TLDR: Researchers introduce Sampling Best-First Search (S-BFS), a novel algorithm for automated planning problems with continuous numeric “control parameters.” Unlike traditional methods that treat these parameters as constraints, S-BFS explicitly handles them as decision points using delayed partial expansions and sampling. The algorithm demonstrates probabilistic completeness and outperforms existing planners in terms of problem coverage, offering a new way to tackle complex planning scenarios with infinite decision spaces.
Automated planning, a core area of artificial intelligence, focuses on finding sequences of actions to achieve specific goals. Traditionally, planning problems deal with a finite number of objects and actions. However, real-world scenarios often involve continuous numeric decision variables, known as ‘control parameters,’ which introduce an infinite number of possibilities into the planning process.
These control parameters represent physical quantities that a planner might adjust to achieve a desired effect, such as the exact amount of fuel to use or the precise duration of an operation. Existing state-of-the-art approaches have largely treated these control parameters as embedded constraints, similar to other temporal or numeric restrictions. This means they are implicitly handled by solvers like Linear Programming or Satisfiability Modulo Theories, rather than being considered as explicit decision points within the search space.
A new research paper, titled Handling Infinite Domain Parameters in Planning Through Best-First Search with Delayed Partial Expansions, proposes an innovative alternative. Authored by ´Angel Aso-Mollar, Diego Aineto, Enrico Scala, and Eva Onaindia, this work introduces an efficient method that explicitly treats control parameters as true decision points within a systematic search framework.
Introducing Sampling Best-First Search (S-BFS)
The core of their proposal is a best-first, heuristic search algorithm called Sampling Best-First Search (S-BFS). This algorithm is designed to operate over the infinite decision spaces created by control parameters. A key innovation in S-BFS is the concept of ‘delayed partial expansion.’ In traditional search algorithms, a state is fully expanded by generating all its successors. However, with infinite successors, this is impossible. S-BFS overcomes this by incrementally expanding only a subset of a state’s successors through a ‘sampling function.’
The sampling function assigns a probability density to every state, defining the likelihood of sampling a particular successor. This allows the algorithm to explore the vast decision space without needing to enumerate every single possibility. Furthermore, partially expanded states are not immediately closed; instead, they are re-added to the ‘open list’ (a priority queue) for potential re-expansion in future iterations. Their evaluation value is adjusted using a ‘rectification function,’ which helps balance exploration and exploitation.
Ensuring Completeness and Quality
Given the infinite nature of the decision space, the paper introduces a relaxed notion of completeness called ‘probabilistic completeness.’ This means that for any solvable problem, the algorithm is guaranteed to find a solution with a probability of 1 as the number of steps approaches infinity. The researchers prove that S-BFS achieves this under specific conditions for its sampling and rectification functions.
For solution quality, S-BFS algorithms, particularly the S-A (A* variant) version, can provide optimality bounds. When a solution is found, its cost is bounded by the current evaluation value of the initial state, influenced by the rectification function. While not guaranteeing absolute optimality, this mechanism allows for controlling the trade-off between exploration depth and solution quality.
Experimental Insights and Comparison
The researchers conducted extensive experiments, comparing S-BFS against existing planners like NextFLAP and Monte-Carlo Tree Search (MCTS) with Progressive Widening. They tested various configurations of S-BFS, exploring different rectification functions (linear, quadratic, logarithmic) and sampling strategies (systematic, uniform, heuristic-guided).
The results showed that logarithmic rectification significantly improved problem coverage for S-BFS, suggesting that emphasizing heuristics over high penalties is beneficial. Systematic and uniform sampling strategies also outperformed heuristic-guided sampling, which was attributed to the heuristic function exhibiting numerous plateaus, making it less effective in guiding the sampling process.
Crucially, S-BFS demonstrated a significantly higher problem-solving coverage compared to NextFLAP and MCTS. While NextFLAP sometimes produced plans with fewer actions for smaller instances, the substantial difference in the number of problems solved by S-BFS highlights its effectiveness in tackling complex planning problems with continuous control parameters.
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Future Directions
This work lays a strong foundation for handling control parameters through systematic search. The authors plan to integrate their framework into temporal planning contexts and extend it to handle continuous-time actions, as described in PDDL+. They also aim to develop new numeric planning heuristics that can effectively account for infinite decision spaces.
By explicitly treating control parameters as decision points and employing a sophisticated sampling-based search, S-BFS offers a powerful new tool for automated planning in domains where continuous numeric variables are essential.
