反向传播算法矩阵表示法的归纳证明
An induction proof of the backpropagation algorithm in matrix notation
摘要：
反向传播（BP）是神经网络的当代深度学习化身的核心组件。简而言之，BP是一种利用神经网络的计算架构来在神经网络参数优化期间有效地评估成本函数的梯度的算法。BP算法的有效性依赖于将多元链式规则应用于神经网络的计算结构及其相关的目标函数。对深度学习理论的介绍通常以矩阵形式呈现神经网络的计算架构，但避免在矩阵微分框架中对BP进行并行公式化和证明。这给深度学习的理论和教学带来了一些缺点。在这项工作中，我们克服了这些限制，提供了一个完整的归纳证明的BP算法的矩阵符号。具体地说，我们将BP算法引入矩阵微分的框架中，引入仿射线性势函数，以归纳形式证明了BP算法的有效性，并给出了矩阵形式BP算法的计算机代码实现。
英文摘要：
Backpropagation (BP) is a core component of the contemporary deep learning incarnation of neural networks. Briefly, BP is an algorithm that exploits the computational architecture of neural networks to efficiently evaluate the gradient of a cost function during neural network parameter optimization. The validity of BP rests on the application of a multivariate chain rule to the computational architecture of neural networks and their associated objective functions. Introductions to deep learning theory commonly present the computational architecture of neural networks in matrix form, but eschew a parallel formulation and justification of BP in the framework of matrix differential calculus. This entails several drawbacks for the theory and didactics of deep learning. In this work, we overcome these limitations by providing a full induction proof of the BP algorithm in matrix notation. Specifically, we situate the BP algorithm in the framework of matrix differential calculus, encompass affine-linear potential functions, prove the validity of the BP algorithm in inductive form, and exemplify the implementation of the matrix form BP algorithm in computer code.
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