梯度是一个数学概念在多维空间中表示函数在某一点处增长最快的方向模长表示变化率大小。对于一个多元函数f(x1,x2,⋯ ,xn)f(x_1, x_2, \cdots, x_n)f(x1,x2,⋯,xn)称nnn维向量[∂∂x1f(xˉ),∂∂x2f(xˉ),⋯ ,∂∂xnf(xˉ)]⊤\left[ \frac{\partial}{\partial x_1} f(\bar{\boldsymbol x}), \frac{\partial}{\partial x_2} f(\bar{\boldsymbol x}), \cdots, \frac{\partial}{\partial x_n} f(\bar{\boldsymbol x}) \right]^{\top}[∂x1∂f(xˉ),∂x2∂f(xˉ),⋯,∂xn∂f(xˉ)]⊤为函数f(x)f({\boldsymbol x})f(x)在xˉ\bar{\boldsymbol x}xˉ处的梯度gradient记为∇f(xˉ)\nabla f(\bar{ {\boldsymbol x}})∇f(xˉ)即∇f(xˉ)[∂∂x1f(xˉ),∂∂x2f(xˉ),⋯ ,∂∂xnf(xˉ)]⊤. \nabla f(\bar{\boldsymbol x}) \left[ \frac{\partial}{\partial x_1} f(\bar{\boldsymbol x}), \frac{\partial}{\partial x_2} f(\bar{\boldsymbol x}), \cdots, \frac{\partial}{\partial x_n} f(\bar{\boldsymbol x}) \right]^{\top}.∇f(xˉ)[∂x1∂f(xˉ),∂x2∂f(xˉ),⋯,∂xn∂f(xˉ)]⊤.称∇f(x)\nabla f({\boldsymbol x})∇f(x)为梯度函数简称梯度。有时将∇f(x)\nabla f({\boldsymbol x})∇f(x)记为g(x)g({\boldsymbol x})g(x)。f:Rn→Rf: \mathbb{R}^n \to \mathbb{R}f:Rn→R是标量值函数。梯度函数指把梯度看作关于x{\boldsymbol x}x的函数即∇f:Rn→Rn\nabla f: \mathbb{R}^n \to \mathbb{R}^n∇f:Rn→Rn。