{x}$ ($\sigma$=ReLU). We train our network with gradient descent on $\mathbf{w}$ to mimic the output of a teacher network with the same architecture and fixed parameters $\mathbf{w}^*$. We show that its population gradient has an analytical formula, leading to interesting theoretical analysis of critical points and convergence behaviors. First, we prove that critical points outside the hyperplane spanned by the teacher parameters ("out-of-plane") are not isolated and form manifolds, and characterize in-plane critical-point-free regions for two ReLU case. On the other hand, convergence to $\mathbf{w}^*$ for one ReLU node is guaranteed with at least $(1-\epsilon)/2$ probability, if weights are initialized randomly with standard deviation upper-bounded by $O(\epsilon/\sqrt{d})$, consistent with empirical practice. For network with many ReLU nodes, we prove that an infinitesimal perturbation of weight initialization results in convergence towards $\mathbf{w}^*$ (or its permutation), a phenomenon known as spontaneous symmetric-breaking (SSB) in physics. We assume no independence of ReLU activations. Simulation verifies our findings.