Informed Search School of Data Science, Fudan University - - PowerPoint PPT Presentation

复旦大学大数据学院

School of Data Science, Fudan University

Informed Search

魏忠钰

March 6th, 2019

Informed Search

▪ In uninformed search, we never “look-ahead” to the goal. E.g., We don't consider the cost of getting to the goal from the end of the current path. ▪ Often we have some other knowledge about the merit of nodes.

Search Heuristics

▪ A heuristic is:

▪ A function that estimates how close a state is to a goal ▪ Designed for a particular search problem ▪ Examples: Manhattan distance, Euclidean distance for pathing

10 5 11.2 𝑦1 − 𝑦2 + |𝑧1 − 𝑧2| (𝑦1 − 𝑦2 )2+(𝑧1 − 𝑧2 )2 Manhattan distance: Euclidean distance:

Heuristic

Example: Pancake Problem

Cost: Number of pancakes flipped

Example: Heuristic Function

Heuristic: the number of the largest pancake that is still out of place

4 3 2 3 3 3 4 4 3 4 4 4

h(x)

Greedy Search

Example: Heuristic Function

Greedy Search

▪ Expand the node that seems closest…

Greedy Search

▪ What can go wrong?

▪ From Lasi to Fagaras

Greedy Search

▪ What can go wrong?

▪ From Lasi to Fagaras

Greedy Search

▪ Strategy: expand a node that you think is closest to a goal state

▪ Heuristic: estimate of distance to nearest goal for each state

▪ A common case:

▪ Best-first takes you straight to the (wrong) goal

▪ Worst-case: like a badly-guided DFS ▪ Not Complete ▪ Not Optimal

… b … b

A* Search

▪ Take into account the cost of getting to the node as well as our estimation of the cost of getting to the goal from the node. ▪ Evaluation function f(n) ▪ f(n) = g(n) + h(n) ▪ g(n) is the cost of the path represented by node n ▪ h(n) is the heuristic estimate of the cost of achieving the goal from n.

Quiz: Combining UCS and Greedy

• Uniform-cost orders by path cost, or backward cost g(n)
• Greedy orders by goal proximity, or forward cost h(n)
• A* Search orders by the sum: f(n) = g(n) + h(n)

S a d b G h=5 h=6 h=2 1 8 1 1 2 h=6 h=0 c h=7 3 e h=1 1

Is A* Optimal?

• What went wrong?
• Actual bad goal cost < estimated good goal cost
• We need estimates to be less than actual costs!

A G S 1 3

h = 6 h = 0

5

h = 7

Admissible Heuristics

Idea: Admissibility

Inadmissible (pessimistic) heuristics break optimality by trapping good plans on the fringe Admissible (optimistic) heuristics slow down bad plans but never

• utweigh true costs

Admissible Heuristics

• A heuristic h is admissible (optimistic) if:

where is the true cost to a nearest goal

• Examples:
• Coming up with admissible heuristics is most of what’s

involved in using A* in practice. 15 4

Optimality of A* Tree Search

Optimality of A* Tree Search Assume:

• A is an optimal goal node
• B is a suboptimal goal node
• h is admissible

Claim:

• A will be visited before B

…

Optimality of A* Tree Search: Blocking Proof: ▪ Imagine B is on the fringe ▪ Some ancestor n of A is on the fringe, too ▪ Claim: n will be expanded before B ▪ f(n) is less or equal to f(A)

Definition of f-cost Admissibility of h

…

h = 0 at a goal

Optimality of A* Tree Search: Blocking Proof: ▪ Imagine B is on the fringe ▪ Some ancestor n of A is on the fringe, too ▪ Claim: n will be expanded before B ▪ f(n) is less or equal to f(A) ▪ f(A) is less than f(B)

…

Optimality of A* Tree Search: Blocking Proof: ▪ Imagine B is on the fringe ▪ Some ancestor n of A is on the fringe, too (maybe A!) ▪ Claim: n will be expanded before B ▪ f(n) is less or equal to f(A) ▪ f(A) is less than f(B) ▪ n expands before B ▪ All ancestors of A expand before B ▪ A expands before B ▪ A* search is optimal

…

Properties of A*

… b … b

Uniform-Cost A*

UCS vs A* Contours

• Uniform-cost expands equally in all

“directions”

• A* expands mainly toward the goal, but

does hedge its bets to ensure optimality

• Expand all the nodes with F less or equal to
• ptimal cost

Start Goal Start Goal

A* History

▪ Peter Hart, Nils Nilsson and Bertram Raphael of Stanford Research Institute (now SRI International) first described the algorithm in 1968. ▪ A1 -> A2 -> A*(Optimal)

A* Applications

• Video games
• Pathing / routing problems
• Resource planning problems
• Robot motion planning
• Language analysis
• Machine translation
• Speech recognition
• …

Creating Heuristics

Creating Admissible Heuristics

▪ Most of the work in solving hard search problems

• ptimally is in coming up with admissible heuristics

▪ Often, admissible heuristics are solutions to relaxed problems, where new actions are available ▪ Inadmissible heuristics are often useful too 15 366

Example: 8 Puzzle Start State Goal State Actions

8 Puzzle I

▪ Heuristic: Number of tiles misplaced ▪ Why is it admissible? ▪ h(start) = 8

Start State Goal State

8 Puzzle II ▪ What if we had an easier 8-puzzle where any tile could slide any direction at any time, ignoring other tiles? ▪ Total Manhattan distance ▪ Why is it admissible? ▪ h(start) = 3 + 1 + 2 + 2 + 2 + 3 + 3 + 2 = 18

Start State Goal State

How to design a admissible heuristic functions ?

From relaxed problem A tile can move from square A to square B if A is adjacent to B and B is blank. ▪ Constraint 1: A and B is adjacent ▪ Constraint 2: B is blank ▪ Problem 1: A tile can move from square A to square B if A is adjacent to B. ▪ Problem 2: A tile can move from square A to square B if B is blank. ▪ Problem 3: A tile can move from square A to square B.

Heuristic function can be generated automatically with formal expression of the original question!

From sub-problems ▪ The task is to get tiles 1, 2, 3, and 4 into their correct positions, without worrying about what happens to the other tiles. ▪ The cost of solving the sub-problem is definitely no more than its

• riginal problem.

The comparison of different heuristic function ▪ Effective Branching Factor (𝑐∗) is computed based on the depth and nodes # in the tree.

▪ h1: # of tiles mis-placed ▪ h2: Total Manhattan distance 𝑂 + 1 = 1 + 𝑐∗ + (𝑐∗)2 + (𝑐∗)2 + ⋯ + (𝑐∗)𝑒

Trivial Heuristics, Dominance ▪ Dominance: ha ≥ hc if ▪ Heuristics form a semi-lattice:

▪ Max of admissible heuristics is admissible

▪ Trivial heuristics

▪ Bottom of lattice is the zero heuristic (what does this give us?) ▪ Top of lattice is the exact heuristic

8 Puzzle III

▪ How about using the actual cost as a heuristic?

▪ Would it be admissible?

▪ Yes

▪ Would we save on nodes expanded?

▪ Yes

▪ What’s wrong with it?

▪ More computational cost.

▪ With A*: a trade-off between quality of estimate and work per node

▪ As heuristics get closer to the true cost, you will expand fewer nodes but usually do more work per node to compute the heuristic itself

Learning for heuristic functions ▪ Learning heuristic functions

𝐼 𝑜 = 𝑑1𝑦1 𝑜 , … , 𝑑𝑛𝑦𝑛(𝑜)

Tree Search: Extra Work!

▪ Failure to detect repeated states can cause exponentially more work. Search Tree State Graph

Graph Search

▪ In BFS, for example, we shouldn’t bother expanding the circled nodes (why?)

S

a b d p a c e p h f r q q c

G

a q e p h f r q q c

G

a

Graph Search ▪ Idea: never expand a state twice ▪ How to implement:

▪ Tree search + set of expanded states (“closed set”) ▪ Expand the search tree node-by-node, but… ▪ Before expanding a node, check to make sure its state has never been expanded before ▪ If not new, skip it, if new add to closed set

▪ Important: store the closed set as a set, not a list ▪ Can graph search wreck completeness? Why/why not?

▪ No

▪ How about optimality?

A* Gone Wrong with admissible function S A B C G

1 1 1 2 3 h=2 h=1 h=4 h=1 h=0

S (0+2) A (1+4) B (1+1) C (2+1) G (5+0) C (3+1) G (6+0)

State space graph Search tree

Consistency of Heuristics ▪ Main idea: estimated heuristic costs ≤ actual costs

▪ Admissibility: heuristic cost ≤ actual cost to goal ▪ h(A) ≤ actual cost from A to G ▪ Consistency: heuristic “arc” cost ≤ actual cost for each arc ▪ h(A) – h(C) ≤ cost(A to C)

▪ Consequences of consistency:

▪ The f value along a path never decreases ▪ h(A) ≤ cost(A to C) + h(C) ▪ A* graph search is optimal

3

A C G

h=4 h=1 1 h=2

Optimality of A* Search

▪ Sketch: consider what A* does with a consistent heuristic:

▪ Fact 1: In tree search, A* expands nodes in increasing total f value (f-contours) ▪ Fact 2: For every state s, nodes that reach s

• ptimally are expanded before nodes that

reach s sub-optimally ▪ Result: A* graph search is optimal

… f  3 f  2 f  1

Optimality

▪ Tree search:

▪ A* is optimal if heuristic is admissible ▪ UCS is a special case (h = 0)

▪ Graph search:

▪ A* optimal if heuristic is consistent ▪ UCS optimal (h = 0 is consistent)

▪ Consistency implies admissibility

A*: Summary

▪ A* uses both backward costs and (estimates of) forward costs ▪ A* is optimal with admissible / consistent heuristics ▪ Heuristic design is key: often use relaxed problems

Tree Search Pseudo-Code

Graph Search Pseudo-Code

Quiz

Consider the state space graph shown below. A is the start state and H is the goal state. The costs for each edge are shown on the graph. Each edge can be traversed in both directions. You are required to use graph search algorithm.

• 1. Provide the search path using depth first search algorithm (using

alphabetical order to break the tie). Specify the status of the frontier (node with depth) along the search.

• 2. Provide the search path using uniform cost search algorithm (using

alphabetical order to break the tie). Specify the status of the frontier (node with cost) along the search.