Asymptotic Convergence Analysis of High-Order Proximal-Point Methods Beyond Sublinear Rates

This paper investigates the asymptotic convergence behavior of the high-order proximal-point algorithm (HiPPA) to global minimizers, extending existing analyses beyond sublinear convergence rates and complexity analysis. Specifically, we study the proximal operator of a proper 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 functions in the class of uniformly quasiconvex functions, which includes 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 $\phi$ admits a power function of degree $q$ as a lower bound, i.e., $\phi(t) \geq c t^q$ for some $c>0$, on an interval $\mathcal{I}$: (i) for $q\in (1,2)$ and $\mathcal{I}=[0,1)$, HiPPA exhibits a local linear rate for $p\in [q,2)$; (ii) HiPPA converges linearly when $p=2$, $q=2$, and also when $p=q>2$, provided that $\mathcal{I}=[0,\infty)$; (iii) for $q\geq 2$ and $\mathcal{I}=[0,\infty)$, HiPPA achieves a 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.