Who is Afraid of Big Bad Minima? Analysis of Gradient-Flow in a Spiked Matrix-Tensor Model

TL;DR

This paper provides a quantitative theory explaining why gradient-flow algorithms can find good minima in high-dimensional non-convex functions, specifically in a spiked matrix-tensor model, by analyzing saddle points and phase transitions.

Contribution

It introduces a framework based on Kac-Rice analysis and statistical physics to explain the success of gradient-flow in avoiding spurious minima in complex landscapes.

Findings

Gradient-flow finds global minima in certain parameter regimes.

Saddles with negative directions facilitate escape from spurious minima.

A BBP-type threshold in the Hessian determines landscape topology.

Abstract

Gradient-based algorithms are effective for many machine learning tasks, but despite ample recent effort and some progress, it often remains unclear why they work in practice in optimising high-dimensional non-convex functions and why they find good minima instead of being trapped in spurious ones. Here we present a quantitative theory explaining this behaviour in a spiked matrix-tensor model. Our framework is based on the Kac-Rice analysis of stationary points and a closed-form analysis of gradient-flow originating from statistical physics. We show that there is a well defined region of parameters where the gradient-flow algorithm finds a good global minimum despite the presence of exponentially many spurious local minima. We show that this is achieved by surfing on saddles that have strong negative direction towards the global minima, a phenomenon that is connected to a BBP-type threshold in the Hessian describing the critical points of the landscapes.