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本文主要包含以下内容：

1. 推导神经网络的误差反向传播过程
2. 使用numpy编写简单的神经网络，并使用iris数据集和california_housing数据集分别进行分类和回归任务，最终将训练过程可视化。

1. BP算法的推导过程

1.1 导入

前向传播和反向传播的总体过程。

神经网络的直接输出记为$Z^{[l]}$，表示激活前的输出，激活后的输出记为$A$。

第一个图像是神经网络的前向传递和反向传播的过程，第二个图像用于解释中间的变量关系，第三个图像是前向和后向过程的计算图，方便进行推导，但是第三个图左下角的$A^{[l-2]}$有错误，应该是$A^{[l-1]}$。

1.2 符号表

为了方便进行推导，有必要对各个符号进行介绍

符号表

记号	含义
$n_l$	第$l$层神经元个数
$f_l(\cdot )$	第$l$层神经元的激活函数
${\mathbf{W}}^l\in {\mathbb{R}}^{n_{l-1}\times n_l}$	第$l-1$层到第$l$层的权重矩阵
${\mathbf{b}}^l\in {\mathbb{R}}^{n_l}$	第$l-1$层到第$l$层的偏置
${\mathbf{Z}}^l\in {\mathbb{R}}^{n_l}$	第$l$层的净输出，没有经过激活的输出
${\mathbf{A}}^l\in {\mathbb{R}}^{n_l}$	第$l$层经过激活函数的输出，$A^0=X$

深层的神经网络都是由一个一个单层网络堆叠起来的，于是我们可以写出神经网络最基本的结构，然后进行堆叠得到深层的神经网络。

于是，我们可以开始编写代码，通过一个类Layer来描述单个神经网络层

class Layer:def __init__(self, input_dim, output_dim):# 初始化参数self.W = np.random.randn(input_dim, output_dim) * 0.01self.b = np.zeros((1, output_dim))def forward(self, X):# 前向计算self.Z = np.dot(X, self.W) + self.bself.A = self.activation(self.Z)return self.Adef backward(self, dA, A_prev, activation_derivative):# 反向传播# 计算公式推导见下方m = A_prev.shape[0]self.dZ = dA * activation_derivative(self.Z)self.dW = np.dot(A_prev.T, self.dZ) / mself.db = np.sum(self.dZ, axis=0, keepdims=True) / mdA_prev = np.dot(self.dZ, self.W.T)return dA_prevdef update_parameters(self, learning_rate):# 参数更新self.W -= learning_rate * self.dWself.b -= learning_rate * self.db# 带有ReLU激活函数的Layer
class ReLULayer(Layer):def activation(self, Z):return np.maximum(0, Z)def activation_derivative(self, Z):return (Z > 0).astype(float)# 带有Softmax激活函数（主要用于分类）的Layer
class SoftmaxLayer(Layer):def activation(self, Z):exp_z = np.exp(Z - np.max(Z, axis=1, keepdims=True))return exp_z / np.sum(exp_z, axis=1, keepdims=True)def activation_derivative(self, Z):# Softmax derivative is more complex, not directly used in this form.return np.ones_like(Z)

1.3 推导过程

权重更新的核心在于计算得到self.dW和self.db，同时，为了将梯度信息不断回传，需要backward函数返回梯度信息dA_prev。

需要用到的公式
$$\begin{gathered} Z^l=W^l{A}^{l-1}+b^l \\ {A}^l=f(Z^l) \\ \frac{dZ}{dW}=(A^{l-1}{)}^T \\ \frac{dZ}{db}=1 \end{gathered}$$
解释：

从上方计算图右侧的反向传播过程可以看到，来自于上一层的梯度信息dA经过dZ之后直接传递到db，也经过dU之后传递到dW，于是我们可以得到dW和db的梯度计算公式如下：
$$\begin{array}{rl}dW& =dA\cdot \frac{dA}{dZ}\cdot \frac{dZ}{dW}\\ & =dA\cdot f^{\prime }(dZ)\cdot A_{prev}^T\end{array}$$
其中，$f(\cdot )$是激活函数，$f^{\prime }(\cdot )$是激活函数的导数，$A_{prev}^T$是当前层上一层激活输出的转置。

同理，可以得到
$$\begin{array}{rl}db& =dA\cdot \frac{dA}{dZ}\cdot \frac{dZ}{db}\\ & =dA\cdot f^{\prime }(dZ)\end{array}$$
需要仅需往前传递的梯度信息：
$$\begin{array}{rl}d{A}_{prev}& =dA\cdot \frac{dA}{dZ}\cdot \frac{dZ}{A_{prev}}\\ & =dA\cdot f^{\prime }(dZ)\cdot W^T\end{array}$$
所以，经过上述推导，我们可以将梯度信息从后向前传递。

分类损失函数

分类过程的损失函数最常见的就是交叉熵损失了，用来计算模型输出分布和真实值之间的差异，其公式如下：
$$L=-\frac{1}{N}\sum _{i=1}^N\sum _{j=1}^C{y}_{ij}log(\widehat{y_{ij}})$$
其中，$N$表示样本个数，$C$表示类别个数，$y_{ij}$表示第i个样本的第j个位置的值，由于使用了独热编码，因此每一行仅有1个数字是1，其余全部是0，所以，交叉熵损失每次需要对第$i$个样本不为0的位置的概率计算对数，然后将所有所有概率取平均值的负数。

交叉熵损失函数的梯度可以简洁地使用如下符号表示：
$${\nabla }_{z}L=\hat{\mathbf{y}}-\mathbf{y}$$

回归损失函数

均方差损失函数由于良好的性能被回归问题广泛采用，其公式如下：
$$L=\frac{1}{N}\sum _{i=1}^N(y_i-\widehat{y_i}{)}^2$$
向量形式：
$$L=\frac{1}{N}||\mathbf{y}-\hat{\mathbf{y}}|{|}_2^2$$
梯度计算：
$${\nabla }_{\hat{y}}L=\frac{2}{N}(\hat{\mathbf{y}}-\mathbf{y})$$

2 代码

2.1 分类代码

import numpy as np
from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import OneHotEncoder
import matplotlib.pyplot as pltclass Layer:def __init__(self, input_dim, output_dim):self.W = np.random.randn(input_dim, output_dim) * 0.01self.b = np.zeros((1, output_dim))def forward(self, X):self.Z = np.dot(X, self.W) + self.b     # 激活前的输出self.A = self.activation(self.Z)        # 激活后的输出return self.Adef backward(self, dA, A_prev, activation_derivative):# 注意：梯度信息是反向传递的: l+1 --> l --> l-1# A_prev是第l-1层的输出，也即A^{l-1}# dA是第l+1的层反向传递的梯度信息# activation_derivative是激活函数的导数# dA_prev是传递给第l-1层的梯度信息m = A_prev.shape[0]self.dZ = dA * activation_derivative(self.Z)self.dW = np.dot(A_prev.T, self.dZ) / mself.db = np.sum(self.dZ, axis=0, keepdims=True) / mdA_prev = np.dot(self.dZ, self.W.T) # 反向传递给下一层的梯度信息return dA_prevdef update_parameters(self, learning_rate):self.W -= learning_rate * self.dWself.b -= learning_rate * self.dbclass ReLULayer(Layer):def activation(self, Z):return np.maximum(0, Z)def activation_derivative(self, Z):return (Z > 0).astype(float)class SoftmaxLayer(Layer):def activation(self, Z):exp_z = np.exp(Z - np.max(Z, axis=1, keepdims=True))return exp_z / np.sum(exp_z, axis=1, keepdims=True)def activation_derivative(self, Z):# Softmax derivative is more complex, not directly used in this form.return np.ones_like(Z)class NeuralNetwork:def __init__(self, layer_dims, learning_rate=0.01):self.layers = []self.learning_rate = learning_ratefor i in range(len(layer_dims) - 2):self.layers.append(ReLULayer(layer_dims[i], layer_dims[i + 1]))self.layers.append(SoftmaxLayer(layer_dims[-2], layer_dims[-1]))def cross_entropy_loss(self, y_true, y_pred):n_samples = y_true.shape[0]y_pred_clipped = np.clip(y_pred, 1e-12, 1 - 1e-12)return -np.sum(y_true * np.log(y_pred_clipped)) / n_samplesdef accuracy(self, y_true, y_pred):y_true_labels = np.argmax(y_true, axis=1)y_pred_labels = np.argmax(y_pred, axis=1)return np.mean(y_true_labels == y_pred_labels)def train(self, X, y, epochs):loss_history = []for epoch in range(epochs):A = X# Forward propagationcache = [A]for layer in self.layers:A = layer.forward(A)cache.append(A)loss = self.cross_entropy_loss(y, A)loss_history.append(loss)# Backward propagation# 损失函数求导dA = A - yfor i in reversed(range(len(self.layers))):layer = self.layers[i]A_prev = cache[i]dA = layer.backward(dA, A_prev, layer.activation_derivative)# Update parametersfor layer in self.layers:layer.update_parameters(self.learning_rate)if (epoch + 1) % 100 == 0:print(f'Epoch {epoch + 1}/{epochs}, Loss: {loss:.4f}')return loss_historydef predict(self, X):A = Xfor layer in self.layers:A = layer.forward(A)return A# 导入数据
iris = load_iris()
X = iris.data
y = iris.target.reshape(-1, 1)# One hot encoding
encoder = OneHotEncoder(sparse_output=False)
y = encoder.fit_transform(y)# 分割数据
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)# 定义并训练神经网络
layer_dims = [X_train.shape[1], 100, 20, y_train.shape[1]]  # Example with 2 hidden layers
learning_rate = 0.01
epochs = 5000nn = NeuralNetwork(layer_dims, learning_rate)
loss_history = nn.train(X_train, y_train, epochs)# 预测和评估
train_predictions = nn.predict(X_train)
test_predictions = nn.predict(X_test)train_acc = nn.accuracy(y_train, train_predictions)
test_acc = nn.accuracy(y_test, test_predictions)print(f'Training Accuracy: {train_acc:.4f}')
print(f'Test Accuracy: {test_acc:.4f}')# 绘制损失曲线
plt.plot(loss_history)
plt.xlabel('Epochs')
plt.ylabel('Loss')
plt.title('Loss Curve')
plt.show()

输出

Epoch 100/1000, Loss: 1.0983
Epoch 200/1000, Loss: 1.0980
Epoch 300/1000, Loss: 1.0975
Epoch 400/1000, Loss: 1.0960
Epoch 500/1000, Loss: 1.0891
Epoch 600/1000, Loss: 1.0119
Epoch 700/1000, Loss: 0.6284
Epoch 800/1000, Loss: 0.3711
Epoch 900/1000, Loss: 0.2117
Epoch 1000/1000, Loss: 0.1290
Training Accuracy: 0.9833
Test Accuracy: 1.0000

可以看到经过1000轮迭代，最终的准确率到达100%。

回归代码

import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
import matplotlib.pyplot as plt
from sklearn.datasets import fetch_california_housingclass Layer:def __init__(self, input_dim, output_dim):self.W = np.random.randn(input_dim, output_dim) * 0.01self.b = np.zeros((1, output_dim))def forward(self, X):self.Z = np.dot(X, self.W) + self.bself.A = self.activation(self.Z)return self.Adef backward(self, dA, X, activation_derivative):m = X.shape[0]self.dZ = dA * activation_derivative(self.Z)self.dW = np.dot(X.T, self.dZ) / mself.db = np.sum(self.dZ, axis=0, keepdims=True) / mdA_prev = np.dot(self.dZ, self.W.T)return dA_prevdef update_parameters(self, learning_rate):self.W -= learning_rate * self.dWself.b -= learning_rate * self.dbclass ReLULayer(Layer):def activation(self, Z):return np.maximum(0, Z)def activation_derivative(self, Z):return (Z > 0).astype(float)class LinearLayer(Layer):def activation(self, Z):return Zdef activation_derivative(self, Z):return np.ones_like(Z)class NeuralNetwork:def __init__(self, layer_dims, learning_rate=0.01):self.layers = []self.learning_rate = learning_ratefor i in range(len(layer_dims) - 2):self.layers.append(ReLULayer(layer_dims[i], layer_dims[i + 1]))self.layers.append(LinearLayer(layer_dims[-2], layer_dims[-1]))def mean_squared_error(self, y_true, y_pred):return np.mean((y_true - y_pred) ** 2)def train(self, X, y, epochs):loss_history = []for epoch in range(epochs):A = X# Forward propagationcache = [A]for layer in self.layers:A = layer.forward(A)cache.append(A)loss = self.mean_squared_error(y, A)loss_history.append(loss)# Backward propagation# 损失函数求导dA = -(y - A)for i in reversed(range(len(self.layers))):layer = self.layers[i]A_prev = cache[i]dA = layer.backward(dA, A_prev, layer.activation_derivative)# Update parametersfor layer in self.layers:layer.update_parameters(self.learning_rate)if (epoch + 1) % 100 == 0:print(f'Epoch {epoch + 1}/{epochs}, Loss: {loss:.4f}')return loss_historydef predict(self, X):A = Xfor layer in self.layers:A = layer.forward(A)return Ahousing = fetch_california_housing()# 导入数据
X = housing.data
y = housing.target.reshape(-1, 1)# 标准化
scaler_X = StandardScaler()
scaler_y = StandardScaler()
X = scaler_X.fit_transform(X)
y = scaler_y.fit_transform(y)# 分割数据
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)# 定义并训练神经网络
layer_dims = [X_train.shape[1], 50, 5, 1]  # Example with 2 hidden layers
learning_rate = 0.8
epochs = 1000nn = NeuralNetwork(layer_dims, learning_rate)
loss_history = nn.train(X_train, y_train, epochs)# 预测和评估
train_predictions = nn.predict(X_train)
test_predictions = nn.predict(X_test)train_mse = nn.mean_squared_error(y_train, train_predictions)
test_mse = nn.mean_squared_error(y_test, test_predictions)print(f'Training MSE: {train_mse:.4f}')
print(f'Test MSE: {test_mse:.4f}')# 绘制损失曲线
plt.plot(loss_history)
plt.xlabel('Epochs')
plt.ylabel('Loss')
plt.title('Loss Curve')
plt.show()

输出

Epoch 100/1000, Loss: 1.0038
Epoch 200/1000, Loss: 0.9943
Epoch 300/1000, Loss: 0.3497
Epoch 400/1000, Loss: 0.3306
Epoch 500/1000, Loss: 0.3326
Epoch 600/1000, Loss: 0.3206
Epoch 700/1000, Loss: 0.3125
Epoch 800/1000, Loss: 0.3057
Epoch 900/1000, Loss: 0.2999
Epoch 1000/1000, Loss: 0.2958
Training MSE: 0.2992
Test MSE: 0.3071