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.
In the stochastic optimization setting, second-order methods have been researched for decades. J. Martens @cite_11 @cite_1 published papers about the Hessian-free optimization method to train neural networks including deep auto-encoder and recurrent neural networks. In these papers, it is argued that using a Gauss-Newton approximation instead of a Hessian approximation is more effective to train neural networks. Also, he employs a damping coefficient to maintain the positive definiteness of a Hessian approximation, which helps restrict a step size. Consequently, with a damping coefficient @math , a linear system is solved, @math , to attain an update step @math by using the conjugate gradient algorithm, at every iterate. The idea to use a damping coefficient is highly similar to that of adopting an adaptive cubic regularization term. But it is noted that adaptive cubic regularization has a theoretical adaptive rule to guarantee its convergence whereas the Newton method with a damping coefficient does not provide.
@cite_1 @cite_11
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In this work we resolve the long-outstanding problem of how to effectively train recurrent neural networks (RNNs) on complex and difficult sequence modeling problems which may contain long-term data dependencies. Utilizing recent advances in the Hessian-free optimization approach (Martens, 2010), together with a novel damping scheme, we successfully train RNNs on two sets of challenging problems. First, a collection of pathological synthetic datasets which are known to be impossible for standard optimization approaches (due to their extremely long-term dependencies), and second, on three natural and highly complex real-world sequence datasets where we find that our method significantly outperforms the previous state-of-the-art method for training neural sequence models: the Long Short-term Memory approach of Hochreiter and Schmidhuber (1997). Additionally, we offer a new interpretation of the generalized Gauss-Newton matrix of Schraudolph (2002) which is used within the HF approach of Martens. We develop a 2nd-order optimization method based on the "Hessian-free" approach, and apply it to training deep auto-encoders. Without using pre-training, we obtain results superior to those reported by Hinton & Salakhutdinov (2006) on the same tasks they considered. Our method is practical, easy to use, scales nicely to very large datasets, and isn't limited in applicability to auto-encoders, or any specific model class. We also discuss the issue of "pathological curvature" as a possible explanation for the difficulty of deep-learning and how 2nd-order optimization, and our method in particular, effectively deals with it.