Algorithmic Instabilities of Accelerated Gradient Descent
This work is significant for researchers and practitioners using Nesterov's accelerated gradient method, as it reveals a fundamental instability issue that was previously underestimated.
This paper disproves a conjecture regarding the stability of Nesterov's accelerated gradient method for general convex and smooth objectives. It demonstrates that the algorithmic stability, under two different notions, deteriorates exponentially fast with the number of gradient steps, contrasting with quadratic growth in quadratic cases and linear growth in non-accelerated methods.
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.