# Polyak Steps for Adaptive Fast Gradient Method

## TL;DR

This paper introduces a new adaptive method for accelerated gradient algorithms that estimates the strong convexity parameter online, eliminating the need for restart schemes and maintaining optimal convergence rates.

## Contribution

It proposes a novel approach to adaptively estimate the strong convexity parameter during optimization, removing the necessity for restart strategies.

## Key findings

- Achieves optimal linear convergence without restarts.
- Demonstrates robustness of the method with estimated bounds on .
- Provides empirical evidence of effectiveness.

## Abstract

Accelerated algorithms for minimizing smooth strongly convex functions usually require knowledge of the strong convexity parameter $\mu$. In the case of an unknown $\mu$, current adaptive techniques are based on restart schemes. When the optimal value $f^*$ is known, these strategies recover the accelerated linear convergence bound without additional grid search. In this paper we propose a new approach that has the same bound without any restart, using an online estimation of strong convexity parameter. We show the robustness of the Fast Gradient Method when using a sequence of upper bounds on $\mu$. We also present a good candidate for this estimate sequence and detail consistent empirical results.