Escape saddle points by a simple gradient-descent based algorithm

• URL: http://arxiv.org/abs/2111.14069v1
• Date: Sun, 28 Nov 2021 07:32:54 GMT
• Title: Escape saddle points by a simple gradient-descent based algorithm
• Authors: Chenyi Zhang, Tongyang Li
• Abstract summary: Escaping saddle points is a central research topic in non optimization. We propose a simple gradient-based algorithm such that for a smooth function $fcolonmathbbRntomathbbR$, it outputs an $epsilonO(log n)2/epsilon4)$. Technically, our main contribution is an idea of implementing a robust Hessian power method using only gradients.
• Score: 6.7298812735467095
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Escaping saddle points is a central research topic in nonconvex optimization. In this paper, we propose a simple gradient-based algorithm such that for a smooth function $f\colon\mathbb{R}^n\to\mathbb{R}$, it outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}(\log n/\epsilon^{1.75})$ iterations. Compared to the previous state-of-the-art algorithms by Jin et al. with $\tilde{O}((\log n)^{4}/\epsilon^{2})$ or $\tilde{O}((\log n)^{6}/\epsilon^{1.75})$ iterations, our algorithm is polynomially better in terms of $\log n$ and matches their complexities in terms of $1/\epsilon$. For the stochastic setting, our algorithm outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}((\log n)^{2}/\epsilon^{4})$ iterations. Technically, our main contribution is an idea of implementing a robust Hessian power method using only gradients, which can find negative curvature near saddle points and achieve the polynomial speedup in $\log n$ compared to the perturbed gradient descent methods. Finally, we also perform numerical experiments that support our results.

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