Bayesian Optimization Overview
Bayesian optimization is an efficient method for hyperparameter tuning that uses probability and previous results to guide the search for the best values. Instead of randomly trying settings or checking every possibility, it builds a model to predict which hyperparameter choices are most promising. This approach focuses your search on areas likely to give better results, helping you find strong hyperparameters with fewer attempts.
Acquisition functions guide Bayesian optimization by scoring which hyperparameters to try next using the current probabilistic model. Popular choices include:
- Expected Improvement (EI): EI(x)=E[max(0,f(x)−f(x+))]
- Probability of Improvement (PI): PI(x)=P(f(x)>f(x+)+ξ)
- Upper Confidence Bound (UCB): UCB(x)=μ(x)+κ⋅σ(x)
Here, f(x+) is the best observed value, μ(x) and σ(x) are the model's predicted mean and standard deviation, and ξ, κ are tunable parameters.
1234567891011121314151617181920212223242526272829303132333435363738# Conceptual workflow for Bayesian optimization (no external libraries) import numpy as np # Suppose we are tuning a single hyperparameter x in [0, 10] space = np.linspace(0, 10, 100) def objective_function(x): # Simulate a function to optimize (e.g., validation error) return (x - 3) ** 2 + np.sin(x) * 2 # Step 1: Initialize with a few random samples history = [] for x in np.random.uniform(0, 10, 3): y = objective_function(x) history.append((x, y)) # Step 2: Fit a simple model to predict objective as a function of x # (Here, we use a moving average just for illustration) def predict(x, history): # Use the closest measured value as prediction closest = min(history, key=lambda h: abs(h[0] - x)) return closest[1] # Step 3: Acquisition - choose next x to evaluate def acquisition(space, history): # Pick the point with lowest predicted objective predictions = [predict(x, history) for x in space] return space[np.argmin(predictions)] # Step 4: Iterate - update history with new evaluation for _ in range(5): x_next = acquisition(space, history) y_next = objective_function(x_next) history.append((x_next, y_next)) # Step 5: Report best found best_x, best_y = min(history, key=lambda h: h[1]) print(f"Best x: {best_x:.2f}, Best objective: {best_y:.2f}")
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Bayesian optimization is an efficient method for hyperparameter tuning that uses probability and previous results to guide the search for the best values. Instead of randomly trying settings or checking every possibility, it builds a model to predict which hyperparameter choices are most promising. This approach focuses your search on areas likely to give better results, helping you find strong hyperparameters with fewer attempts.
Acquisition functions guide Bayesian optimization by scoring which hyperparameters to try next using the current probabilistic model. Popular choices include:
- Expected Improvement (EI): EI(x)=E[max(0,f(x)−f(x+))]
- Probability of Improvement (PI): PI(x)=P(f(x)>f(x+)+ξ)
- Upper Confidence Bound (UCB): UCB(x)=μ(x)+κ⋅σ(x)
Here, f(x+) is the best observed value, μ(x) and σ(x) are the model's predicted mean and standard deviation, and ξ, κ are tunable parameters.
1234567891011121314151617181920212223242526272829303132333435363738# Conceptual workflow for Bayesian optimization (no external libraries) import numpy as np # Suppose we are tuning a single hyperparameter x in [0, 10] space = np.linspace(0, 10, 100) def objective_function(x): # Simulate a function to optimize (e.g., validation error) return (x - 3) ** 2 + np.sin(x) * 2 # Step 1: Initialize with a few random samples history = [] for x in np.random.uniform(0, 10, 3): y = objective_function(x) history.append((x, y)) # Step 2: Fit a simple model to predict objective as a function of x # (Here, we use a moving average just for illustration) def predict(x, history): # Use the closest measured value as prediction closest = min(history, key=lambda h: abs(h[0] - x)) return closest[1] # Step 3: Acquisition - choose next x to evaluate def acquisition(space, history): # Pick the point with lowest predicted objective predictions = [predict(x, history) for x in space] return space[np.argmin(predictions)] # Step 4: Iterate - update history with new evaluation for _ in range(5): x_next = acquisition(space, history) y_next = objective_function(x_next) history.append((x_next, y_next)) # Step 5: Report best found best_x, best_y = min(history, key=lambda h: h[1]) print(f"Best x: {best_x:.2f}, Best objective: {best_y:.2f}")
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