Algorithmic Instabilities of Accelerated Gradient Descent

TL;DR

This paper investigates the stability of Nesterov's accelerated gradient method, revealing that its stability deteriorates exponentially with the number of steps, contrasting with previous quadratic case results and non-accelerated methods.

Contribution

It disproves the conjecture that stability grows quadratically in the general convex case and shows exponential deterioration, highlighting fundamental differences in accelerated methods.

Findings

Stability of Nesterov's method deteriorates exponentially with steps.

Contrasts with quadratic case where stability grows quadratically.

Differs from non-accelerated methods with linear stability growth.

Abstract

We study the algorithmic stability of Nesterov's accelerated gradient method. For convex quadratic objectives, Chen et al. (2018) proved that the uniform stability of the method grows quadratically with the number of optimization steps, and conjectured that the same is true for the general convex and smooth case. We disprove this conjecture and show, for two notions of algorithmic stability (including uniform stability), that the stability of Nesterov's accelerated method in fact deteriorates exponentially fast with the number of gradient steps. This stands in sharp contrast to the bounds in the quadratic case, but also to known results for non-accelerated gradient methods where stability typically grows linearly with the number of steps.