Transcript [ppt]
Informed Search Next time: Search Application Reading: Machine Translation paper under Links Username and password will be mailed to class Questions on the HW? 2 Agenda A* example Hill climbing Example: n-Queens Online search Depth first Example: maze 3 A* Search OPEN = start node; CLOSED = empty While OPEN is not empty do Remove leftmost state from OPEN, call it X If X = goal state, return success Put X on CLOSED SUCCESSORS = Successor function (X) Remove any successors on OPEN or CLOSED Compute f(n)= g(n) + h(n) Put remaining successors on either end of OPEN Sort nodes on OPEN by value of heuristic function End while 4 5 Heuristics for other problems Problems Shortest path from one city to another Challenge: Is there an admissable heuristic for sodoku? 6 Romania with step costs in km 7 Local Search Algorithms Operate using a single current state Move only to neighbors of the state Paths followed by search are not retained Iterative improvement Keep a single current state and try to improve it 8 Advantages to local search Use very little memory – usually a constant amount Can often find reasonable solutions in large or infinite state spaces (e.g., continuous) Unsuitable for systematic search Useful for pure optimatization problems Find the best state according to an objective function Traveling salesman 9 10 11 Hill-climbing aka gradient ascent/descent; steepest ascent Function Hill-Climbing(problem) Returns a local maximum Inputs: problem Local variables: current (a node) neighbor (a node) Current <- make-node(initial-state[problem]) Loop do End. Neighbor <- highest valued successor of current If value(neighbor) <- value(current) return state(current) current <- neighbor 12 What we think hill-climbing looks like What we learn hill-climbing is Usually like 13 14 Problems for hill climbing When the higher the heuristic function the better: maxima (objective fns); when the lower the function the better: minima (cost fns) Local maxima: A local maximum is a peak that is higher than each of its neighboring states, but lower than the global maximum Ridges: a sequence of local maxima Plateaux: an area of the state space landscape where the evaluation function is flat 15 Hill-climbing search: 8-queens problem A local minimum with h = 1 16 Some solutions Stochastic hill-climbing First-choice hill climbing Generates successors randomly until one is generated that is better than current state Simulated annealing Chose at random from among the uphill moves Generate a random move. Accept if improvement. Otherwise accept with continually decreasing probability. Local beam search Keep track of k states rather than just 1 17 Another point of view Wrong peak has been climbed for over 22 years According to the official Nepalese peak list the highest Trekking Peak, Mera Peak -6654m- has never been climbed. Despite having been a popular climb for over 22 years, all expeditions attacked a peak that has now been revealed as the wrong one. Both new Nepalese maps and the official peak list confirm this new information. A first attempt to climb the new peak was made in May 2000, but the peak remains unclimbed. You can read more in Rock and Ice Super Guide 104 and Climber UK Sept 2000. http://www.abc.net.au/science/k2/moments/s1086384.htm 18 Avoiding climbing the wrong peak Random-restart hill climbing Keep restarting from randomly generated initial states, stopping when goal is found 19 ROOMBA 20 Online Search Agent operates by interleaving computation and action No time for thinking The agent only knows Actions (s) The step-cost function c(s,a,s’) Goal-test (s) Cannot access the successors of a state without trying all actions 21 Assumptions Agent recognizes a state it has seen before Actions are deterministic Competitive ratio: Compare cost that agent actually travels with cost of the actual shortest path 22 What properties of search are desirable? Will A* work? Expand nodes in a local order Depth first Variant of greedy search Difference from offline search: Agent must physically backtrack Record states to which agent can backtrack and has not yet explored 23 Depth-first OPEN = start node; CLOSED = empty While OPEN is not empty do Remove leftmost state from OPEN, call it X If X = goal state, return success Put X on CLOSED SUCCESSORS = Successor function (X) Remove any successors on OPEN or CLOSED Put remaining successors on left end of OPEN End while 24 Online DFS - setup Inputs: s’, a percept that identifies the current state Static: result, a table indexed by action and state, initially empty unexplored: a table that lists, for each visited state, the actions not yet tried unbacktracked: a table that lists, for each visited state, the backtracks not yet tried s,a: the previous state and action, initially null 25 Online DFS – the algorithm If Goal-test(s’) then return stop If s’ is a new state then unexplored[s’] actions(s’) If s is not null then do result[a,s]s’, result[reverse(a),s’)-< s Add s to the front of unbacktracked[s’] If unexplored[s’] is empty Then return stop Else a action b such that result[b,s’]=pop(unbacktracked[s’]) Else apop(unexplored[s’]) s s’ Return a 26 Learning Real Time A* (LRTA*) Augment hill-climbing with memory Store current best estimate of cost from node to goal: H(s) Initially, H(s) = h(s) Update H(s) through experience Estimated cost to reach the goal through neighbor s’ H(s) = c(s,a,s’)+ H(s’) 27 End of Class Questions 28