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We consider a variant of inexact Newton Method, called Newton-MR, in whichthe least-squares sub-problems are solved approximately using Minimum Residualmethod. By construction, Newton-MR can be readily applied for unconstrainedoptimization of a class of non-convex problems known as invex, which subsumesconvexity as a sub-class. For invex optimization, instead of the classicalLipschitz continuity assumptions on gradient and Hessian, Newton-MR's globalconvergence can be guaranteed under a weaker notion of joint regularity ofHessian and gradient. We also obtain Newton-MR's problem-independent localconvergence to the set of minima. We show that fast local/global convergencecan be guaranteed under a novel inexactness condition, which, to our knowledge,is much weaker than the prior related works. Numerical results demonstrate theperformance of Newton-MR as compared with several other Newton-typealternatives on a few machine learning problems.

Newton-MR: Inexact Newton Method With Minimum Residual Sub-problem Solver