课程前言一元复合链式是单层网络梯度多元复合链式法则是大模型反向传播的底层核心。一元函数是一个因素对结果的影响多元复合链式法则是多个因素相互作用对结果的影响。我们为了寻求最佳结果必须对影响结果因素的相互作用进行计算求导找到最优路径。l二元复合通用模型zf(u,v),\ u\varphi(x,y),\ v\psi(x,y)l链式公式\frac{\partial z}{\partial x}\frac{\partial z}{\partial u}\cdot\frac{\partial u}{\partial x}\frac{\partial z}{\partial v}\cdot\frac{\partial v}{\partial x}\frac{\partial z}{\partial y}\frac{\partial z}{\partial u}\cdot\frac{\partial u}{\partial y}\frac{\partial z}{\partial v}\cdot\frac{\partial v}{\partial y}AI对应逻辑z 损失函数u、v 隐藏层输出x、y 网络权重参数多层网络梯度回传就是多层多元链式连加连乘Transformer、CNN全部依靠该公式完成参数更新。一、核心计算规则通俗解读多路径求和损失可通过多条中间变量影响权重梯度全部相加逐层相乘外层函数偏导 × 内层变量偏导对应跨层梯度传递复合层数越多链式计算步骤越多深层大模型梯度链路更长。一件事情多个影响结果的因素我们如何分析因素达到最佳结果。二、10道多元复合链式计算题完整步骤AI场景解读题1zu^2v,\ u2x,\ v3y求\dfrac{\partial z}{\partial x},\dfrac{\partial z}{\partial y}解\dfrac{\partial z}{\partial u}2u,\ \dfrac{\partial z}{\partial v}1\dfrac{\partial u}{\partial x}2,\dfrac{\partial u}{\partial y}0;\ \dfrac{\partial v}{\partial x}0,\dfrac{\partial v}{\partial y}3\dfrac{\partial z}{\partial x}2u\cdot 24u8x\dfrac{\partial z}{\partial y}1\cdot 33AI解读两层简单网络两条独立特征支路权重梯度互不干扰浅层全连接基础梯度计算。题2zuv^2,\ uxy,\ vx-y对x、y求偏导解\dfrac{\partial z}{\partial u}1,\dfrac{\partial z}{\partial v}2v\dfrac{\partial u}{\partial x}1,\dfrac{\partial u}{\partial y}1;\ \dfrac{\partial v}{\partial x}1,\dfrac{\partial v}{\partial y}-1\dfrac{\partial z}{\partial x}1\cdot1 2v\cdot112(x-y)\dfrac{\partial z}{\partial y}1\cdot1 2v\cdot(-1)1-2(x-y)AI解读特征融合网络x、y两个权重同时作用两条中间特征梯度叠加求和。题3ze{uv}, ux2,\ v2xy求\partial z/\partial x解\dfrac{\partial z}{\partial u}\dfrac{\partial z}{\partial v}e^{uv}\dfrac{\partial u}{\partial x}2x,\ \dfrac{\partial v}{\partial x}2y\dfrac{\partial z}{\partial x}e^{uv}\cdot2x e{uv}\cdot2y2(xy)e{x^22xy}AI解读指数激活融合双特征注意力模块特征加权梯度计算模板。题4z\sin(u),\ ux^23y求两个一阶偏导解\dfrac{\partial z}{\partial u}\cos u\dfrac{\partial u}{\partial x}2x,\ \dfrac{\partial u}{\partial y}3\dfrac{\partial z}{\partial x}2x\cos(x^23y)\dfrac{\partial z}{\partial y}3\cos(x^23y)AI解读RoPE旋转位置编码复合函数位置参数多层嵌套梯度求解。题5zuv,\ u3x-y,\ vx2y求偏导数解\dfrac{\partial z}{\partial u}v,\dfrac{\partial z}{\partial v}u\dfrac{\partial u}{\partial x}3,\dfrac{\partial u}{\partial y}-1;\ \dfrac{\partial v}{\partial x}1,\dfrac{\partial v}{\partial y}2\dfrac{\partial z}{\partial x}3vu3(x2y)3x-y6x5y\dfrac{\partial z}{\partial y}-v2u-(x2y)2(3x-y)5x-4yAI解读特征交叉相乘层权重耦合梯度大模型多头注意力交互项底层运算。题6zu3, ux-y2求\partial z/\partial x、\partial z/\partial y解\dfrac{\partial z}{\partial u}3u^2\dfrac{\partial u}{\partial x}1,\ \dfrac{\partial u}{\partial y}-2y\dfrac{\partial z}{\partial x}3(x-y2)2\dfrac{\partial z}{\partial y}-6y(x-y2)2AI解读高次损失嵌套线性特征权重y的梯度带二次衰减对应权重衰减正则效果。题7z\ln(uv),\ uxy,\ vx求对x偏导解\dfrac{\partial z}{\partial u}\dfrac{\partial z}{\partial v}\dfrac{1}{uv}\dfrac{\partial u}{\partial x}y,\dfrac{\partial v}{\partial x}1\dfrac{\partial z}{\partial x}\dfrac{y1}{xyx}AI解读交叉熵对数损失复合函数分类任务反向传播核心梯度。题8三层嵌套复合ze^t,\ tuv,\ u2x,\ vxy求\partial z/\partial x解\dfrac{\partial z}{\partial t}e^t,\ \dfrac{\partial t}{\partial u}1,\dfrac{\partial t}{\partial v}1\dfrac{\partial u}{\partial x}2,\dfrac{\partial v}{\partial x}y\dfrac{\partial z}{\partial x}e^{2xxy}(2y)AI解读三层复合对应三层隐藏神经网络梯度三层连乘深层模型反向传播标准流程。题9zu\cos v,\ ux^2,\ v3y求两个偏导解\dfrac{\partial z}{\partial u}\cos v,\ \dfrac{\partial z}{\partial v}-u\sin v\dfrac{\partial u}{\partial x}2x,\dfrac{\partial u}{\partial y}0;\ \dfrac{\partial v}{\partial x}0,\dfrac{\partial v}{\partial y}3\dfrac{\partial z}{\partial x}2x\cos3y\dfrac{\partial z}{\partial y}-3x^2\sin3yAI解读时序特征加权周期位置编码复合结构大模型时序预测梯度模板。题10 压轴综合多路径复合zu2ev,\ ux-y,\ vxy完整求\dfrac{\partial z}{\partial x},\dfrac{\partial z}{\partial y}解\dfrac{\partial z}{\partial u}2u,\ \dfrac{\partial z}{\partial v}e^v\dfrac{\partial u}{\partial x}1,\dfrac{\partial u}{\partial y}-1;\ \dfrac{\partial v}{\partial x}y,\dfrac{\partial v}{\partial y}x\frac{\partial z}{\partial x}2(x-y)\cdot 1 e^{xy}\cdot y2(x-y)ye^{xy}\frac{\partial z}{\partial y}2(x-y)\cdot(-1)e^{xy}\cdot x-2(x-y)xe^{xy}AI解读同时包含二次损失、指数激活、交叉特征项完整复刻Transformer损失梯度计算逻辑是工程最通用复合梯度模型。三、课程核心总结衔接35课二阶混合偏导4. 多元链式两大核心多路径梯度相加、每层导数相乘5. 大模型反向传播本质对海量权重求解多元复合偏导6. 路径越多、嵌套层数越深梯度表达式越复杂深层网络易出现梯度消失/爆炸7. 下一课程学习多元二阶、混合偏导为海塞矩阵、多元极值判定打基础。多元复合链式求导很容易漏算某一条梯度路径你做题时怎么避免丢项深层大模型梯度消失和链式连乘有什么关系欢迎评论区交流