Newton-MR: Inexact Newton Method With Minimum Residual Sub-problem Solver
This work addresses optimization challenges in machine learning for non-convex invex problems, offering a method with weaker convergence requirements, though it appears incremental as it builds on existing Newton-type frameworks.
The authors tackled the problem of unconstrained optimization for non-convex invex functions by proposing Newton-MR, an inexact Newton method using Minimum Residual sub-problem solving, which guarantees global and local convergence under weaker assumptions than classical methods. Numerical results show its performance compared to other Newton-type methods on machine learning tasks.
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.