如何绘制梯度下降法线性回归后的损失曲面？

我已经用以下Python代码实现了基于梯度下降的线性回归，得到了最优斜率m和截距c。现在想绘制损失曲面，要求以斜率为X轴、截距为Y轴、损失函数为Z轴，请问该怎么实现？

# Making the imports
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = (12.0, 9.0)

# Preprocessing Input data
data = pd.read_csv('data.csv')
X = data.iloc[:, 0]
Y = data.iloc[:, 1]
plt.scatter(X, Y)
# Building the model
m = 0
c = 0

L = 0.0001  # The learning Rate
epochs = 1000  # The number of iterations to perform gradient descent

n = float(len(X)) # Number of elements in X

# Performing Gradient Descent 
for i in range(epochs): 
    Y_pred = m*X + c  # The current predicted value of Y
    D_m = (-2/n) * sum(X * (Y - Y_pred))  # Derivative wrt m
    D_c = (-2/n) * sum(Y - Y_pred)  # Derivative wrt c
    m = m - L * D_m  # Update m
    c = c - L * D_c  # Update c
    
print (m, c)

# Making predictions
Y_pred = m*X + c

plt.scatter(X, Y) 
plt.plot([min(X), max(X)], [min(Y_pred), max(Y_pred)], color='red')  # regression line
plt.show()

实现方案 #

要绘制损失曲面，核心是生成覆盖m和c合理范围的网格点，计算每个点对应的损失值，再通过3D绘图工具可视化。以下是具体步骤和代码：

1. 核心思路 #

• 定义损失函数：用线性回归的均方误差(MSE)作为损失指标，公式为：
  $$J(m,c) = \frac{1}{n}\sum_{i=1}^n (Y_i - (mX_i + c))^2$$
• 生成参数网格：围绕最终得到的最优m和c，取合理范围生成密集网格点
• 批量计算损失：遍历网格点，计算每个点对应的损失值
• 3D可视化：用matplotlib的3D模块绘制曲面，可额外标记最优解和梯度下降路径

2. 完整代码示例 #

在原代码基础上添加以下内容：

# 导入3D绘图模块
from mpl_toolkits.mplot3d import Axes3D

# 定义均方误差损失函数
def compute_loss(m, c, X, Y):
    n = len(X)
    Y_pred = m * X + c
    return np.sum((Y - Y_pred)**2) / n

# 生成m和c的网格范围（可根据实际最优值调整范围）
m_range = np.linspace(m - 2, m + 2, 100)
c_range = np.linspace(c - 20, c + 20, 100)
M, C = np.meshgrid(m_range, c_range)

# 计算所有网格点的损失值
Z = np.array([compute_loss(m_val, c_val, X, Y) for m_val, c_val in zip(np.ravel(M), np.ravel(C))])
Z = Z.reshape(M.shape)

# 创建3D绘图
fig = plt.figure(figsize=(12, 8))
ax = fig.add_subplot(111, projection='3d')

# 绘制损失曲面
ax.plot_surface(M, C, Z, cmap='viridis', alpha=0.7)

# 标记最优解位置
ax.scatter(m, c, compute_loss(m, c, X, Y), color='red', s=100, label='最优解')

# 设置坐标轴与标题
ax.set_xlabel('斜率 m')
ax.set_ylabel('截距 c')
ax.set_zlabel('损失 J(m,c)')
ax.set_title('线性回归损失曲面')
ax.legend()

plt.show()

3. 额外优化：添加梯度下降路径 #

若想展示梯度下降过程中参数的变化轨迹，可在原梯度下降循环中记录每一步的m、c和损失值，再在3D图中绘制：

修改梯度下降循环代码：

# 初始化列表保存迭代过程
m_history = []
c_history = []
loss_history = []

# Performing Gradient Descent 
for i in range(epochs): 
    Y_pred = m*X + c  # The current predicted value of Y
    D_m = (-2/n) * sum(X * (Y - Y_pred))  # Derivative wrt m
    D_c = (-2/n) * sum(Y - Y_pred)  # Derivative wrt c
    m = m - L * D_m  # Update m
    c = c - L * D_c  # Update c
    # 记录每一步的参数和损失
    m_history.append(m)
    c_history.append(c)
    loss_history.append(compute_loss(m, c, X, Y))
    
print (m, c)

在3D绘图部分添加路径绘制：

# 绘制梯度下降路径
ax.plot(m_history, c_history, loss_history, color='orange', linewidth=2, label='梯度下降路径')
ax.legend()

内容的提问来源于stack exchange，提问作者방준호