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In this work and the supporting Part II, we examine the performance ofstochastic sub-gradient learning strategies under weaker conditions thanusually considered in the literature. The new conditions are shown to beautomatically satisfied by several important cases of interest including SVM,LASSO, and Total-Variation denoising formulations. In comparison, theseproblems do not satisfy the traditional assumptions used in prior analyses and,therefore, conclusions derived from these earlier treatments are not directlyapplicable to these problems. The results in this article establish thatstochastic sub-gradient strategies can attain linear convergence rates, asopposed to sub-linear rates, to the steady-state regime. A realizableexponential-weighting procedure is employed to smooth the intermediate iteratesand guarantee useful performance bounds in terms of convergence rate andexcessive risk performance. Part I of this work focuses on single-agentscenarios, which are common in stand-alone learning applications, while Part IIextends the analysis to networked learners. The theoretical conclusions areillustrated by several examples and simulations, including comparisons with theFISTA procedure.

Performance Limits of Stochastic Sub-Gradient Learning, Part I: Single Agent Case