Outline. Informed Search. Recall: Uninformed Search. An Idea. Heuristics Informed search techniques More on heuristics Iterative improvement

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1 Outline Informed Search EE457 pplied rtificial Intelligence Spring 8 Lecture # Heuristics Informed search techniques More on heuristics Iterative improvement Russell & Norvig, chapter 4 Skip Genetic algorithms pages 6- (will be covered in Lecture ) EE457 pplied rtificial Intelligence R. Khoury (8) Page Recall: Uninformed Search Travel blindly until they reach ucharest n Idea It would be better if the agent knew whether or not the city it is travelling to gets it closer to ucharest Of course, the agent doesn t know the exact distance or path to ucharest (it wouldn t need to search otherwise!) The agent must estimate the distance to ucharest EE457 pplied rtificial Intelligence R. Khoury (8) Page EE457 pplied rtificial Intelligence R. Khoury (8) Page 4

2 Heuristic Function More generally: We want the search algorithm to be able to estimate the path cost from the current node to the goal This estimate is called a heuristic function annot be done based on problem formulation Need to add additional information Informed search EE457 pplied rtificial Intelligence R. Khoury (8) Page 5 Heuristic Function Heuristic function h(n) h(n): estimated cost from node n to goal h(n ) < h(n ) means it s probably cheaper to get to the goal from n h(n goal ) = Path cost g(n) Evaluation function f(n) f(n) = g(n) Uniform ost f(n) = h(n) Greedy est-first f(n) = g(n) + h(n) * EE457 pplied rtificial Intelligence R. Khoury (8) Page 6 Greedy est-first Search f(n) = h(n) lways expand the node closest to the goal and ignore path cost omplete only if m is finite Rarely true in practice Not optimal an go down a long path of cheap actions Time complexity = O(b m ) Space complexity = O(b m ) Greedy est-first Search Upper-bound case: goal is last node of the tree Number of nodes generated: b nodes for each node of m levels (entire tree) Time and space complexity: all generated nodes O(b m ) EE457 pplied rtificial Intelligence R. Khoury (8) Page 7 EE457 pplied rtificial Intelligence R. Khoury (8) Page 8

3 * Search f(n) = g(n) + h(n) est-first search omplete Optimal, given admissible heuristic Never overestimates the cost to the goal Optimally efficient No other optimal algorithm will expand less nodes Time complexity = O(b */є + ) Space complexity = O(b */є + ) * Search Upper-bound case: heuristic is the trivial h(n) = * becomes Uniform ost Search Goal has path cost *, all other actions have minimum cost of є epth explored before taking action *: */є epth of fringe nodes: */є + Space & time complexity: all generated nodes: O(b */є + ) * є є є є є є є є є є є є є є є EE457 pplied rtificial Intelligence R. Khoury (8) Page 9 EE457 pplied rtificial Intelligence R. Khoury (8) Page * Search Using a good heuristic can reduce time complexity an go down to O(bm) However, space complexity will always be exponential * runs out of memory before running out of time Iterative eepening * Search Like Iterative eepening Search, but cut-off limit is f-value instead of depth Next iteration limit is the smallest f-value of any node that exceeded the cut-off of current iteration Properties omplete and optimal like * Space complexity of depth-first search (because it s possible to delete nodes and paths from memory when we explore down to the cut-off limit) Performs poorly if small action cost (small step in each iteration) EE457 pplied rtificial Intelligence R. Khoury (8) Page EE457 pplied rtificial Intelligence R. Khoury (8) Page

4 Simplified Memory-ounded * Uses all available memory When memory limit reached, delete worst leaf node (highest f-value) If equality, delete oldest leaf node SM memory problem If the entire optimal path fills the memory and there is only one non-goal leaf node SM cannot continue expanding Goal is not reachable Simplified Memory-ounded * Space complexity known and controlled by system designer omplete if shallowest goal depth less than memory size Shallowest goal is reachable Optimal if optimal goal is reachable EE457 pplied rtificial Intelligence R. Khoury (8) Page EE457 pplied rtificial Intelligence R. Khoury (8) Page 4 Example: Greedy Search Example: * Search h(n) = straight-line distance rad 66 Zerind 74 Sibiu 5 Timisoara 9 Fagaras 76 Rimnicu 9 ucharest h(n) = straight-line distance rad 66 Zerind 449 Sibiu 9 Timisoara 447 Fagaras 45 Rimnicu 4 Pitesti 47 raiova 56 ucharest EE457 pplied rtificial Intelligence R. Khoury (8) Page 5 EE457 pplied rtificial Intelligence R. Khoury (8) Page 6 4

5 Heuristic Function Properties dmissible Never overestimate the cost onsistency / Monotonicity h(n p ) h(n c ) + cost(n p,n c ) h(n p ) + g(n p ) h(n c ) + cost(n p,n c ) + g(n p ) h(n p ) + g(n p ) h(n c ) + g(n c ) f(n p ) f(n c ) f(n) never decreases as we get closer to the goal omination h (n) h (n) for all n EE457 pplied rtificial Intelligence R. Khoury (8) Page 7 reating Heuristic Functions Found by relaxing the problem Straight-line distance to ucharest Eliminate constraint of traveling on roads 8-puzzle Move each square that s out of place (7) Move by the number of squares to get to place () Move some tiles in place EE457 pplied rtificial Intelligence R. Khoury (8) Page 8 reating Heuristic Functions lock world Move a block on the table or on another block If there s nothing on top of it Possible heuristics for this game. + for each block in the wrong position. + for each block on top of the wrong block. + for every block in the support structure of each block with incorrect support for every block with the correct support structure 5. Partial solving (get to --?-?) EE457 pplied rtificial Intelligence R. Khoury (8) Page 9 reating Heuristic Functions State h h h h4 h5 ½(h+h4) EE457 pplied rtificial Intelligence R. Khoury (8) Page 5

6 reating Heuristic Functions Path to the Goal State Sometimes the path to the goal is irrelevant Only the solution matters h h n-queen puzzle h h4 h5 ½(h+h4).5.5 EE457 pplied rtificial Intelligence R. Khoury (8) Page EE457 pplied rtificial Intelligence R. Khoury (8) Page ifferent Search Problem No longer minimizing path cost Improve quality of state Minimize state cost Maximize state payoff Iterative improvement Example: Iterative Improvement Minimize cost: number of attacks EE457 pplied rtificial Intelligence R. Khoury (8) Page EE457 pplied rtificial Intelligence R. Khoury (8) Page 4 6

7 Example: Travelling Salesman Tree search method Start with home city Visit next city until optimal round trip Iterative improvement method Start with random round trip Swap cities until optimal round trip EE457 pplied rtificial Intelligence R. Khoury (8) Page 5 Value Graphic Visualisation State value / state plot: state space State axis can be states or specific properties Neighbouring states on the axis are states linked by actions or with similar property values State values are computed using a heuristic and do not include path cost State EE457 pplied rtificial Intelligence R. Khoury (8) Page 6 Graphic Visualisation Value State value / state plot: state space Global maximum Local maxima Plateau Global minimum Graphic Visualisation If state payoff is a complex mathematical function depending on one state property - x * x + sin (x)/x + (-x)*cos(5x)/5x x/ State space: x œ [, 8] Max: x = 74 payoff = 66.9 Local minima State EE457 pplied rtificial Intelligence R. Khoury (8) Page 7 EE457 pplied rtificial Intelligence R. Khoury (8) Page 8 7

8 Graphic Visualisation More complex state spaces can have several dimensions Example: States are X-Y coordinates, state value is Z coordinate EE457 pplied rtificial Intelligence R. Khoury (8) Page 9 Graphic Visualisation Each state is a point on the map Each state s value is the distance to the N Tower Locations in water always have the worst value because we can t swim state space X-Y coordinates of the agent Z coordinate for state value Red = minimum distance lue = maximum distance EE457 pplied rtificial Intelligence R. Khoury (8) Page Hill limbing (Gradient escent) Simple but efficient local optimization strategy lways take the action that most improves the state Hill limbing (Gradient escent) Generate random initial state Each iteration Generate and evaluate neighbours at step size Move to neighbour with greatest increase/decrease (i.e. take one step) End when there are no better neighbours EE457 pplied rtificial Intelligence R. Khoury (8) Page EE457 pplied rtificial Intelligence R. Khoury (8) Page 8

9 Example: Travelling to Toronto Trying to get to downtown Toronto Take steps toward the N Tower EE457 pplied rtificial Intelligence R. Khoury (8) Page Hill limbing (Gradient escent) dvantages Fast No search tree isadvantages Gets stuck in local optimum oes not allow worse moves Solution dependant on initial state Selecting step size ommon improvements Random restarts Intelligently-chosen initial state ecreasing step size EE457 pplied rtificial Intelligence R. Khoury (8) Page 4 Simulated nnealing Problem with hill climbing: local best move doesn t lead to optimal goal Solution: allow bad moves Simulated annealing is a popular way of doing that Stochastic search method Simulates annealing process in metallurgy nnealing Tempering technique in metallurgy Weakness and defects come from atoms of crystals freezing in the wrong place (local optimum) Heating to unstuck the atoms (escape local optimum) Slow cooling to allow atoms to get to better place (global optimum) EE457 pplied rtificial Intelligence R. Khoury (8) Page 5 EE457 pplied rtificial Intelligence R. Khoury (8) Page 6 9

10 Simulated nnealing nnealing toms moving towards minimum-energy location in crystal while avoiding bad position. toms are more likely to move out of a bad position if the metal s temperature is high. Simulated nnealing gent modifying state towards state with global optimal value while avoiding local optimum. gents are more likely to accept bad moves if the temperature control parameter has a high value. EE457 pplied rtificial Intelligence R. Khoury (8) Page 7 Simulated nnealing nnealing The metal s temperature starts hot, then it cools off continuously over time until the metal is room temperature Simulated nnealing The temperature control parameter starts with a high value, then it decreases incrementally with each iteration of the search until it reaches a pre-set threshold. EE457 pplied rtificial Intelligence R. Khoury (8) Page 8 Simulated nnealing llow some bad moves ad enough to get out of local optimum Not so bad as to get out of global optimum Probability of accepting bad moves given adness of the move (i.e. variation in state value V) Temperature T P = e - V/T Stochastic search technique EE457 pplied rtificial Intelligence R. Khoury (8) Page 9 Simulated nnealing Generate random initial state and high temperature Each iteration Generate and evaluate a random neighbour If neighbour better than current state ccept Else (if neighbour worse than current state) ccept with probability e - V/T Reduce temperature End when temperature less than threshold EE457 pplied rtificial Intelligence R. Khoury (8) Page 4

11 Simulated nnealing dvantages voids local optima Very good at finding high-quality solutions Very good for hard problems with complex state value functions isadvantage an be very slow in practice Simulated nnealing pplication Traveling-wave tube (TWT) Uses focused electron beam to amplify electromagnetic communication waves Produces high-power radio frequency (RF) signals ritical components in deep-space probes and communication satellites Power efficiency becomes a key issue TWT research group at NS working for over years on improving power efficiency EE457 pplied rtificial Intelligence R. Khoury (8) Page 4 EE457 pplied rtificial Intelligence R. Khoury (8) Page 4 Simulated nnealing pplication Optimizing TWT efficiency Synchronize electron velocity and phase velocity of RF wave Using phase velocity tapper to control and decrease RF wave phase velocity Improving tapper design improves synchronization, improves efficiency of TWT Tapper with simulated annealing algorithm to optimize synchronization oubled TWT efficiency More flexible then past tappers Maximize overall power efficiency Maximize efficiency over various bandwidth Maximize efficiency while minimize signal distortion ssumptions Goal-based agent Environment Fully observable eterministic Sequential Static iscrete Single agent EE457 pplied rtificial Intelligence R. Khoury (8) Page 4 EE457 pplied rtificial Intelligence R. Khoury (8) Page 44

12 ssumptions Updated Utility-based agent Environment Fully observable eterministic Sequential Static iscrete / ontinuous Single agent EE457 pplied rtificial Intelligence R. Khoury (8) Page 45