3. 您的第一次优化#
备注
本教程将逐步引导您使用 Bgolearn 完成第一次贝叶斯优化。
3.1. 问题设置#
让我们优化一个简单的一维函数来理解工作流程。我们将找到以下函数的最小值：
\[f(x) = (x-2)^2 + 0.1 \sin(10x)\]
该函数在 \(x ≈ 1.75\) 附近有一个全局最小值（正弦项使其从 \(x = 2\) 略有偏移），并包含振荡以使其更有趣。
3.2. 步骤 1：导入库#
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from Bgolearn import BGOsampling
# Set random seed for reproducibility
np.random.seed(42)
3.3. 步骤 2：定义目标函数#
def objective_function(x):
"""
Our test function to minimize: f(x) = (x-2)^2 + 0.1*sin(10x)
Global minimum at x ≈ 1.77 (sine term shifts it from x=2).
"""
return (x - 2)**2 + 0.1 * np.sin(10 * x)
# Visualize the function
x_plot = np.linspace(0, 4, 200)
y_plot = [objective_function(x) for x in x_plot]
plt.figure(figsize=(10, 6))
plt.plot(x_plot, y_plot, 'b-', linewidth=2, label='Objective function')
plt.axvline(x=1.75, color='red', linestyle='--', alpha=0.7, label='True minimum')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title('Objective Function to Minimize')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()
print(f"True minimum: f(1.75) = {objective_function(1.75):.4f}")
3.4. 步骤 3：生成初始数据#
在实际优化中，我们从几个初始实验开始：
# Generate initial training data
n_initial = 5
X_train = np.random.uniform(0, 4, n_initial).reshape(-1, 1)
y_train = np.array([objective_function(x[0]) for x in X_train])
# Add some experimental noise
noise_level = 0.02
y_train += noise_level * np.random.randn(len(y_train))
# Convert to pandas (Bgolearn's preferred format)
X_train_df = pd.DataFrame(X_train, columns=['x'])
y_train_series = pd.Series(y_train)
print(f"Initial data: {len(X_train_df)} points")
print(f"Current best: f({X_train_df.iloc[y_train_series.argmin()]['x']:.3f}) = {y_train_series.min():.4f}")
3.5. 步骤 4：定义候选点#
创建搜索空间 - 我们可能下一步采样的位置：
# Generate candidate points
X_candidates = np.linspace(0, 4, 100).reshape(-1, 1)
X_candidates_df = pd.DataFrame(X_candidates, columns=['x'])
print(f"Search space: {len(X_candidates_df)} candidate points")
3.6. 步骤 5：拟合 Bgolearn 模型#
# Initialize and fit Bgolearn
optimizer = BGOsampling.Bgolearn()
model = optimizer.fit(
data_matrix=X_train_df,
Measured_response=y_train_series,
virtual_samples=X_candidates_df,
min_search=True, # We want to minimize
CV_test=3, # 3-fold cross-validation
Normalize=True # Normalize features
)
print("Bgolearn model fitted successfully!")
3.7. 步骤 6：可视化模型#
# Get model predictions
mean_pred = model.virtual_samples_mean
std_pred = model.virtual_samples_std
# Create visualization
plt.figure(figsize=(12, 8))
# Plot true function
plt.plot(x_plot, y_plot, 'b-', linewidth=2, alpha=0.7, label='True function')
# Plot GP mean prediction
plt.plot(X_candidates, mean_pred, 'g-', linewidth=2, label='GP mean')
# Plot uncertainty (±2σ)
plt.fill_between(X_candidates.flatten(),
mean_pred - 2*std_pred,
mean_pred + 2*std_pred,
alpha=0.3, color='green', label='GP uncertainty (±2σ)')
# Plot training data
plt.scatter(X_train, y_train, c='red', s=100, zorder=5,
edgecolors='black', linewidth=1, label='Training data')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title('Gaussian Process Model')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()
3.8. 步骤 7：使用采集函数#
找到下一个最佳采样点：
# Expected Improvement
ei_values, ei_point = model.EI()
print(f"Expected Improvement recommends: x = {ei_point[0][0]:.3f}")
# Upper Confidence Bound
ucb_values, ucb_point = model.UCB(alpha=2.0)
print(f"Upper Confidence Bound recommends: x = {ucb_point[0][0]:.3f}")
# Probability of Improvement
poi_values, poi_point = model.PoI(tao=0.01)
print(f"Probability of Improvement recommends: x = {poi_point[0][0]:.3f}")
3.9. 步骤 8：可视化采集函数#
# Create comprehensive visualization
fig, axes = plt.subplots(2, 2, figsize=(15, 10))
# Plot 1: GP Model
axes[0,0].plot(x_plot, y_plot, 'b-', linewidth=2, alpha=0.7, label='True function')
axes[0,0].plot(X_candidates, mean_pred, 'g-', linewidth=2, label='GP mean')
axes[0,0].fill_between(X_candidates.flatten(),
mean_pred - 2*std_pred, mean_pred + 2*std_pred,
alpha=0.3, color='green', label='GP ±2σ')
axes[0,0].scatter(X_train, y_train, c='red', s=80, zorder=5,
edgecolors='black', label='Training data')
axes[0,0].set_title('Gaussian Process Model')
axes[0,0].set_xlabel('x')
axes[0,0].set_ylabel('f(x)')
axes[0,0].legend()
axes[0,0].grid(True, alpha=0.3)
# Plot 2: Expected Improvement
axes[0,1].plot(X_candidates, ei_values, 'purple', linewidth=2)
axes[0,1].axvline(x=ei_point[0][0], color='red', linestyle='--',
label=f'EI max (x={ei_point[0][0]:.3f})')
axes[0,1].set_title('Expected Improvement')
axes[0,1].set_xlabel('x')
axes[0,1].set_ylabel('EI(x)')
axes[0,1].legend()
axes[0,1].grid(True, alpha=0.3)
# Plot 3: Upper Confidence Bound
axes[1,0].plot(X_candidates, ucb_values, 'orange', linewidth=2)
axes[1,0].axvline(x=ucb_point[0][0], color='red', linestyle='--',
label=f'UCB max (x={ucb_point[0][0]:.3f})')
axes[1,0].set_title('Upper Confidence Bound')
axes[1,0].set_xlabel('x')
axes[1,0].set_ylabel('UCB(x)')
axes[1,0].legend()
axes[1,0].grid(True, alpha=0.3)
# Plot 4: Probability of Improvement
axes[1,1].plot(X_candidates, poi_values, 'brown', linewidth=2)
axes[1,1].axvline(x=poi_point[0][0], color='red', linestyle='--',
label=f'PoI max (x={poi_point[0][0]:.3f})')
axes[1,1].set_title('Probability of Improvement')
axes[1,1].set_xlabel('x')
axes[1,1].set_ylabel('PoI(x)')
axes[1,1].legend()
axes[1,1].grid(True, alpha=0.3)
plt.tight_layout()
plt.show()
3.10. 步骤 9：模拟下一次实验#
# Simulate conducting the experiment recommended by EI
next_x = ei_point[0][0]
true_value = objective_function(next_x)
measured_value = true_value + noise_level * np.random.randn()
print(f"\n🔬 Next Experiment:")
print(f"Recommended point: x = {next_x:.3f}")
print(f"True value: f({next_x:.3f}) = {true_value:.4f}")
print(f"Measured value (with noise): {measured_value:.4f}")
print(f"Current best: {y_train_series.min():.4f}")
if measured_value < y_train_series.min():
improvement = y_train_series.min() - measured_value
print(f"Improvement found! Δf = {improvement:.4f}")
else:
print("No improvement this iteration (normal in optimization)")
3.11. 步骤 10：完整优化循环#
def run_optimization_loop(n_iterations=10):
"""Run a complete optimization loop."""
# Start with initial data
X_current = X_train_df.copy()
y_current = y_train_series.copy()
history = {
'iteration': [],
'x_new': [],
'y_new': [],
'best_so_far': []
}
for i in range(n_iterations):
print(f"\n--- Iteration {i+1} ---")
# Fit model
model = optimizer.fit(
data_matrix=X_current,
Measured_response=y_current,
virtual_samples=X_candidates_df,
min_search=True,
Normalize=True
)
# Get next point using EI
ei_values, ei_point = model.EI()
next_x = ei_point[0][0]
# Evaluate function (simulate experiment)
true_y = objective_function(next_x)
measured_y = true_y + noise_level * np.random.randn()
# Update data
new_row = pd.DataFrame({'x': [next_x]})
X_current = pd.concat([X_current, new_row], ignore_index=True)
y_current = pd.concat([y_current, pd.Series([measured_y])], ignore_index=True)
# Track progress
current_best = y_current.min()
history['iteration'].append(i+1)
history['x_new'].append(next_x)
history['y_new'].append(measured_y)
history['best_so_far'].append(current_best)
print(f"New point: x = {next_x:.3f}, f(x) = {measured_y:.4f}")
print(f"Best so far: {current_best:.4f}")
return history, X_current, y_current
# Run optimization
history, X_final, y_final = run_optimization_loop(n_iterations=8)
print(f"\n Final Results:")
print(f"Best found: f({X_final.iloc[y_final.argmin()]['x']:.3f}) = {y_final.min():.4f}")
print(f"True minimum: f(1.750) = {objective_function(1.75):.4f}")
print(f"Error: {abs(y_final.min() - objective_function(1.75)):.4f}")
3.12. 步骤 11：分析结果#
# Plot optimization progress
plt.figure(figsize=(12, 5))
# Convergence plot
plt.subplot(1, 2, 1)
plt.plot(history['iteration'], history['best_so_far'], 'bo-', linewidth=2, markersize=8)
plt.axhline(y=objective_function(1.75), color='red', linestyle='--',
label=f'True minimum ({objective_function(1.75):.4f})')
plt.xlabel('Iteration')
plt.ylabel('Best Value Found')
plt.title('Optimization Convergence')
plt.legend()
plt.grid(True, alpha=0.3)
# Sampling locations
plt.subplot(1, 2, 2)
plt.plot(x_plot, y_plot, 'b-', linewidth=2, alpha=0.7, label='True function')
plt.scatter(X_train, y_train, c='red', s=100, marker='o',
edgecolors='black', label='Initial points', zorder=5)
plt.scatter(history['x_new'], history['y_new'], c='green', s=100, marker='s',
edgecolors='black', label='BO points', zorder=5)
plt.axvline(x=1.75, color='red', linestyle='--', alpha=0.7, label='True minimum')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title('Sampling Locations')
plt.legend()
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()
3.13. 关键要点#
我们学到了什么：
贝叶斯优化是迭代的 - 每个新点都改进我们的模型
采集函数平衡探索与利用 - EI 找到了良好的平衡
不确定性指导采样 - 高不确定性的点被探索
需要的评估次数少 - 我们只用几次实验就找到了最小值
下一步：
尝试不同的采集函数：采集函数指南
探索材料应用：使用 Bgolearn 进行材料发现
了解多目标优化：MultiBgolearn：多目标贝叶斯全局优化
尝试 GUI 界面：BgoFace：Bgolearn 的图形用户界面
小技巧
这个简单的示例演示了核心的 Bgolearn 工作流程。相同的原理适用于具有多个目标和约束的复杂材料发现问题！