Noise and Fluctuation of Finite Learning Rate Stochastic Gradient Descent
This work provides a foundational theoretical understanding of SGD dynamics for researchers and practitioners using non-vanishing learning rates, particularly highlighting the underestimation of fluctuations by continuous-time models.
This paper investigates the properties of Stochastic Gradient Descent (SGD) and its variants when using a non-vanishing learning rate. It derives exact stationary distributions for discrete-time SGD on a quadratic loss, revealing that fluctuations are significantly larger and distorted compared to continuous-time predictions.
In the vanishing learning rate regime, stochastic gradient descent (SGD) is now relatively well understood. In this work, we propose to study the basic properties of SGD and its variants in the non-vanishing learning rate regime. The focus is on deriving exactly solvable results and discussing their implications. The main contributions of this work are to derive the stationary distribution for discrete-time SGD in a quadratic loss function with and without momentum; in particular, one implication of our result is that the fluctuation caused by discrete-time dynamics takes a distorted shape and is dramatically larger than a continuous-time theory could predict. Examples of applications of the proposed theory considered in this work include the approximation error of variants of SGD, the effect of minibatch noise, the optimal Bayesian inference, the escape rate from a sharp minimum, and the stationary covariance of a few second-order methods including damped Newton's method, natural gradient descent, and Adam.