From Information to Generative Exponent: Learning Rate Induces Phase Transitions in SGD

TL;DR

This paper explores how the learning rate in gradient-based algorithms influences phase transitions in sample complexity for neural network feature learning, revealing a critical relationship between learning rate and efficiency.

Contribution

It characterizes the impact of learning rate on phase transitions in sample complexity, unifies prior analyses, and introduces a new layer-wise training algorithm leveraging two-timescales.

Findings

Identifies phase transition from information to generative exponent regimes based on learning rate.

Demonstrates the importance of learning rate choice in statistical and computational efficiency.

Introduces a novel layer-wise training method with two different learning rates.

Abstract

To understand feature learning dynamics in neural networks, recent theoretical works have focused on gradient-based learning of Gaussian single-index models, where the label is a nonlinear function of a latent one-dimensional projection of the input. While the sample complexity of online SGD is determined by the information exponent of the link function, recent works improved this by performing multiple gradient steps on the same sample with different learning rates -- yielding a non-correlational update rule -- and instead are limited by the (potentially much smaller) generative exponent. However, this picture is only valid when these learning rates are sufficiently large. In this paper, we characterize the relationship between learning rate(s) and sample complexity for a broad class of gradient-based algorithms that encapsulates both correlational and non-correlational updates. We demonstrate that, in certain cases, there is a phase transition from an "information exponent regime" with small learning rate to a "generative exponent regime" with large learning rate. Our framework covers prior analyses of one-pass SGD and SGD with batch reuse, while also introducing a new layer-wise training algorithm that leverages a two-timescales approach (via different learning rates for each layer) to go beyond correlational queries without reusing samples or modifying the loss from squared error. Our theoretical study demonstrates that the choice of learning rate is as important as the design of the algorithm in achieving statistical and computational efficiency.