📗 Additional information can be given in the form of a heuristic cost from any state to the goal state: this is an estimate or guess of the minimum cost from \(s\) to a goal state: Wikipedia.
➩ The cost from the initial state to a state \(s\) is denoted by \(g\left(s\right)\).
➩ The cost from the state \(s\) to the goal state is denoted by \(h^\star\left(s\right)\), which is unknown during the search.
➩ The estimated value of \(h^\star\left(s\right)\) is called the heuristic cost, denoted by \(h\left(s\right)\).
📗 Heuristic costs can be used to speed up the search.
TopHat Discussion
📗 Search for a page containing an image. Describe how you find the image.
📗 UCS (Uniform Cost Search, or Dijkstra's Algorithm) expands the state with the lowest current cost \(g\left(s\right)\): Wikipedia.
📗 It is BFS with a priority queue based on \(g\left(s\right)\), and it is equivalent to BFS if the cost of every action is 1.
📗 UCS is complete.
📗 UCS is optimal.
TopHat Quiz
📗 [4 points] Given initial state \(S\) and goal state \(G\), write down the expansion path (list of expanded nodes) for . In case the diagram is not clear, the edge costs are (heuristic on the diagonal).
📗 (Best First) Greedy Search expands the state with the lowest heuristic cost \(h\left(s\right)\): Wikipedia.
📗 BFGS is not an abbreviation of Greedy Search, since it usually stands for Broyden Fletcher Goldfarb Shanno algorithm (a version of a gradient descent algorithm).
📗 Greedy uses a priority queue based on \(h\left(s\right)\).
📗 Greedy is incomplete.
📗 Greedy is not optimal.
TopHat Quiz
📗 [4 points] Given initial state \(S\) and goal state \(G\), write down the expansion path (list of expanded nodes) for . In case the diagram is not clear, the edge costs are (heuristic on the diagonal).
📗 A Search expands the state with the lowest total cost \(g\left(s\right) + h\left(s\right)\): Wikipedia (for A*).
📗 A Search uses a priority queue based on \(g\left(s\right) + h\left(s\right)\).
📗 A is complete.
📗 A is not optimal.
TopHat Quiz
📗 [4 points] Given initial state \(S\) and goal state \(G\), write down the expansion path (list of expanded nodes) for . In case the diagram is not clear, the edge costs are (heuristic on the diagonal).
📗 Answer: .
TopHat Quiz
📗 [4 points] Given initial state \(S\) and goal state \(G\), write down the expansion path (list of expanded nodes) for . In case the diagram is not clear, the edge costs are (heuristic on the diagonal).
📗 \(h_{2}\) dominates \(h_{1}\) (or \(h_{1}\) is dominated by \(h_{2}\)) if \(h_{1}\left(s\right) \leq h_{2}\left(s\right) \leq h^\star\left(s\right)\) for every state \(s\).
📗 A* with a dominated heuristic is less informed and worse for A*.
📗 If optimality is not required, then a heuristic that is close but slightly over estimates the true cost can be preferred.
TopHat Quiz
(Past Exam Question) ID:
📗 [3 points] Let \(h_{1}\) be an admissible heuristic from a state to the optimal goal, A* search with which ones of the following \(h\) will be admissible?
📗 Iterative Deepening A* (IDA*) expands states with limit on \(g\left(s\right) + h\left(s\right)\) instead of depth in IDS: Link, Wikipedia.
📗 Beam Search keeps a priority queue with a limited size: Wikipedia.
Example
📗 A* search is often used to control robotic arms. States are usually represented by rotations angles instead of positions. The search space is called the configuration space: Link.
Review Note
Review of search algorithms:
📗 Ignore edge cost (equivalently, assume edge costs are all 1)
BFS:
Expand the shallowest node first
Complete, optimal (if cost is all 1)
Time complexity ~ \(b^{d}\)
Space complexity ~ \(b^{d}\)
DFS:
Expand deepest node first
Incomplete (when D is infinite), hence not optimal
Time complexity ~ \(b^{D}\)
Space complexity ~ \(b D\)
IDS:
"Search like BFS, fringe like DFS"
Complete, optimal (if cost is all 1)
Time complexity: ~ \(b^{d}\)
Space complexity: ~ \(b d\)
Preferred algorithms if heuristic is unknown
📗 Consider edge cost
UCS:
(Generalization of BFS)
Expand least-cost node first (first half cost \(g\left(s\right)\))
Complete, optimal
\(g\left(s\right)\): first half cost, known.
\(h^\star\left(s\right)\): second half cost, unknown.
\(h\left(s\right)\): an estimate/guess for second half cost; user-specified
BFG:
Expand node with lowest \(h\left(s\right)\)
Incomplete, not optimal.
Issue: ignores \(g\left(s\right)\)
A Search:
Expand node with lowest \(g\left(s\right) + h\left(s\right)\)
Complete, but not optimal.
Issue: \(h\left(s\right)\) may be an overestimate of \(h^\star\left(s\right)\), preventing us to expand a node with low true \(h^\star\left(s\right)\).
testucs,bfg,as,asf,adq
📗 Notes and code adapted from the course taught by Professors Jerry Zhu, Yudong Chen, Yingyu Liang, and Charles Dyer.
📗 Content from note blocks marked "optional" and content from Wikipedia and other demo links are helpful for understanding the materials, but will not be explicitly tested on the exams.
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