牛客题解 | 具有反向传播的单神经元

题目

题目链接

具有反向传播的单神经元（Single Neuron with Backpropagation）是神经网络中的最常见的基本单元。
本算法的步骤如下：

1. 初始化权重和偏置
   \[w = \begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix} \]
   \[b = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix} \]

2. 前向传播，计算预测值
   \[z = w \cdot x + b \]
   \[predictions = sigmoid(z) \]

3. 反向传播，计算梯度
   \[errors = predictions - labels \]
   \[weight_gradients = (2/len(labels)) * np.dot(features.T, errors * predictions * (1 - predictions)) \]
   \[bias_gradient = (2/len(labels)) * np.sum(errors * predictions * (1 - predictions)) \]
   本题选择的均方误差作为损失函数，公式为
   \[mse = \frac{1}{n} \sum_{i=1}^{n} (predictions_i - labels_i)^2 \]
   因此，梯度计算公式为
   \[weight_gradients = (2/len(labels)) * np.dot(features.T, errors * predictions * (1 - predictions)) \]
   \[bias_gradient = (2/len(labels)) * np.sum(errors * predictions * (1 - predictions)) \]

4. 更新权重和偏置
   \[w = w - learning_rate * weight_gradients \]
   \[b = b - learning_rate * bias_gradient \]

5. 重复步骤2-4，直到达到最大迭代次数

标准代码如下

def sigmoid(x):
    return 1 / (1 + np.exp(-x))

def train_neuron(features, labels, initial_weights, initial_bias, learning_rate, epochs):
    weights = np.array(initial_weights)
    bias = initial_bias
    features = np.array(features)
    labels = np.array(labels)
    mse_values = []

    for _ in range(epochs):
        z = np.dot(features, weights) + bias
        predictions = sigmoid(z)
        
        mse = np.mean((predictions - labels) ** 2)
        mse_values.append(round(mse, 4))

        # Gradient calculation for weights and bias
        errors = predictions - labels
        weight_gradients = (2/len(labels)) * np.dot(features.T, errors * predictions * (1 - predictions))
        bias_gradient = (2/len(labels)) * np.sum(errors * predictions * (1 - predictions))
        
        # Update weights and bias
        weights -= learning_rate * weight_gradients
        bias -= learning_rate * bias_gradient

        # Round weights and bias for output
        updated_weights = np.round(weights, 4)
        updated_bias = round(bias, 4)

    return updated_weights.tolist(), updated_bias, mse_values