Simplified Quadratic Gradient: A Unified Framework Bridging Gradient Descent and Newton-Type Methods by Synthesizing Hessians and Gradients

Abstract

Accelerating the convergence of second-order optimization, particularly Newton-type methods, remains a pivotal challenge in algorithmic research. In this paper, we extend previous work on the \textbf{Quadratic Gradient (QG)} and rigorously validate its applicability to general convex numerical optimization problems. We introduce a novel variant of the Quadratic Gradient that departs from the conventional fixed Hessian Newton framework. We present a new way to build a new version of the quadratic gradient. This new quadratic gradient doesn't satisfy the convergence conditions of the fixed Hessian Newton's method. However, experimental results show that it sometimes has a better performance than the original one in convergence rate. While this variant relaxes certain classical convergence constraints, it maintains a positive-definite Hessian proxy and demonstrates comparable, or in some cases superior, empirical performance in convergence rates. Furthermore, we demonstrate that both the original and the proposed QG variants can be effectively applied to non-convex optimization landscapes. A key motivation of our work is the limitation of traditional scalar learning rates. We argue that a diagonal matrix can more effectively accelerate gradient elements at heterogeneous rates. Our findings establish the Quadratic Gradient as a versatile and potent framework for modern optimization. Furthermore, we integrate Hutchinson's Estimator to estimate the Hessian diagonal efficiently via Hessian-vector products. Notably, we demonstrate that the proposed Quadratic Gradient variant is highly effective for Deep Learning architectures, providing a robust second-order alternative to standard adaptive optimizers.