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Machine learning (ML) problems are often posed as highly nonlinear andnonconvex unconstrained optimization problems. Methods for solving ML problemsbased on stochastic gradient descent are easily scaled for very large problemsbut may involve fine-tuning many hyper-parameters. Quasi-Newton approachesbased on the limited-memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) updatetypically do not require manually tuning hyper-parameters but suffer fromapproximating a potentially indefinite Hessian with a positive-definite matrix.Hessian-free methods leverage the ability to perform Hessian-vectormultiplication without needing the entire Hessian matrix, but each iteration'scomplexity is significantly greater than quasi-Newton methods. In this paper wepropose an alternative approach for solving ML problems based on a quasi-Newtontrust-region framework for solving large-scale optimization problems that allowfor indefinite Hessian approximations. Numerical experiments on a standardtesting data set show that with a fixed computational time budget, the proposedmethods achieve better results than the traditional limited-memory BFGS and theHessian-free methods.

Trust-Region Algorithms for Training Responses: Machine Learning Methods Using Indefinite Hessian Approximations