AI 中文总结
本文提出广义二次梯度（GQG）这一统一框架，将二次梯度原理扩展至更广泛牛顿型优化算法，突破了现有二次梯度方法对特定海森近似的限制，为感知曲率的优化算法开发提供了更广泛基础。
AI 中文摘要
二次梯度（Quadratic Gradient, QG）是一种牛顿型优化框架，它将曲率信息融入梯度更新，从而连接了一阶梯度下降与二阶优化。简化二次梯度（Simplified Quadratic Gradient, SQG）在保留优化能力的同时降低了QG构建的复杂度；而准二次梯度（Quasi-Quadratic Gradient, QQG）则将二次梯度原理扩展至BFGS等拟牛顿方法。本文提出广义二次梯度（Generalized Quadratic Gradient, GQG），这是一种将二次梯度原理扩展至更广泛牛顿型优化算法的统一框架。通过抽象现有二次梯度方法的共同结构，研究表明二次梯度构建的基本要求并不局限于特定的海森近似，如常数海森矩阵、对角海森近似或基于BFGS的海森替代矩阵，而是可推广至任何满足局部二次模型平稳条件的正定曲率矩阵。基于该视角，研究利用BFGS之外的各类正定海森替代矩阵探究广义二次梯度的构建，为开发感知曲率的优化算法提供了更广泛的基础。
英文摘要
Quadratic Gradient (QG) is a Newton-type optimization framework that bridges first-order gradient descent and second-order optimization by incorporating curvature information into gradient updates. Simplified Quadratic Gradient (SQG) reduces the complexity of QG construction while preserving its optimization capability, whereas Quasi-Quadratic Gradient (QQG) extends the quadratic gradient principle to quasi-Newton methods such as BFGS. In this paper, we propose **Generalized Quadratic Gradient (GQG)**, a unified framework that extends the quadratic gradient principle to a broader class of Newton-type optimization algorithms. By abstracting the common structure of existing quadratic gradient methods, we show that the fundamental requirement of quadratic gradient construction is not limited to specific Hessian approximations, such as constant Hessian matrices, diagonal Hessian approximations, or BFGS-based Hessian surrogates. Instead, it can be generalized to any positive-definite curvature matrix satisfying the stationary condition of a local quadratic model. Based on this perspective, we investigate the construction of generalized quadratic gradients using various positive-definite Hessian surrogates beyond BFGS, providing a broader foundation for developing curvature-aware optimization algorithms.
