Loss functions with a large number of saddle points are one of the main obstacles to training many modern machine learning models. Gradient descent (GD) is a fundamental algorithm for machine learning and converges to a saddle point for certain initial data. We call the region formed by these initial values the "attraction region." For quadratic functions, GD converges to a saddle point if the initial data is in a subspace of up to n-1 dimensions. In this paper, we prove that a small modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher, et al, arXiv:1806.06317] contributes to avoiding saddle points without sacrificing the convergence rate of GD. In particular, we show that the dimension of the LSGD's attraction region is at most floor((n-1) 2) for a class of quadratic functions which is significantly smaller than GD's (n-1)-dimensional attraction region.
The Hessian is utilized to distinguish saddle point from local minima. By using the Hessian, Nesterov & Polyak @cite_20 designed a cubic regularization algorithm which converges to an @math -second-order stationary point in @math iterations; @cite_25 developed a class of trust region algorithms and achieved the same complexity. By using only the product of Hessian with any vector, @cite_3 and Carmon & Duchi @cite_2 show that it is also possible to find an @math -approximate local minimum.
@cite_3 @cite_25 @cite_20 @cite_2
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This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general smooth, nonconvex functions in only ˜ O (ϵ−3.5) stochastic gradient and stochastic Hessian-vector product evaluations. The latter can be computed as efficiently as stochastic gradients. This improves upon the ˜ O (ϵ−4) rate of stochastic gradient descent. Our rate matches the best-known result for finding local minima without requiring any delicate acceleration or variance-reduction techniques. We consider variants of trust-region and cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated. Under mild conditions on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve @math -approximate second-order optimality which have shown to be tight. Our Hessian approximation conditions constitute a major relaxation over the existing ones in the literature. Consequently, we are able to show that such mild conditions allow for the construction of the approximate Hessian through various random sampling methods. In this light, we consider the canonical problem of finite-sum minimization, provide appropriate uniform and non-uniform sub-sampling strategies to construct such Hessian approximations, and obtain optimal iteration complexity for the corresponding sub-sampled trust-region and cubic regularization methods. An Adaptive Regularisation algorithm using Cubics (ARC) is proposed for unconstrained optimization, generalizing at the same time an unpublished method due to Griewank (Technical Report NA 12, 1981, DAMTP, University of Cambridge), an algorithm by Nesterov and Polyak (Math Program 108(1):177–205, 2006) and a proposal by (Optim Methods Softw 22(3):413–431, 2007). At each iteration of our approach, an approximate global minimizer of a local cubic regularisation of the objective function is determined, and this ensures a significant improvement in the objective so long as the Hessian of the objective is locally Lipschitz continuous. The new method uses an adaptive estimation of the local Lipschitz constant and approximations to the global model-minimizer which remain computationally-viable even for large-scale problems. We show that the excellent global and local convergence properties obtained by Nesterov and Polyak are retained, and sometimes extended to a wider class of problems, by our ARC approach. Preliminary numerical experiments with small-scale test problems from the CUTEr set show encouraging performance of the ARC algorithm when compared to a basic trust-region implementation. Two classes of methods have been proposed for escaping from saddle points with one using the second-order information carried by the Hessian and the other adding the noise into the first-order information. The existing analysis for algorithms using noise in the first-order information is quite involved and hides the essence of added noise, which hinder further improvements of these algorithms. In this paper, we present a novel perspective of noise-adding technique, i.e., adding the noise into the first-order information can help extract the negative curvature from the Hessian matrix, and provide a formal reasoning of this perspective by analyzing a simple first-order procedure. More importantly, the proposed procedure enables one to design purely first-order stochastic algorithms for escaping from non-degenerate saddle points with a much better time complexity (almost linear time in terms of the problem's dimensionality). In particular, we develop a first-order stochastic algorithm based on our new technique and an existing algorithm that only converges to a first-order stationary point to enjoy a time complexity of @math for finding a nearly second-order stationary point @math such that @math and @math (in high probability), where @math denotes the objective function and @math is the dimensionality of the problem. To the best of our knowledge, this is the best theoretical result of first-order algorithms for stochastic non-convex optimization, which is even competitive with if not better than existing stochastic algorithms hinging on the second-order information.