目錄

時間-精度權衡曲線（不同模型復雜度）

訓練與驗證損失對比

帕累托前沿分析（3D）?

---

在機器學習實踐中，理解模型收斂所需時間及其與精度的關系至關重要。下面介紹如何分析模型收斂時間與精度之間的權衡，并找到最佳性價比點。

時間-精度權衡曲線（不同模型復雜度）

# 1. 時間-精度曲線（不同模型復雜度）import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import make_regression
from sklearn.ensemble import GradientBoostingRegressor
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
import time
import seaborn as sns# 設置樣式
plt.rcParams['font.sans-serif'] = ['SimHei']
plt.rcParams['axes.unicode_minus'] = False
sns.set_palette("viridis")X, y = make_regression(n_samples=10000, n_features=50, noise=0.2, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)# 模擬不同訓練策略的結果
def simulate_training(model, max_iter=100):"""模擬訓練過程并記錄時間和精度"""train_losses = []val_losses = []times = []# 初始模型model.warm_start = Truemodel.n_estimators = 1model.fit(X_train, y_train)start_time = time.time()for i in range(1, max_iter + 1):model.n_estimators = imodel.fit(X_train, y_train)# 記錄時間elapsed = time.time() - start_timetimes.append(elapsed)# 記錄訓練損失train_pred = model.predict(X_train)train_loss = mean_squared_error(y_train, train_pred)train_losses.append(train_loss)# 記錄驗證損失val_pred = model.predict(X_test)val_loss = mean_squared_error(y_test, val_pred)val_losses.append(val_loss)return np.array(times), np.array(train_losses), np.array(val_losses)model_simple = GradientBoostingRegressor(learning_rate=0.1, max_depth=3, random_state=42)
model_medium = GradientBoostingRegressor(learning_rate=0.05, max_depth=5, random_state=42)
model_complex = GradientBoostingRegressor(learning_rate=0.01, max_depth=8, random_state=42)# 模擬訓練過程
times_simple, train_loss_simple, val_loss_simple = simulate_training(model_simple, 100)
times_medium, train_loss_medium, val_loss_medium = simulate_training(model_medium, 100)
times_complex, train_loss_complex, val_loss_complex = simulate_training(model_complex, 100)plt.figure(figsize=(12, 8))
plt.plot(times_simple, val_loss_simple, 'o-', label='簡單模型 (depth=3, lr=0.1)', alpha=0.7)
plt.plot(times_medium, val_loss_medium, 's-', label='中等模型 (depth=5, lr=0.05)', alpha=0.7)
plt.plot(times_complex, val_loss_complex, 'd-', label='復雜模型 (depth=8, lr=0.01)', alpha=0.7)# 標記最佳平衡點（最大曲率點）
def find_elbow_point(times, losses):"""找到曲線的拐點（最大曲率點）"""# 計算一階導數（梯度）grad = np.gradient(losses, times)# 計算二階導數grad2 = np.gradient(grad, times)# 計算曲率curvature = np.abs(grad2) / (1 + grad ** 2) ** 1.5# 找到最大曲率點（排除前10%的點）exclude_n = max(5, int(len(curvature) * 0.1))max_idx = np.argmax(curvature[exclude_n:-5]) + exclude_nreturn times[max_idx], losses[max_idx]# 標記各模型的最佳點
elbow_time_simple, elbow_loss_simple = find_elbow_point(times_simple, val_loss_simple)
elbow_time_medium, elbow_loss_medium = find_elbow_point(times_medium, val_loss_medium)
elbow_time_complex, elbow_loss_complex = find_elbow_point(times_complex, val_loss_complex)plt.scatter(elbow_time_simple, elbow_loss_simple, s=150, c='red', marker='*', zorder=10)
plt.scatter(elbow_time_medium, elbow_loss_medium, s=150, c='red', marker='*', zorder=10)
plt.scatter(elbow_time_complex, elbow_loss_complex, s=150, c='red', marker='*', zorder=10)plt.annotate('最佳平衡點',(elbow_time_medium, elbow_loss_medium),xytext=(elbow_time_medium + 0.5, elbow_loss_medium + 5),arrowprops=dict(facecolor='red', shrink=0.05, width=2),fontsize=12, color='red')plt.xlabel('訓練時間 (秒)', fontsize=12)
plt.ylabel('驗證集 MSE', fontsize=12)
plt.title('不同復雜度模型的時間-精度權衡', fontsize=14)
plt.legend()
plt.grid(True, linestyle='--', alpha=0.7)
plt.tight_layout()
plt.show()

訓練與驗證損失對比

# 2. 訓練與驗證損失對比
import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import make_regression
from sklearn.ensemble import GradientBoostingRegressor
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
import time
import seaborn as sns# 設置樣式
plt.rcParams['font.sans-serif'] = ['SimHei']
plt.rcParams['axes.unicode_minus'] = False
sns.set_palette("viridis")X, y = make_regression(n_samples=10000, n_features=50, noise=0.2, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)# 模擬不同訓練策略的結果
def simulate_training(model, max_iter=100):"""模擬訓練過程并記錄時間和精度"""train_losses = []val_losses = []times = []# 初始模型model.warm_start = Truemodel.n_estimators = 1model.fit(X_train, y_train)start_time = time.time()for i in range(1, max_iter + 1):model.n_estimators = imodel.fit(X_train, y_train)# 記錄時間elapsed = time.time() - start_timetimes.append(elapsed)# 記錄訓練損失train_pred = model.predict(X_train)train_loss = mean_squared_error(y_train, train_pred)train_losses.append(train_loss)# 記錄驗證損失val_pred = model.predict(X_test)val_loss = mean_squared_error(y_test, val_pred)val_losses.append(val_loss)return np.array(times), np.array(train_losses), np.array(val_losses)model_simple = GradientBoostingRegressor(learning_rate=0.1, max_depth=3, random_state=42)
model_medium = GradientBoostingRegressor(learning_rate=0.05, max_depth=5, random_state=42)
model_complex = GradientBoostingRegressor(learning_rate=0.01, max_depth=8, random_state=42)# 模擬訓練過程
times_simple, train_loss_simple, val_loss_simple = simulate_training(model_simple, 100)
times_medium, train_loss_medium, val_loss_medium = simulate_training(model_medium, 100)
times_complex, train_loss_complex, val_loss_complex = simulate_training(model_complex, 100)# 標記最佳平衡點（最大曲率點）
def find_elbow_point(times, losses):"""找到曲線的拐點（最大曲率點）"""# 計算一階導數（梯度）grad = np.gradient(losses, times)# 計算二階導數grad2 = np.gradient(grad, times)# 計算曲率curvature = np.abs(grad2) / (1 + grad ** 2) ** 1.5# 找到最大曲率點（排除前10%的點）exclude_n = max(5, int(len(curvature) * 0.1))max_idx = np.argmax(curvature[exclude_n:-5]) + exclude_nreturn times[max_idx], losses[max_idx]elbow_time_simple, elbow_loss_simple = find_elbow_point(times_simple, val_loss_simple)
elbow_time_medium, elbow_loss_medium = find_elbow_point(times_medium, val_loss_medium)
elbow_time_complex, elbow_loss_complex = find_elbow_point(times_complex, val_loss_complex)plt.figure(figsize=(12, 8))
plt.plot(times_medium, train_loss_medium, 'b-', label='訓練損失')
plt.plot(times_medium, val_loss_medium, 'r-', label='驗證損失')# 標記過擬合開始點
diff = val_loss_medium - train_loss_medium
overfit_idx = np.where(diff > np.percentile(diff, 90))[0][0]
overfit_time = times_medium[overfit_idx]
overfit_val_loss = val_loss_medium[overfit_idx]plt.axvline(x=overfit_time, color='purple', linestyle='--', alpha=0.7)
plt.scatter(overfit_time, overfit_val_loss, s=150, c='purple', marker='x')plt.annotate('過擬合開始點',(overfit_time, overfit_val_loss),xytext=(overfit_time+0.5, overfit_val_loss+5),arrowprops=dict(facecolor='purple', shrink=0.05, width=2),fontsize=12, color='purple')# 標記最佳點
plt.scatter(elbow_time_medium, elbow_loss_medium, s=150, c='red', marker='*', zorder=10)
plt.annotate('最佳平衡點',(elbow_time_medium, elbow_loss_medium),xytext=(elbow_time_medium+0.5, elbow_loss_medium+2),arrowprops=dict(facecolor='red', shrink=0.05, width=2),fontsize=12, color='red')plt.xlabel('訓練時間 (秒)', fontsize=12)
plt.ylabel('MSE', fontsize=12)
plt.title('訓練與驗證損失隨時間變化', fontsize=14)
plt.legend()
plt.grid(True, linestyle='--', alpha=0.7)
plt.tight_layout()
plt.show()

帕累托前沿分析（3D）?

# 3. 帕累托前沿分析 (3D)
# 模擬不同超參數配置的結果import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import make_regression
from sklearn.ensemble import GradientBoostingRegressor
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
import time
import pandas as pd
import seaborn as sns# 設置樣式
plt.rcParams['font.sans-serif'] = ['SimHei']
plt.rcParams['axes.unicode_minus'] = False
sns.set_palette("viridis")X, y = make_regression(n_samples=10000, n_features=50, noise=0.2, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
np.random.seed(42)
results = []
for _ in range(10):depth = np.random.randint(3, 10)lr = 10 ** np.random.uniform(-2, -0.5)n_estimators = np.random.randint(20, 200)model = GradientBoostingRegressor(max_depth=depth,learning_rate=lr,n_estimators=n_estimators,random_state=42)start_time = time.time()model.fit(X_train, y_train)train_time = time.time() - start_timey_pred = model.predict(X_test)mse = mean_squared_error(y_test, y_pred)results.append({'depth': depth,'lr': lr,'n_estimators': n_estimators,'time': train_time,'mse': mse,'cost_score': 0.7 * mse + 0.3 * np.log1p(train_time)  # 自定義成本函數})# 轉換為DataFrame
df = pd.DataFrame(results)# 找到帕累托前沿點
def is_pareto_efficient(costs):"""找到帕累托最優解"""is_efficient = np.ones(costs.shape[0], dtype=bool)for i, c in enumerate(costs):if is_efficient[i]:# 所有成本都小于等于當前點的點is_efficient[is_efficient] = np.any(costs[is_efficient] < c, axis=1)is_efficient[i] = True  # 保持當前點return is_efficientcosts = df[['mse', 'time']].values
pareto_mask = is_pareto_efficient(costs)
pareto_points = df[pareto_mask]# 3D可視化
fig = plt.figure(figsize=(14, 10))
ax = fig.add_subplot(111, projection='3d')# 繪制所有點
scatter = ax.scatter(df['depth'],df['lr'],df['n_estimators'],c=df['cost_score'],cmap='viridis',s=50,alpha=0.7,label='所有配置'
)# 繪制帕累托前沿點
pareto_scatter = ax.scatter(pareto_points['depth'],pareto_points['lr'],pareto_points['n_estimators'],c='red',s=100,edgecolors='black',label='帕累托前沿'
)# 標記最佳性價比點
best_idx = df['cost_score'].idxmin()
best_point = df.loc[best_idx]
ax.scatter(best_point['depth'],best_point['lr'],best_point['n_estimators'],c='gold',s=200,edgecolors='black',marker='*',label='最佳性價比'
)ax.set_xlabel('樹深度', fontsize=12)
ax.set_ylabel('學習率', fontsize=12)
ax.set_zlabel('樹數量', fontsize=12)
ax.set_title('超參數空間的帕累托前沿分析', fontsize=14)
ax.legend()# 添加顏色條
cbar = fig.colorbar(scatter, pad=0.1)
cbar.set_label('成本分數 (MSE + 時間成本)', fontsize=12)plt.tight_layout()
plt.show()