Analysis of biased stochastic gradient descent using sequential semidefinite programs

We present a convergence rate analysis for biased stochastic gradient descent (SGD), where individual gradient updates are corrupted by computation errors. We develop stochastic quadratic constraints to formulate a small linear matrix inequality (LMI) whose feasible points lead to convergence bounds of biased SGD. Based on this LMI condition, we develop a sequential minimization approach to analyze the intricate trade-offs that couple stepsize selection, convergence rate, optimization accuracy, and robustness to gradient inaccuracy. We also provide feasible points for this LMI and obtain theoretical formulas that quantify the convergence properties of biased SGD under various assumptions on the loss functions.

Automated summary

A convergence rate analysis for biased stochastic gradient descent is presented, with stochastic quadratic constraints and a small linear matrix inequality formulated to provide convergence bounds. A sequential minimization approach is developed to analyze trade-offs involving stepsize selection, convergence rate, optimization accuracy, and robustness to gradient inaccuracy. Theoretical formulas are obtained to quantify the convergence properties of biased SGD under various assumptions on the loss functions.