Analysis of biased stochastic gradient descent using sequential semidefinite programs

Springer Science and Business Media LLC - Tập 187 - Trang 383-408 - 2020
Bin Hu1, Peter Seiler2, Laurent Lessard3
1Department of Electrical and Computer Engineering, University of Illinois at Urbana–Champaign, Urbana, USA
2Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, USA
3Department of Electrical and Computer Engineering, Wisconsin Institute for Discovery, University of Wisconsin–Madison, Madison, USA

Tóm tắt

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.

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