# Faster Subgradient Methods for Functions with H\"olderian Growth

## TL;DR

This paper introduces new convergence results for subgradient methods applied to nonsmooth convex functions with H"olderian growth, including faster rates and adaptive stepsize strategies for improved efficiency.

## Contribution

It presents novel convergence analyses for subgradient methods with fixed, decaying, and adaptive stepsizes, achieving faster rates and linear convergence under certain conditions.

## Key findings

- Linear convergence with small constant stepsize
- Faster convergence with decaying stepsize when parameters are known
- Adaptive stepsize method matches faster convergence without parameter knowledge

## Abstract

The purpose of this manuscript is to derive new convergence results for several subgradient methods applied to minimizing nonsmooth convex functions with H\"olderian growth. The growth condition is satisfied in many applications and includes functions with quadratic growth and weakly sharp minima as special cases. To this end there are three main contributions. First, for a constant and sufficiently small stepsize, we show that the subgradient method achieves linear convergence up to a certain region including the optimal set, with error of the order of the stepsize. Second, if appropriate problem parameters are known, we derive a decaying stepsize which obtains a much faster convergence rate than is suggested by the classical $O(1/\sqrt{k})$ result for the subgradient method. Thirdly we develop a novel "descending stairs" stepsize which obtains this faster convergence rate and also obtains linear convergence for the special case of weakly sharp functions. We also develop an adaptive variant of the "descending stairs" stepsize which achieves the same convergence rate without requiring an error bound constant which is difficult to estimate in practice.