Information Theory in Hyperparameter Optimization

Explore information theory in hyperparameter optimization. Learn about entropy, mutual information, and information-theoretic acquisition functions.

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Introduction

Information theory provides powerful tools for hyperparameter optimization, especially in Bayesian methods. Entropy and mutual information quantify uncertainty and guide exploration-exploitation trade-offs.

Entropy

Shannon Entropy

Measures uncertainty in distribution:

H(X) = -Σ P(x) log P(x)

Differential Entropy

For continuous distributions:

h(X) = -∫ p(x) log p(x) dx

Entropy in Optimization

High entropy indicates high uncertainty, guiding exploration:

Regions with high entropy need exploration
Low entropy regions are well-understood
Balance entropy reduction with improvement

Mutual Information

Definition

Measures information shared between variables:

I(X;Y) = H(X) - H(X|Y) = H(Y) - H(Y|X)

Mutual Information for Optimization

Measures information gain from evaluating hyperparameters:

I(λ; f*) = H(f*) - H(f*|λ)

Where f* is optimal function value.

Information-Theoretic Acquisition Functions

Entropy Search

Maximizes reduction in entropy of optimum location:

ES(λ) = H(λ*) - H(λ*|y_λ)

Predictive Entropy Search

More tractable approximation:

Uses predictive distribution
Computationally efficient
Good exploration

Max-Value Entropy Search

Maximizes entropy reduction of optimum value:

MES(λ) = H(f*) - H(f*|y_λ)

Information Gain

Expected Information Gain

Expected reduction in uncertainty:

IG(λ) = E_{y_λ}[H(λ*) - H(λ*|y_λ)]

Computational Considerations

Requires entropy computation
Monte Carlo approximations
Computational overhead

Key Insight

Information-theoretic methods maximize information gain about the optimum, providing principled exploration strategies. They balance uncertainty reduction with performance improvement.

Applications

Active Learning

Select most informative samples:

Maximize information gain
Efficient exploration
Reduced evaluations

Experimental Design

Optimize measurement locations:

Information-theoretic criteria
Uncertainty reduction
Efficient design

Comparison with Other Methods

vs Expected Improvement

Information-theoretic methods:

Focus on uncertainty reduction
Better exploration
More computationally expensive

vs Upper Confidence Bound

Information methods:

More principled
Consider full distribution
Better theoretical guarantees

Frequently Asked Questions

What is entropy in hyperparameter optimization?

Entropy measures uncertainty in distributions. High entropy indicates high uncertainty, guiding exploration. Entropy reduction measures information gain from evaluations.

What is mutual information?

Mutual information I(X;Y) = H(X) - H(X|Y) measures information shared between variables. In optimization, it measures information gain about optimum from evaluating hyperparameters.

How does information theory help optimization?

Information theory provides principled methods to maximize information gain about the optimum. Entropy-based acquisition functions guide exploration more systematically than heuristics.

What is Entropy Search?

Entropy Search maximizes reduction in entropy of optimum location: ES(λ) = H(λ*) - H(λ*|y_λ). It seeks evaluations that most reduce uncertainty about where optimum lies.

What is Max-Value Entropy Search?

MES maximizes entropy reduction of optimum value: MES(λ) = H(f*) - H(f*|y_λ). It focuses on learning the optimal value rather than location.