Continuized Quasar-Convex Acceleration via an Unbiased Finite-Mixing Network Oracle

Abstract

We study decentralized minimization of a smooth network average that is globally quasar-convex with respect to a fixed minimizer; the node functions may be nonconvex and need not share a minimizer. A unit-rate Poisson clock drives an exactly simulable continuized process: between events the state follows a closed-form accelerated flow, and each event uses one oracle vector. One master theorem covers oracles with a predictable bias and a state-dependent variance and keeps the inverse-quadratic term $\mathcal{O}(LR_0^2/(γ^2T^2))$; one selected-row lemma supplies the oracle moments of three network implementations. CQA-Net keeps a common state, mixes local rows for finitely many rounds, and disseminates one uniformly selected row; its oracle is unbiased at every mixing depth, so incomplete averaging affects only the variance. A walking token removes dissemination, the shared clock, and the shared root, and an importance weight keeps it unbiased on fixed graphs. D-CQA keeps local states and mixes them; under bounded heterogeneity it needs $\widetilde{\mathcal{O}}(\sqrtχ\sqrt{L}R_0/(γθ\sqrt{\varepsilon}))$ communication rounds and no dissemination. Changing graphs enter through certified operator norms. Experiments on a certified nonconvex regression on static and changing graphs compare the three implementations.