Cross-Entropy Method #

Name #

Cross-Entropy Method, CEM

Taxonomy #

The Cross-Entropy Method is a stochastic optimization technique that falls under the broader category of Computational Intelligence and Biologically Inspired Computation. It is closely related to the field of Optimization and shares similarities with other metaheuristic algorithms such as Evolutionary Algorithms and Estimation of Distribution Algorithms.

• Computational Intelligence
  • Biologically Inspired Computation
    • Optimization
      • Metaheuristics
        • Stochastic Optimization
          • Cross-Entropy Method

Strategy #

The Cross-Entropy Method is a iterative optimization algorithm that aims to find the optimal solution to a problem by minimizing the cross-entropy between a target distribution and a parametric distribution. The target distribution represents the optimal solution, while the parametric distribution is used to generate candidate solutions.

Sampling and Evaluation #

At each iteration, the CEM generates a batch of candidate solutions by sampling from the parametric distribution. These candidate solutions are then evaluated using a fitness function or objective function, which measures how well each solution solves the given problem.

Elite Selection #

After evaluating the candidate solutions, the CEM selects a subset of the best-performing solutions, known as the elite solutions. The elite solutions are used to update the parameters of the parametric distribution, bringing it closer to the target distribution.

Distribution Update #

The parameters of the parametric distribution are updated based on the elite solutions. This is typically done by minimizing the cross-entropy between the empirical distribution of the elite solutions and the parametric distribution. The updated distribution is then used to generate candidate solutions in the next iteration.

The CEM continues this process of sampling, evaluation, elite selection, and distribution update until a stopping criterion is met, such as reaching a maximum number of iterations or finding a satisfactory solution.

Procedure #

1. Initialize the parameters of the parametric distribution.
2. Repeat until a stopping criterion is met:
   1. Generate a batch of candidate solutions by sampling from the parametric distribution.
   2. Evaluate the candidate solutions using the fitness function.
   3. Select the elite solutions based on their fitness values.
   4. Update the parameters of the parametric distribution by minimizing the cross-entropy between the empirical distribution of the elite solutions and the parametric distribution.
3. Return the best solution found.

Data Structures #

• Candidate Solutions: A batch of potential solutions generated by sampling from the parametric distribution.
• Elite Solutions: A subset of the best-performing candidate solutions based on their fitness values.
• Parametric Distribution: A probability distribution used to generate candidate solutions, typically a Gaussian distribution for continuous problems or a Bernoulli distribution for discrete problems.

Parameters #

• Batch Size: The number of candidate solutions generated at each iteration.
• Elite Fraction: The proportion of candidate solutions selected as elite solutions.
• Learning Rate: The step size used to update the parameters of the parametric distribution.
• Stopping Criteria: Conditions for terminating the algorithm, such as a maximum number of iterations or a threshold for the fitness value.

Considerations #

Advantages #

• Effective for solving complex optimization problems with high-dimensional search spaces.
• Does not require gradient information, making it suitable for black-box optimization.
• Can handle both continuous and discrete optimization problems.

Disadvantages #

• Performance depends on the choice of the parametric distribution and its initial parameters.
• May require a large number of function evaluations, which can be computationally expensive.
• The elite fraction and learning rate need to be carefully tuned for optimal performance.

Heuristics #

Parametric Distribution Selection #

• For continuous optimization problems, use a Gaussian distribution with adaptive mean and variance.
• For discrete optimization problems, use a Bernoulli distribution with adaptive probability parameters.

Batch Size and Elite Fraction #

• Use a batch size that balances exploration and exploitation. A larger batch size promotes exploration but increases computational cost.
• Set the elite fraction between 0.1 and 0.2 to focus on the best-performing solutions while maintaining diversity.

Learning Rate #

• Use a learning rate that allows for sufficient exploration in the early iterations and exploitation in the later iterations.
• Adapt the learning rate based on the progress of the optimization process, reducing it as the algorithm converges.

Stopping Criteria #

• Set a maximum number of iterations based on the computational budget and problem complexity.
• Monitor the improvement in the best fitness value over a fixed number of iterations and stop if the improvement is below a threshold.

Problem-Specific Considerations #

• Incorporate domain knowledge into the initialization of the parametric distribution and the design of the fitness function.
• Use problem-specific constraints and boundary handling techniques to ensure the generation of feasible solutions.