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利用梯度下降求一个凸函数的最小值

news 2026/5/12 1:58:01

一、流程

求函数的导数
更新x，x_new = x - learning_rate * gradient
检查收敛性： |gradient| < tolerance

二、为什么沿着梯度方向就一定能达到最值？

需要用泰勒展开式进行推导。过程省略

import numpy as np
import matplotlib.pyplot as pltdef f(x):"""目标函数: f(x) = (x+3)^2 + 1"""return (x + 3) ** 2 + 1def grad_f(x):"""梯度(导数): f'(x) = 2(x+3)"""return 2 * (x + 3)def gradient_descent(learning_rate=0.1, max_iters=100, tolerance=1e-6, x0=0):"""梯度下降算法参数:learning_rate: 学习率max_iters: 最大迭代次数tolerance: 收敛容差x0: 初始点返回:x_history: x的历史值f_history: 函数值的历史"""x = x0x_history = [x]f_history = [f(x)]print(f"初始值: x0 = {x0:.4f}, f(x0) = {f(x0):.4f}")print("-" * 50)for i in range(max_iters):# 计算梯度gradient = grad_f(x)# 更新xx_new = x - learning_rate * gradient# 记录历史
        x_history.append(x_new)f_history.append(f(x_new))# 打印迭代信息if i < 10 or i % 10 == 9 or i == max_iters - 1:print(f"迭代 {i + 1:3d}: x = {x_new:8.6f}, f(x) = {f(x_new):8.6f}, 梯度 = {gradient:8.6f}")# 检查收敛if abs(gradient) < tolerance:print(f"\n在 {i + 1} 次迭代后收敛!")breakx = x_newreturn x_history, f_historydef plot_results(x_history, f_history):"""绘制优化过程"""fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))# 绘制函数曲线和优化路径x_vals = np.linspace(-5, 1, 100)y_vals = f(x_vals)ax1.plot(x_vals, y_vals, 'b-', label='f(x) = (x+3)² + 1', linewidth=2)ax1.plot(x_history, f_history, 'ro-', label='优化路径', markersize=4)ax1.set_xlabel('x')ax1.set_ylabel('f(x)')ax1.set_title('梯度下降优化过程')ax1.legend()ax1.grid(True)# 绘制函数值收敛情况ax2.semilogy(range(len(f_history)), f_history, 'g-o', markersize=4)ax2.set_xlabel('迭代次数')ax2.set_ylabel('f(x) (对数坐标)')ax2.set_title('函数值收敛情况')ax2.grid(True)plt.tight_layout()plt.show()# 运行梯度下降
print("梯度下降法求解 f(x) = (x+3)² + 1 的最小值")
print("=" * 50)# 不同学习率的比较
learning_rates = [0.01, 0.1, 0.3]
results = {}for lr in learning_rates:print(f"\n学习率 α = {lr}:")x_hist, f_hist = gradient_descent(learning_rate=lr, x0=0)results[lr] = (x_hist, f_hist)print(f"最终结果: x = {x_hist[-1]:.6f}, f(x) = {f_hist[-1]:.6f}")# 绘制最后一个学习率的结果
plot_results(results[0.1][0], results[0.1][1])# 比较不同学习率的收敛速度
print("\n" + "=" * 50)
print("不同学习率的收敛情况比较:")
print("=" * 50)for lr in learning_rates:x_hist, f_hist = results[lr]iterations = len(x_hist) - 1final_x = x_hist[-1]final_f = f_hist[-1]print(f"α = {lr:.2f}: {iterations:2d} 次迭代, x = {final_x:7.4f}, f(x) = {final_f:7.4f}")# 理论最小值
print(f"\n理论最小值: x = -3.0000, f(x) = 1.0000")