Artificial Intelligence: Local Beam Search

Local beam search is an optimization algorithm used in artificial intelligence to find high-quality solutions in large search spaces, particularly for problems like scheduling, planning, or natural language processing. It’s a heuristic search strategy that balances exploration and exploitation, improving on greedy hill-climbing by maintaining multiple promising paths.


How It Works

  • Initialization: Start with k randomly generated states (solutions), where k is the beam width.
  • Successor Generation: For each of the k states, generate all possible successor states (neighbors) by applying valid actions or transformations.
  • Selection: Evaluate all successors using a heuristic function (e.g., a cost or fitness score). Select the top k successors with the best scores across all states, discarding the rest.
  • Iteration: Repeat the process, using the k selected states as the new beam, until a termination condition is met (e.g., a goal state is found, a maximum number of iterations is reached, or no better solutions emerge).
  • Output: Return the best state found, which may be a local or global optimum.


Key Features

  • Beam Width (k): Controls the number of states kept at each step. A larger k explores more of the search space but increases computational cost.
  • Heuristic-Driven: Relies on a heuristic function to rank states, guiding the search toward promising areas.
  • Parallel Exploration: Unlike hill-climbing (which follows a single path), local beam search explores k paths simultaneously, reducing the risk of getting stuck in local optima.
  • Memory Efficiency: Only stores k states at a time, making it more memory-efficient than exhaustive searches like breadth-first search.


Example

Imagine optimizing a travel itinerary:

  • States: Different itineraries (e.g., city visit sequences).
  • Successors: Variations of an itinerary (e.g., swapping two cities).
  • Heuristic: Total cost (flights, hotels, time).
  • Beam Width: k = 3.
  • Start with 3 random itineraries.
  • Generate all possible next itineraries (e.g., by swapping cities).
  • Pick the 3 best-scoring itineraries (lowest cost) from all successors.
  • Repeat until the cost stops decreasing or a time limit is reached.

LEARN-ONE-RULE Algorithm: A General-to-Specific Beam Search Approach


The LEARN_ONE_RULE function is a practical implementation of beam search designed to derive a single rule that covers a subset of examples. It utilizes a general-to-specific greedy search, guided by a performance metric to identify the most effective rule.

Algorithm Execution Flow
Start:

* Initialize the node to Root_Node and Found to False.

Search Loop:

* If the current node is the goal, set Found to True.
* Otherwise, find successors and estimate costs, storing them in the OPEN list.
* Continuously select the top W elements from the OPEN list for expansion.

Evaluation:

* If the goal is found during expansion, return Yes.
* If the search concludes without finding the goal, return No.


Pseudocode

LEARN_ONE_RULE(Target_attribute, Attributes, Examples, k)
    Initialize Best_hypothesis to the most general hypothesis (θ)
    Initialize Candidate_hypotheses to {Best_hypothesis}

    While Candidate_hypotheses is not empty:
        All_constraints <- Set of all constraints (a = v) where:
            a ∈ Attributes
            v = value of a that occurs in Examples

        New_candidate_hypotheses <- Empty Set
        For each h in Candidate_hypotheses:
            For each c in All_constraints:
                new_h <- h + c
                // Create specialization of h by adding the constraint c
                If new_h is not duplicate, inconsistent, and maximally specific:
                    Add new_h to New_candidate_hypotheses

        // Evaluate and update the best hypothesis
        For each h in New_candidate_hypotheses:
            If PERFORMANCE(h, Examples, Target_attribute) > PERFORMANCE(Best_hypothesis, Examples, Target_attribute):
                Best_hypothesis <- h

        // Narrow down to the k best hypotheses
        Candidate_hypotheses <- Select the k best New_candidate_hypotheses based on PERFORMANCE

    // Formulate the final rule
    Return "IF Best_hypothesis THEN prediction"
    Where prediction is the most frequent value of Target_attribute among Examples that match Best_hypothesis ​


Advantages

  • Efficiency: Scales better than exhaustive search for large problems.
  • Parallelism: Explores multiple paths, avoiding some local optima.
  • Flexibility: Works with any problem where states and heuristics can be defined.


Limitations

  • Local Optima: Can still get trapped in local optima if the beam width is too small or the heuristic is misleading.
  • Loss of Diversity: Early pruning of states may discard potentially better solutions, especially if initial states are poor.
  • Heuristic Dependence: Performance relies heavily on the quality of the heuristic function.


Comparison

  • Vs. Hill-Climbing: Local beam search is less greedy, as it tracks k paths instead of one, improving solution quality.
  • Vs. Genetic Algorithms: Both maintain multiple solutions, but beam search is deterministic and focuses on local improvements, while genetic algorithms use crossover and mutation for broader exploration.
  • Vs. A*: Beam search is less memory-intensive (fixed k states) but doesn’t guarantee optimality, unlike A* with an admissible heuristic.