Escape saddle points by a simple gradient-descent based algorithm

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 : R n → R, it outputs an -approximate second-order stationary point in Õ(log n/ 1.75 ) iterations. Compared to the previous state-of-the-art algorithms by Jin et al. with Õ(log 4 n/ 2 ) or Õ(log 6 n/ 1.75 ) iterations, our algorithm is polynomially better in terms of log n and matches their complexities in terms of 1/ . For the stochastic setting, our algorithm outputs an -approximate second-order stationary point in Õ(log 2 n/ 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.

Quantum simulation of real-space dynamics

“…Then for any ǫ > 0, Algorithm 1 outputs an ǫ-approximate local minimum with success probability at least 2/3 using Õ f (x 0 )−f * ǫ 1.75 log d queries to the evaluation oracle U f , where x 0 is the initial point of the algorithm, f * is the global minimum of f , and the Õ notation omits poly-logarithmic factors as in Footnote 1. For comparison, the previous result [68] uses Õ f (x 0 )−f * ǫ 1.75 log 2 d quantum evaluation queries, so our simulation approach achieves a quadratic speedup in terms of log d. Compared to classical algorithms for escaping from saddle points, we achieve polynomial speedup over the seminal work of Jin et al [37] which makes Õ f (x 0 )−f * ǫ 1.75 log 6 d gradient queries, and match the iteration number of the state-of-the-art result [69] which makes Õ f (x 0 )−f * ǫ 1.75 log d classical gradient queries. The fact that this simulation-based quantum algorithm uses only evaluation queries instead of gradient queries enables a larger range of applications than classical approaches [37,69], especially for problems where the gradient values are not directly available.…”

“…For comparison, the previous result [68] uses Õ f (x 0 )−f * ǫ 1.75 log 2 d quantum evaluation queries, so our simulation approach achieves a quadratic speedup in terms of log d. Compared to classical algorithms for escaping from saddle points, we achieve polynomial speedup over the seminal work of Jin et al [37] which makes Õ f (x 0 )−f * ǫ 1.75 log 6 d gradient queries, and match the iteration number of the state-of-the-art result [69] which makes Õ f (x 0 )−f * ǫ 1.75 log d classical gradient queries. The fact that this simulation-based quantum algorithm uses only evaluation queries instead of gradient queries enables a larger range of applications than classical approaches [37,69], especially for problems where the gradient values are not directly available. Although in principle one can use Jordan's algorithm [39] to replace the classical gradient queries in [69] by quantum evaluation queries with logarithmic overhead in query complexity, Jordan's algorithm must be implemented with high precision to detect feasible directions for escaping from saddle points since the gradients near saddles have small norms.…”

Quantum simulation of real-space dynamics

“…where H is the Hessian matrix of f and ρ, are constants,. Then for any > 0, Algorithm 1 outputs an -approximate local minimum with success probability at least 2/3 using Õ f (x 0 )−f * For comparison, the previous result [68] uses Õ f (x 0 )−f * 1.75 log 2 d quantum evaluation queries, so our simulation approach achieves a quadratic speedup in terms of log d. Compared to classical algorithms for escaping from saddle points, we achieve polynomial speedup over the seminal work of Jin et al [37] which makes Õ f (x 0 )−f * 1.75 log 6 d gradient queries, and match the iteration number of the state-of-the-art result [69] which makes Õ f (x 0 )−f * 1.75 log d classical gradient queries.…”

“…The fact that this simulation-based quantum algorithm uses only evaluation queries instead of gradient queries enables a larger range of applications than classical approaches [37,69], especially for problems where the gradient values are not directly available. Although in principle one can use Jordan's algorithm [39] to replace the classical gradient queries in [69] by quantum evaluation queries with logarithmic overhead in query complexity, Jordan's algorithm must be implemented with high precision to detect feasible directions for escaping from saddle points since the gradients near saddles have small norms. Therefore, the number of qubits required in an approach based on Jordan's algorithm may be large.…”