AI基础 | 前向传播与反向传播(2)

# 接续前向传播的代码和结果y_true = 2.5y_pred = result['y_pred']a1 = result['a1']z1 = result['z1']x = x_inputprint(f"真实值 y_true = {y_true}")print(f"预测值 y_pred = {y_pred:.4f}")print(f"初始损失 Loss = {(y_true - y_pred)**2:.4f}")print("\n--- 开始反向传播计算梯度 ---")# 1. 计算损失对y_pred的梯度dL_dy_pred = -2 * (y_true - y_pred)print(f"梯度 ∂L/∂y_pred = -2*(y_true - y_pred) = {dL_dy_pred:.4f}")# 2. 计算输出层参数 w2, b2 的梯度dL_dw2 = dL_dy_pred * a1dL_db2 = dL_dy_pred * 1print(f"梯度 ∂L/∂w2 = (∂L/∂y_pred) * a1 = {dL_dw2:.4f}")print(f"梯度 ∂L/∂b2 = (∂L/∂y_pred) * 1 = {dL_db2:.4f}")# 3. 计算损失对隐藏层输出a1的梯度dL_da1 = dL_dy_pred * w2print(f"梯度 ∂L/∂a1 = (∂L/∂y_pred) * w2 = {dL_da1:.4f}")# 4. 计算Sigmoid函数的导数在z1处的值def sigmoid_derivative(x):    """Sigmoid函数的导数"""    s = sigmoid(x)    return s * (1 - s)sigma_prime_z1 = sigmoid_derivative(z1)print(f"Sigmoid导数 σ'(z1) = σ(z1)*(1-σ(z1)) = {sigma_prime_z1:.4f}")# 5. 计算隐藏层参数 w1, b1 的梯度dL_dw1 = dL_da1 * sigma_prime_z1 * xdL_db1 = dL_da1 * sigma_prime_z1 * 1print(f"梯度 ∂L/∂w1 = (∂L/∂a1) * σ'(z1) * x = {dL_dw1:.4f}")print(f"梯度 ∂L/∂b1 = (∂L/∂a1) * σ'(z1) * 1 = {dL_db1:.4f}")

真实值 y_true = 2.5预测值 y_pred = 1.9899初始损失 Loss = 0.2602--- 开始反向传播计算梯度 ---梯度 ∂L/∂y_pred = -2*(y_true - y_pred) = -1.0202梯度 ∂L/∂w2 = (∂L/∂y_pred) * a1 = -1.0134梯度 ∂L/∂b2 = (∂L/∂y_pred) * 1 = -1.0202梯度 ∂L/∂a1 = (∂L/∂y_pred) * w2 = -1.5303Sigmoid导数 σ'(z1) = σ(z1)*(1-σ(z1)) = 0.0066梯度 ∂L/∂w1 = (∂L/∂a1) * σ'(z1) * x = -0.0304梯度 ∂L/∂b1 = (∂L/∂a1) *σ'(z1) * 1 = -0.0101

参数 = 参数 - 学习率 * 该参数的梯度

learning_rate = 0.1# 更新参数w1_new = w1 - learning_rate * dL_dw1b1_new = b1 - learning_rate * dL_db1w2_new = w2 - learning_rate * dL_dw2b2_new = b2 - learning_rate * dL_db2print("--- 更新后的参数 ---")print(f"w1: {w1:.4f} -> {w1_new:.4f}")print(f"b1: {b1:.4f} -> {b1_new:.4f}")print(f"w2: {w2:.4f} -> {w2_new:.4f}")print(f"b2: {b2:.4f} -> {b2_new:.4f}")# 用新参数做一次前向传播，验证损失是否减小def forward_pass_with_params(x, w1, b1, w2, b2):    z1 = w1 * x + b1    a1 = sigmoid(z1)    y_pred = w2 * a1 + b2    return y_predy_pred_new = forward_pass_with_params(x_input, w1_new, b1_new, w2_new, b2_new)loss_new = (y_true - y_pred_new)**2print(f"\n用新参数预测: y_pred_new = {y_pred_new:.4f}")print(f"更新后的损失 New Loss = {loss_new:.4f}")print(f"损失变化: {loss_new - (y_true-y_pred)**2:.4f} (负值表示损失减小)")