Asymptotic Convergence Analysis of High-Order Proximal-Point Methods Beyond Sublinear Rates
Masoud Ahookhosh, Alfredo Iusem, Alireza Kabgani, Felipe Lara
TL;DR
The paper develops a rigorous framework for high-order proximal-point methods (HiPPA) applied to uniformly quasiconvex optimization, proving convergence to a unique global minimizer and deriving precise linear and superlinear convergence rates under a power-law modulus bound $\phi(t)\ge \rho_q t^q$. By establishing structural properties of uniformly quasiconvex functions—such as the absence of saddle points and coercivity—the authors justify the use of high-order proximal subproblems and Moreau envelopes in nonconvex settings. The results unify and extend classical linear convergence theory from strongly convex to a broader class of generalized convex problems, including uniformly convex/quasiconvex cases, with several novel rate regimes. The findings have potential impact on large-scale nonsmooth optimization where saddle points hinder convergence, offering a principled path to faster, globally convergent algorithms. The work also outlines directions for future work, including inexact proximal computations and numerical validation of HiPPA on generalized convex problems.
Abstract
This paper investigates the asymptotic convergence behavior of high-order proximal-point algorithms (HiPPA) toward global minimizers, extending the analysis beyond sublinear convergence rate results. Specifically, we consider the proximal operator of a lower semicontinuous function augmented with a $p$th-order regularization for $p>1$, and establish the convergence of HiPPA to a global minimizer with a particular focus on its convergence rate. To this end, we focus on minimizing the class of uniformly quasiconvex functions, including strongly convex, uniformly convex, and strongly quasiconvex functions as special cases. Our analysis reveals the following convergence behaviors of HiPPA when the uniform quasiconvexity modulus admits a power function of degree $q$ as a lower bound on an interval $\mathcal{I}$: (i) for $q\in (1,2]$ and $\mathcal{I}=[0,1)$, HiPPA exhibits local linear rate for $p\in (1,2)$; (ii) for $q=2$ and $\mathcal{I}=[0,\infty)$, HiPPA converges linearly for $p=2$; (iii) for $p=q>2$ and $\mathcal{I}=[0,\infty)$, HiPPA converges linearly; (iv) for $q\geq 2$ and $\mathcal{I}=[0,\infty)$, HiPPA achieves superlinear rate for $p>q$. Notably, to our knowledge, some of these results are novel, even in the context of strongly or uniformly convex functions, offering new insights into optimizing generalized convex problems.