pareto-optimization

# Pareto Optimization Pareto optimization deals with multi-objective optimization where you want to optimize multiple conflicting objectives simultaneously. ## Key Concepts ### Pareto Dominance Point A **dominates** point B if: - A is at least as good as B in all objectives - A is strictly better than B in at least one objective ### Pareto Frontier (Pareto Front) The set of all non-dominated points. These represent optimal trade-offs where improving one objective requires sacrificing another. ## Computing the Pareto Frontier ### Using the paretoset Library ```python from paretoset import paretoset import pandas as pd # Data with two objectives (e.g., model accuracy vs inference time) df = pd.DataFrame({ 'accuracy': [0.95, 0.92, 0.88, 0.85, 0.80], 'latency_ms': [120, 95, 75, 60, 45], 'model_size': [100, 80, 60, 40, 20], 'learning_rate': [0.001, 0.005, 0.01, 0.05, 0.1] }) # Compute Pareto mask # sense: "max" for objectives to maximize, "min" for objectives to minimize objectives = df[['accuracy', 'latency_ms']] pareto_mask = paretoset(objectives, sense=["max", "min"]) # Get Pareto-optimal points pareto_points = df[pareto_mask] ``` ### Manual Implementation ```python import numpy as np def is_dominated(point, other_points, maximize_indices, minimize_indices): """Check if point is dominated by any point in other_points.""" for other in other_points: dominated = True strictly_worse = False for i in maximize_indices: if point[i] > other[i]: dominated = False break if point[i] < other[i]: strictly_worse = True if dominated: for i in minimize_indices: if point[i] < other[i]: dominated = False break if point[i] > other[i]: strictly_worse = True if dominated and strictly_worse: return True return False def compute_pareto_frontier(points, maximize_indices=[0], minimize_indices=[1]): """Compute Pareto frontier from array of points.""" pareto = [] points_list = list(points) for i, point in enumerate(points_list): others = points_list[:i] + points_list[i+1:] if not is_dominated(point, others, maximize_indices, minimize_indices): pareto.append(point) return np.array(pareto) ``` ## Example: Model Selection ```python import pandas as pd from paretoset import paretoset # Results from model training experiments results = pd.DataFrame({ 'accuracy': [0.95, 0.92, 0.90, 0.88, 0.85], 'inference_time': [150, 120, 100, 80, 60], 'batch_size': [32, 64, 128, 256, 512], 'hidden_units': [512, 256, 128, 64, 32] }) # Filter by minimum accuracy threshold results = results[results['accuracy'] >= 0.85] # Compute Pareto frontier (maximize accuracy, minimize inference time) mask = paretoset(results[['accuracy', 'inference_time']], sense=["max", "min"]) pareto_frontier = results[mask].sort_values('accuracy', ascending=False) # Save to CSV pareto_frontier.to_csv('pareto_models.csv', index=False) ``` ## Visualization ```python import matplotlib.pyplot as plt # Plot all points plt.scatter(results['inference_time'], results['accuracy'], alpha=0.5, label='All models') # Highlight Pareto frontier plt.scatter(pareto_frontier['inference_time'], pareto_frontier['accuracy'], color='red', s=100, marker='s', label='Pareto frontier') plt.xlabel('Inference Time (ms) - minimize') plt.ylabel('Accuracy - maximize') plt.legend() plt.show() ``` ## Properties of Pareto Frontiers 1. **Trade-off curve**: Moving along the frontier improves one objective while worsening another 2. **No single best**: All Pareto-optimal solutions are equally "good" in a multi-objective sense 3. **Decision making**: Final choice depends on preference between objectives