# Behavior of Accelerated Gradient Methods Near Critical Points of Nonconvex Functions

arXiv: https://arxiv.org/abs/1706.07993 · HTML: https://mlsanity.com/paper/1706.07993/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jun 2017.

## Built on by (most cited first, 7 of 7)

- [Restarted Nonconvex Accelerated Gradient Descent: No More Polylogarithmic Factor in the O(ε-7/4) Complexity](https://mlsanity.com/paper/2201.11411/) (2022)
- [Nesterov acceleration in benignly non-convex landscapes](https://mlsanity.com/paper/2410.08395/) (2024)
- [Gradient descent provably escapes saddle points in the training of shallow ReLU networks](https://mlsanity.com/paper/2208.02083/) (2022)
- [Inertial Newton Algorithms Avoiding Strict Saddle Points](https://mlsanity.com/paper/2111.04596/) (2021)
- [Accelerated Gradient Methods for Nonconvex Optimization: Escape Trajectories From Strict Saddle Points and Convergence to Local Minima](https://mlsanity.com/paper/2307.07030/) (2023)
- [Dimension-Free Saddle-Point Escape in Muon](https://mlsanity.com/paper/2605.09331/) (2026)
- [Nonlinear Matrix Approximation with Radial Basis Function Components](https://mlsanity.com/paper/2106.02018/) (2021)

## Abstract (arXiv)

We examine the behavior of accelerated gradient methods in smooth nonconvex unconstrained optimization, focusing in particular on their behavior near strict saddle points. Accelerated methods are iterative methods that typically step along a direction that is a linear combination of the previous step and the gradient of the function evaluated at a point at or near the current iterate. (The previous step encodes gradient information from earlier stages in the iterative process.) We show by means of the stable manifold theorem that the heavy-ball method method is unlikely to converge to strict saddle points, which are points at which the gradient of the objective is zero but the Hessian has at least one negative eigenvalue. We then examine the behavior of the heavy-ball method and other accelerated gradient methods in the vicinity of a strict saddle point of a nonconvex quadratic function, showing that both methods can diverge from this point more rapidly than the steepest-descent method.