Inexact subgradient methods for semialgebraic functions

Motivated by the extensive application of approximate gradients in machine learning and optimization, we investigate inexact subgradient methods subject to persistent additive errors. Within a nonconvex semialgebraic framework, assuming boundedness or coercivity, we establish that the method yields iterates that eventually fluctuate near the critical set at a proximity characterized by an $O(\epsilon^\rho)$ distance, where $\epsilon$ denotes the magnitude of subgradient evaluation errors, and $\rho$ encapsulates geometric characteristics of the underlying problem. Our analysis comprehensively addresses both vanishing and constant step-size regimes. Notably, the latter regime inherently enlarges the fluctuation region, yet this enlargement remains on the order of $\epsilon^\rho$. In the convex scenario, employing a universal error bound applicable to coercive semialgebraic functions, we derive novel complexity results concerning averaged iterates. Additionally, our study produces auxiliary results of independent interest, including descent-type lemmas for nonsmooth nonconvex functions and an invariance principle governing the behavior of algorithmic sequences under small-step limits.