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In this paper, we develop a unified framework for analyzing the trackingerror and dynamic regret of inexact online optimization methods under a varietyof settings. Specifically, we leverage the quadratic constraint approach fromcontrol theory to formulate sequential semidefinite programs (SDPs) whosefeasible points naturally correspond to tracking error bounds of variousinexact online optimization methods including the inexact online gradientdescent (OGD) method, the online gradient descent-ascent method, the onlinestochastic gradient method, and the inexact proximal online gradient method. Weprovide exact analytical solutions for our proposed sequential SDPs, and obtainfine-grained tracking error bounds for the online algorithms studied in thispaper. We also provide a simple routine to convert the obtained tracking errorbounds into dynamic regret bounds. The main novelty of our analysis is that wederive exact analytical solutions for our proposed sequential SDPs undervarious inexact oracle assumptions in a unified manner.

Bounds for the Tracking Error and Dynamic Regret of Inexact Online Optimization Methods: A General Analysis via Sequential Semidefinite Programs