Study of the behaviour of Nesterov Accelerated Gradient in a non convex setting: the strongly quasar convex case

TL;DR

This paper analyzes the convergence behavior of Nesterov Accelerated Gradient on strongly quasar convex functions, revealing conditions for acceleration and the impact of geometric properties.

Contribution

It extends understanding of NAG's convergence in non-convex settings by analyzing strongly quasar convex functions and identifying key geometric factors affecting acceleration.

Findings

NAG achieves accelerated convergence on strongly quasar convex functions.

Negative friction can occur but does not prevent acceleration.

Dropping certain geometric properties cancels acceleration.

Abstract

We study the convergence of Nesterov Accelerated Gradient (NAG) minimization algorithmapplied to a class of non convex functions called strongly quasar convex functions. We show thatNAG can achieve an accelerated convergence speed at the cost of a lower curvature assumption.We provide a continuous analysis through high resolution ODEs, where we show that despite thatnegative friction may appear, the solution of the system achieves accelerated rate of convergenceto the minimum. Finally, we identify the key geometrical property that, if dropped, theoreticallycancels the acceleration phenomenon.