Title:Using Stochastic Gradient Descent to Smooth Nonconvex Functions: Analysis of Implicit Graduated Optimization

Abstract:The graduated optimization approach is a heuristic method for finding global optimal solutions for nonconvex functions by using a function smoothing operation with stochastic noise. We show that stochastic noise in stochastic gradient descent (SGD) has the effect of smoothing the objective function, the degree of which is determined by the learning rate, batch size, and variance of the stochastic gradient. Using this finding, we propose and analyze a new graduated optimization algorithm that varies the degree of smoothing by varying the learning rate and batch size, and provide experimental results on image classification tasks with ResNets that support our theoretical findings. We further show that there is an interesting correlation between the degree of smoothing by SGD's stochastic noise, the well-studied ``sharpness'' indicator, and the generalization performance of the model.