Algorithmic Instabilities of Accelerated Gradient Descent

Amit Attia, Tomer Koren

Keywords: Accelerated Gradient Descent, Nesterov Acceleration, Algorithmic Stability, Uniform Stability, Convex Optimization

TL;DR: We show that the algorithmic stability of Nesterov's accelerated method increases exponentially fast with the number of steps, disproving previous conjectures.

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.

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