Convergence, Sticking and Escape: Stochastic Dynamics Near Critical Points in SGD

TL;DR

This paper analyzes how stochastic gradient descent (SGD) behaves near critical points in one-dimensional landscapes, revealing how noise type and initial position influence convergence, sticking, and escape dynamics.

Contribution

It provides a detailed theoretical analysis of SGD's convergence and escape times near critical points, considering both finite- and infinite-variance noise scenarios.

Findings

SGD converges to local minima unless starting near a maximum.

Near sharp maxima, SGD is unlikely to stay stuck and can escape to neighboring minima.

Noise characteristics and landscape geometry significantly affect SGD's transition dynamics.

Abstract

We study the convergence properties and escape dynamics of Stochastic Gradient Descent (SGD) in one-dimensional landscapes, separately considering infinite- and finite-variance noise. Our main focus is to identify the time scales on which SGD reliably moves from an initial point to the local minimum in the same ''basin''. Under suitable conditions on the noise distribution, we prove that SGD converges to the basin's minimum unless the initial point lies too close to a local maximum. In that near-maximum scenario, we show that SGD can linger for a long time in its neighborhood. For initial points near a ''sharp'' maximum, we show that SGD does not remain stuck there, and we provide results to estimate the probability that it will reach each of the two neighboring minima. Overall, our findings present a nuanced view of SGD's transitions between local maxima and minima, influenced by both noise characteristics and the underlying function geometry.