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.
Carmon @cite_15 used negative curvature directions at the first phase of iterates and then switched it to accelerated stochastic gradient descent method when an iterate reaches an almost convex region.
@cite_15
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We present an accelerated gradient method for nonconvex optimization problems with Lipschitz continuous first and second derivatives. In a time @math , the method finds an @math -stationary point, meaning a point @math such that @math . The method improves upon the @math complexity of gradient descent and provides the additional second-order guarantee that @math for the computed @math . Furthermore, our method is Hessian free, i.e., it only requires gradient computations, and is therefore suitable for large-scale applications.