Convergence diagnostics for stochastic gradient descent with constant step size

TL;DR

This paper introduces a statistical diagnostic for detecting phase transitions in stochastic gradient descent with constant step size, enabling adaptive learning rate adjustments to improve convergence efficiency.

Contribution

It develops a diagnostic test for phase transition detection in SGD, derives a closed-form for the convergence region, and proposes an adaptive learning rate method based on stationarity detection.

Findings

Diagnostic accurately detects convergence region

Adaptive learning rate improves convergence speed

Method performs comparably to state-of-the-art algorithms

Abstract

Many iterative procedures in stochastic optimization exhibit a transient phase followed by a stationary phase. During the transient phase the procedure converges towards a region of interest, and during the stationary phase the procedure oscillates in that region, commonly around a single point. In this paper, we develop a statistical diagnostic test to detect such phase transition in the context of stochastic gradient descent with constant learning rate. We present theory and experiments suggesting that the region where the proposed diagnostic is activated coincides with the convergence region. For a class of loss functions, we derive a closed-form solution describing such region. Finally, we suggest an application to speed up convergence of stochastic gradient descent by halving the learning rate each time stationarity is detected. This leads to a new variant of stochastic gradient descent, which in many settings is comparable to state-of-art.