# Sharpness, Restart and Acceleration

## TL;DR

This paper explores how sharpness bounds, derived from the { extbackslash}Lojasiewicz inequality, influence the performance of restart schemes in convex optimization, showing that optimal restart strategies are robust and can be efficiently identified.

## Contribution

It demonstrates that optimal restart strategies are robust and can be found via a simple log-scale grid search, leading to generic acceleration of first-order methods.

## Key findings

- Sharpness bounds hold generically for convex problems.
- Optimal restart schemes are robust and easily identifiable.
- Restart schemes accelerate first-order methods effectively.

## Abstract

The {\L}ojasiewicz inequality shows that sharpness bounds on the minimum of convex optimization problems hold almost generically. Sharpness directly controls the performance of restart schemes, as observed by Nemirovsky and Nesterov (1985). The constants quantifying these sharpness bounds are of course unobservable, but we show that optimal restart strategies are robust, in the sense that, in some important cases, finding the best restart scheme only requires a log scale grid search. Overall then, restart schemes generically accelerate accelerated first-order methods.