Hessian技术

令C =C(w),假设目标是最小化函数C 根据泰勒公式得:

\begin{eqnarray} C(w+\Delta w) & = & C(w) + \sum_j \frac{\partial C}{\partial w_j} \Delta w_j \nonumber \\ & & + \frac{1}{2} \sum_{jk} \Delta w_j \frac{\partial^2 C}{\partial w_j \partial w_k} \Delta w_k + \ldots \tag{103} \\ & = & C(w) + \nabla C \cdot \Delta w + \frac{1}{2} \Delta w^T H \Delta w + \ldots, \tag{104}\end{eqnarray}
\begin{eqnarray} C(w+\Delta w) \approx C(w) + \nabla C \cdot \Delta w + \frac{1}{2} \Delta w^T H \Delta w. \tag{105}\end{eqnarray}

因此,基于Hessian技术,w的更新的策略为:

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