3,328 research outputs found
A symmetric rank-one Quasi-Newton line-search method using negative curvature directions
We propose a quasi-Newton line-search method that uses negative curvature directions for solving unconstrained optimization problems. In this method, the symmetric rank-one (SR1) rule is used to update the Hessian approximation. The SR1 update rule is known to have a good numerical performance; however, it does not guarantee positive definiteness of the updated matrix. We first discuss the details of the proposed algorithm and then concentrate on its numerical efficiency. Our extensive computational study shows the potential of the proposed method from different angles, such as; its second order convergence behavior, its exceeding performance when compared to two other existing packages, and its computation profile illustrating the possible bottlenecks in the execution time. We then conclude the paper with the convergence analysis of the proposed method
Identifying and attacking the saddle point problem in high-dimensional non-convex optimization
A central challenge to many fields of science and engineering involves
minimizing non-convex error functions over continuous, high dimensional spaces.
Gradient descent or quasi-Newton methods are almost ubiquitously used to
perform such minimizations, and it is often thought that a main source of
difficulty for these local methods to find the global minimum is the
proliferation of local minima with much higher error than the global minimum.
Here we argue, based on results from statistical physics, random matrix theory,
neural network theory, and empirical evidence, that a deeper and more profound
difficulty originates from the proliferation of saddle points, not local
minima, especially in high dimensional problems of practical interest. Such
saddle points are surrounded by high error plateaus that can dramatically slow
down learning, and give the illusory impression of the existence of a local
minimum. Motivated by these arguments, we propose a new approach to
second-order optimization, the saddle-free Newton method, that can rapidly
escape high dimensional saddle points, unlike gradient descent and quasi-Newton
methods. We apply this algorithm to deep or recurrent neural network training,
and provide numerical evidence for its superior optimization performance.Comment: The theoretical review and analysis in this article draw heavily from
arXiv:1405.4604 [cs.LG
Newton-MR: Inexact Newton Method With Minimum Residual Sub-problem Solver
We consider a variant of inexact Newton Method, called Newton-MR, in which
the least-squares sub-problems are solved approximately using Minimum Residual
method. By construction, Newton-MR can be readily applied for unconstrained
optimization of a class of non-convex problems known as invex, which subsumes
convexity as a sub-class. For invex optimization, instead of the classical
Lipschitz continuity assumptions on gradient and Hessian, Newton-MR's global
convergence can be guaranteed under a weaker notion of joint regularity of
Hessian and gradient. We also obtain Newton-MR's problem-independent local
convergence to the set of minima. We show that fast local/global convergence
can be guaranteed under a novel inexactness condition, which, to our knowledge,
is much weaker than the prior related works. Numerical results demonstrate the
performance of Newton-MR as compared with several other Newton-type
alternatives on a few machine learning problems.Comment: 35 page
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