We focus on minimizing nonconvex finite-sum functions that typically arise in machine learning problems. In an attempt to solve this problem, the adaptive cubic regularized Newton method has shown its strong global convergence guarantees and ability to escape from strict saddle points. This method uses a trust region-like scheme to determine if an iteration is successful or not, and updates only when it is successful. In this paper, we suggest an algorithm combining negative curvature with the adaptive cubic regularized Newton method to update even at unsuccessful iterations. We call this new method Stochastic Adaptive cubic regularization with Negative Curvature (SANC). Unlike the previous method, in order to attain stochastic gradient and Hessian estimators, the SANC algorithm uses independent sets of data points of consistent size over all iterations. It makes the SANC algorithm more practical to apply for solving large-scale machine learning problems. To the best of our knowledge, this is the first approach that combines the negative curvature method with the adaptive cubic regularized Newton method. Finally, we provide experimental results including neural networks problems supporting the efficiency of our method.
There have been relatively little works on the negative curvature methods. Curtis and Robinson @cite_26 proposed several algorithms exploiting negative curvature for solving deterministic and stochastic optimization problems. In this work, a current iterate is updated with a direction of (stochastic) gradient descent and negative curvature. The 'dynamic method' is also proposed which adaptively estimates a Lipschitz constant of gradient continuity to estimate a step length in the stochastic optimization framework. They showed that the 'dynamic method' is efficient in performance in stochastic optimization problems, but they did not provide a proof of its convergence.
@cite_26
2594169002 2891935861 2655474054 2772279812
This paper addresses the question of whether it can be beneficial for an optimization algorithm to follow directions of negative curvature. Although prior work has established convergence results for algorithms that integrate both descent and negative curvature steps, there has not yet been extensive numerical evidence showing that such methods offer consistent performance improvements. In this paper, we present new frameworks for combining descent and negative curvature directions: alternating two-step approaches and dynamic step approaches. The aspect that distinguishes our approaches from ones previously proposed is that they make algorithmic decisions based on (estimated) upper-bounding models of the objective function. A consequence of this aspect is that our frameworks can, in theory, employ fixed stepsizes, which makes the methods readily translatable from deterministic to stochastic settings. For deterministic problems, we show that instances of our dynamic framework yield gains in performance compared to related methods that only follow descent steps. We also show that gains can be made in a stochastic setting in cases when a standard stochastic-gradient-type method might make slow progress. Negative curvature descent (NCD) method has been utilized to design deterministic or stochastic algorithms for non-convex optimization aiming at finding second-order stationary points or local minima. In existing studies, NCD needs to approximate the smallest eigen-value of the Hessian matrix with a sufficient precision (e.g., ϵ2≪1) in order to achieve a sufficiently accurate second-order stationary solution (i.e., λmin(∇2f( ))≥−ϵ2). One issue with this approach is that the target precision ϵ2 is usually set to be very small in order to find a high quality solution, which increases the complexity for computing a negative curvature. To address this issue, we propose an adaptive NCD to allow for an adaptive error dependent on the current gradient's magnitude in approximating the smallest eigen-value of the Hessian, and to encourage competition between a noisy NCD step and gradient descent step. We consider the applications of the proposed adaptive NCD for both deterministic and stochastic non-convex optimization, and demonstrate that it can help reduce the the overall complexity in computing the negative curvatures during the course of optimization without sacrificing the iteration complexity. We focus on nonconvex and nonsmooth minimization problems with a composite objective, where the differentiable part of the objective is freed from the usual and restrictive global Lipschitz gradient continuity assumption. This longstanding smoothness restriction is pervasive in first order methods (FOM), and was recently circumvent for convex composite optimization by Bauschke, Bolte and Teboulle, through a simple and elegant framework which captures, all at once, the geometry of the function and of the feasible set. Building on this work, we tackle genuine nonconvex problems. We first complement and extend their approach to derive a full extended descent lemma by introducing the notion of smooth adaptable functions. We then consider a Bregman-based proximal gradient methods for the nonconvex composite model with smooth adaptable functions, which is proven to globally converge to a critical point under natural assumptions on the problem's data. To illustrate the power and potential of our general framework and results, we consider a broad class of quadratic inverse problems with sparsity constraints which arises in many fundamental applications, and we apply our approach to derive new globally convergent schemes for this class. We propose a family of nonconvex optimization algorithms that are able to save gradient and negative curvature computations to a large extent, and are guaranteed to find an approximate local minimum with improved runtime complexity. At the core of our algorithms is the division of the entire domain of the objective function into small and large gradient regions: our algorithms only perform gradient descent based procedure in the large gradient region, and only perform negative curvature descent in the small gradient region. Our novel analysis shows that the proposed algorithms can escape the small gradient region in only one negative curvature descent step whenever they enter it, and thus they only need to perform at most @math negative curvature direction computations, where @math is the number of times the algorithms enter small gradient regions. For both deterministic and stochastic settings, we show that the proposed algorithms can potentially beat the state-of-the-art local minima finding algorithms. For the finite-sum setting, our algorithm can also outperform the best algorithm in a certain regime.