Unified Convergence Theory of Stochastic and Variance-Reduced Cubic Newton Methods

El Mahdi Chayti, Martin Jaggi, Nikita Doikov

Keywords: Cubic Newton, auxiliary information, variance reduction, gradient-dominated functions, non-convex optimization

TL;DR: We study Cubic Newton's method when we have access to helper functions, our theory includes stochastic, variance reduced and Lazy methods. It also can apply to auxiliary learning.

Abstract: We study stochastic Cubic Newton methods for solving general possibly non-convex minimization problems. We propose a new framework, which we call the {\em helper framework}, that provides a unified view of the stochastic and variance-reduced second-order algorithms equipped with global complexity guarantees. It can also be applied to learning with auxiliary information. Our helper framework offers the algorithm designer high flexibility for constructing and analysis of the stochastic Cubic Newton methods, allowing arbitrary size batches, and the use of noisy and possibly biased estimates of the gradients and Hessians, incorporating both the variance reduction and the lazy Hessian updates. We recover the best-known complexities for the stochastic and variance-reduced Cubic Newton, under weak assumptions on the noise and avoiding artificial logarithms. A direct consequence of our theory is the new lazy stochastic second-order method, which significantly improves the arithmetic complexity for large dimension problems. We also establish complexity bounds for the classes of gradient-dominated objectives, that include convex and strongly convex problems. For Auxiliary Learning, we show that using a helper (auxiliary function) can outperform training alone if a given similarity measure is small.

Submission Number: 1893