AI笔记: 数学基础之导数的应用:泰勒Taylor公式

Taylor公式的应用

  • 机器学习中广泛应用,数学建模,线性回归,预测等领域

关于Taylor公式

  • Taylor公式是用一个函数在某点的信息描述其附近取值的公式,如果函数足够平滑,在已知函数在某一点的各阶导数值的情况下
  • Taylor公式可以利用这些导数值来做系数构建一个多项式近似函数在这一点的邻域中的值
  • 若函数f(x)在包含x0x_0x
泰勒展开在AI工程中的四大实战应用 泰勒展开是深度学习中连接数学原理与工程实践的核心桥梁,其本质是在局部范围内用多项式近似非线性函数,从而实现可解析、可干预的系统行为建模。从自动微分的高阶导数计算,到二阶优化器对损失曲面的曲率利用;从对抗样本生成中的一阶最坏扰动方向,到模型压缩时基于Jacobian的敏感度驱动蒸馏——它贯穿AI研发全链路。尤其在梯度不稳定、loss震荡、FP16溢出等典型工程问题中,泰勒一阶/二阶展开提供低开销、高解释性的调试路径。本文聚焦泰勒在AI落地中的真实价值:不是理论推导,而是作为可嵌入、可量化、可部署的‘局部近似术 阅读详情

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的信息描述其附近的取值,信息用该点处的各阶导数表示。从物理学的角度,若复杂函数表示某一运动,当得知运动某一时刻的位置及其一阶导数(速度)、二阶导数(加速度)…尽管指数函数具有良好的导数性质,但其计算量较大。可以通过泰勒公式对其进行多项式展开,把非线性函数近似为分段线性函数,以减少计算开销。注意到泰勒公式中的每一项是逐渐减小的,若余项(近似误差项)比展开的最后一项还要小,则可以忽略。神经网络中的激活函数能够为网络增加非线性,从而提高网络的表示能力。函数进行近似,从而将非线性问题转化为线性问题的方法。

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% Dropout fraction opts.numepochs = 1; % Number of full sweeps through dataopts.batchsize = 100; % Take a mean gradient step over this many samplesnn = nntrain(nn, train_x, train_y, opts);[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.1, 'Too big error');%% ex4 neural net with sigmoid activation functionrand('state',0)nn = nnsetup([784 100 10]);nn.activation_function = 'sigm'; % Sigmoid activation functionnn.learningRate = 1; % Sigm require a lower learning rateopts.numepochs = 1; % Number of full sweeps through dataopts.batchsize = 100; % Take a mean gradient step over this many samplesnn = nntrain(nn, train_x, train_y, opts);[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.1, 'Too big error');%% ex5 plotting functionalityrand('state',0)nn = nnsetup([784 20 10]);opts.numepochs = 5; % Number of full sweeps through datann.output = 'softmax'; % use softmax outputopts.batchsize = 1000; % Take a mean gradient step over this many samplesopts.plot = 1; % enable plottingnn = nntrain(nn, train_x, train_y, opts);[er, bad] = nntest(nn, test_x, test_y);assert(er < 0.1, 'Too big error');%% ex6 neural net with sigmoid activation and plotting of validation and training error% split training data into training and validation datavx = train_x(1:10000,:);tx = train_x(10001:end,:);vy = train_y(1:10000,:);ty = train_y(10001:end,:);rand('state',0)nn = nnsetup([784 20 10]); 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