标题： 协变梯度下降 显示英文标题

标题： Covariant Gradient Descent

摘要： 我们提出了梯度下降法的一种显式协变表述，确保了其在任意坐标系和一般的可训练弯曲空间中的一致性。 优化动力学通过协变力向量和协变度规张量来定义，这两个量均由梯度的一阶和二阶统计矩计算得到。 这些矩通过带指数权重函数的时间平均来估计，从而保持了线性计算复杂度。 我们表明，常用的优化方法如RMSProp、Adam和AdaBelief对应于协变梯度下降（CGD）的特殊极限，并展示了这些方法如何可以进一步被推广和改进。 显示英文摘要

摘要： We present a manifestly covariant formulation of the gradient descent method, ensuring consistency across arbitrary coordinate systems and general curved trainable spaces. The optimization dynamics is defined using a covariant force vector and a covariant metric tensor, both computed from the first and second statistical moments of the gradients. These moments are estimated through time-averaging with an exponential weight function, which preserves linear computational complexity. We show that commonly used optimization methods such as RMSProp, Adam and AdaBelief correspond to special limits of the covariant gradient descent (CGD) and demonstrate how these methods can be further generalized and improved.