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We investigate the use of regularized Newton methods with adaptive norms foroptimizing neural networks. This approach can be seen as a second-ordercounterpart of adaptive gradient methods, which we here show to beinterpretable as first-order trust region methods with ellipsoidal constraints.In particular, we prove that the preconditioning matrix used in RMSProp andAdam satisfies the necessary conditions for provable convergence ofsecond-order trust region methods with standard worst-case complexities ongeneral non-convex objectives. Furthermore, we run experiments across differentneural architectures and datasets to find that the ellipsoidal constraintsconstantly outperform their spherical counterpart both in terms of number ofbackpropagations and asymptotic loss value. Finally, we find comparableperformance to state-of-the-art first-order methods in terms ofbackpropagations, but further advances in hardware are needed to render Newtonmethods competitive in terms of computational time.

Adaptive norms for deep learning with regularized Newton methods