# Population-Based Incremental Learning #

## Name #

Population-Based Incremental Learning (PBIL)

## Taxonomy #

Population-Based Incremental Learning is a technique in the field of Evolutionary Computation, which is a subfield of Computational Intelligence. PBIL is closely related to Genetic Algorithms and Estimation of Distribution Algorithms.

- Computational Intelligence
- Evolutionary Computation
- Evolutionary Algorithms
- Genetic Algorithms
- Estimation of Distribution Algorithms
- Population-Based Incremental Learning

- Evolutionary Algorithms

- Evolutionary Computation

## Strategy #

### Probabilistic Model #

Population-Based Incremental Learning maintains a probabilistic model of the solution space, typically represented as a probability vector. This vector captures the probability of each variable in the solution taking on a particular value. The probabilistic model is used to generate new candidate solutions by sampling from the probability distribution.

### Population Sampling and Evaluation #

PBIL generates a population of candidate solutions by sampling from the probabilistic model. Each candidate solution is evaluated using a fitness function that measures its quality or performance in solving the given problem. The fitness values of the candidate solutions are used to guide the update of the probabilistic model.

### Probabilistic Model Update #

The probabilistic model is updated based on the fitness values of the candidate solutions. The update process involves adjusting the probabilities in the vector to increase the likelihood of generating high-quality solutions in subsequent iterations. This is typically done by shifting the probabilities towards the values of the best-performing candidate solutions.

### Iteration and Convergence #

The process of sampling candidate solutions, evaluating their fitness, and updating the probabilistic model is repeated for a specified number of iterations or until a convergence criterion is met. As the iterations progress, the probabilistic model is expected to converge towards promising regions of the solution space, increasing the likelihood of generating high-quality solutions.

## Procedure #

Data Structures:

- Probability Vector: A vector of probabilities representing the probabilistic model of the solution space. Each element of the vector corresponds to a variable in the solution and captures the probability of that variable taking on a particular value.
- Population: A set of candidate solutions generated by sampling from the probability vector.

Parameters:

- Population Size: The number of candidate solutions generated in each iteration.
- Learning Rate: The rate at which the probabilistic model is updated based on the best-performing candidate solutions.
- Mutation Probability: The probability of applying mutation to the probability vector to maintain diversity and explore new regions of the solution space.

Procedure:

- Initialize the probability vector with equal probabilities for each variable.
- Repeat for a specified number of iterations or until convergence:
- Generate a population of candidate solutions by sampling from the probability vector.
- Evaluate the fitness of each candidate solution using the fitness function.
- Select the best-performing candidate solutions based on their fitness values.
- Update the probability vector based on the selected candidate solutions:
- For each variable in the solution:
- Calculate the proportion of the selected candidate solutions that have a particular value for that variable.
- Update the corresponding probability in the vector by moving it towards the calculated proportion using the learning rate.

- For each variable in the solution:
- Apply mutation to the probability vector:
- For each element in the probability vector:
- With a probability equal to the mutation probability, modify the element by a small random value.

- For each element in the probability vector:

- Return the best candidate solution found during the iterations.

## Considerations #

Advantages:

- PBIL is computationally efficient compared to traditional Genetic Algorithms since it does not require the overhead of genetic operators like crossover and mutation on a population of solutions.
- The probabilistic model in PBIL provides a compact representation of the solution space, allowing for efficient sampling and exploration.
- PBIL is well-suited for problems with high-dimensional search spaces and can effectively capture the dependencies between variables.

Disadvantages:

- PBIL may suffer from premature convergence if the probabilistic model becomes too focused on a particular region of the solution space, losing diversity and potentially missing global optima.
- The performance of PBIL is sensitive to the choice of learning rate and mutation probability, requiring careful tuning of these parameters for each problem instance.
- PBIL may struggle with problems that have complex multimodal fitness landscapes or deceptive optima, as the probabilistic model may not capture the necessary information to escape local optima.

## Heuristics #

### Population Size #

- Start with a population size that is at least 10 times the number of variables in the solution space.
- Increase the population size for problems with a large search space or complex fitness landscape to ensure sufficient exploration.
- Consider reducing the population size in later iterations to focus the search on promising regions.

### Learning Rate #

- Use a small learning rate (e.g., 0.1 or 0.05) to allow for gradual updates to the probabilistic model and maintain diversity.
- Larger learning rates can lead to faster convergence but may result in premature convergence to suboptimal solutions.
- Adapt the learning rate during the search, starting with a larger value for exploration and reducing it over time for exploitation.

### Mutation Probability #

- Set the mutation probability to a small value (e.g., 0.01 or 0.05) to introduce minor variations in the probabilistic model.
- Higher mutation probabilities can help escape local optima but may slow down convergence.
- Adjust the mutation probability based on the problem complexity and the desired balance between exploration and exploitation.

### Elitism #

- Incorporate elitism by always preserving the best candidate solution found so far across iterations.
- Elitism ensures that the best solution is not lost during the probabilistic model update and provides a reference point for the search.

### Termination Criteria #

- Define a maximum number of iterations based on the available computational resources and the problem complexity.
- Monitor the improvement in the best fitness value over a number of iterations and terminate if no significant improvement is observed.
- Set a target fitness value as a termination criterion if the optimal solution quality is known in advance.

### Problem-Specific Considerations #

- Analyze the problem structure and identify any dependencies or correlations between variables that can be exploited in the probabilistic model.
- Incorporate domain knowledge or problem-specific heuristics to guide the initialization and update of the probabilistic model.
- Experiment with different representation schemes for the candidate solutions and the probability vector to find the most suitable encoding for the problem at hand.