Convergence of difference inclusions via a diameter criterion

Abstract

We study discrete dynamics governed by a difference inclusion whose increment is the sum of a selection from a set-valued map and a noise term. For any bounded realization, convergence follows once the inter-iterate diameter is controlled by the variation of a continuous potential. The limit point is then critical for a scaled outer limit of the update map. To certify this diameter criterion, we develop a stratified descent framework: we project iterates onto a suitable stratification and track a potential that decreases up to a summable error. Combining the diameter criterion with a diameter estimate obtained from this framework yields convergence of common first-order optimization methods under step sizes of order $1/k$. The guarantees cover inexact and stochastic subgradient methods, as well as the momentum method, for locally Lipschitz objectives definable in polynomially bounded o-minimal structures. Our arguments are entirely discrete, with no appeal to continuous-time approximations.