General Holder Smooth Convergence Rates Follow From Specialized Rates Assuming Growth Bounds

TL;DR

This paper introduces meta-theorems that derive general convergence rates for first-order optimization methods from specialized growth-bound cases, unifying and extending existing results.

Contribution

It provides a unified framework to obtain general convergence rates from growth-bound assumptions, simplifying proofs and deriving new results for various optimization algorithms.

Findings

Recovered known convergence rates for multiple methods from specialized cases

Derived new convergence results for bundle methods, dual averaging, and Frank-Wolfe

Provided simple proofs of optimal lower bounds under H"older growth

Abstract

Often in the analysis of first-order methods for both smooth and nonsmooth optimization, assuming the existence of a growth/error bound or KL condition facilitates much stronger convergence analysis. Hence separate analysis is typically needed for the general case and for the growth bounded cases. We give meta-theorems for deriving general convergence rates from those assuming a growth lower bound. Applying this simple but conceptually powerful tool to the proximal point, subgradient, bundle, dual averaging, gradient descent, Frank-Wolfe, and universal accelerated methods immediately recovers their known convergence rates for general convex optimization problems from their specialized rates. New convergence results follow for bundle methods, dual averaging, and Frank-Wolfe. Our results can lift any rate based on H\"older continuous gradients and H\"older growth bounds. Moreover, our theory provides simple proofs of optimal convergence lower bounds under H\"older growth from textbook examples without growth bounds.