Beyond Classical Search

Transcription

1 Beyond Classical Search

2 Local Search Iterative Improvement Algorithms qfor many problems, the solution path is irrelevant, othe goal state is itself the solution qstate space = set of complete configurations ogoal is to find an optimal configuration qiterative improvement algorithms okeep a single current state otry to improve it qconstant space. Local Search CSL30 - ARTIFICIAL INTELLIGENCE

3 Example: TSP qstart with any complete tour, perform pairwise exchanges Local Search CSL30 - ARTIFICIAL INTELLIGENCE 3

4 Example: n-queens qmove a queen to reduce number of conflicts Local Search CSL30 - ARTIFICIAL INTELLIGENCE 4

5 Local Search qhill Climbing qsimulated Annealing qlocal Beam Search qgenetic Algorithms Local Search CSL30 - ARTIFICIAL INTELLIGENCE 5

6 Hill Climbing qalso referred to as steepest ascent ogreedy local algorithm qlike climbing Mt. Everest in thick fog with amnesia Local Search CSL30 - ARTIFICIAL INTELLIGENCE 6

7 Hill Climbing Example (1) Local Search CSL30 - ARTIFICIAL INTELLIGENCE 7

8 Hill Climbing Example () qrandomly generated 8-queens starting states q14% of the time it solves the problem q86% of the time it gets stuck at a local minimum qhowever otakes only 4 steps on average when it succeeds oand 3 on average when it gets stuck ostate-space size 8 " ~17M states Local Search CSL30 - ARTIFICIAL INTELLIGENCE 8

9 Hill Climbing qconsider the state space landscape Ridges qrandom sideways moves qrandom-restart hill climbing qstochastic Hill Climbing Local Search CSL30 - ARTIFICIAL INTELLIGENCE 9

10 Random Sideways Moves qif no downhill moves, allow sideways moves in hope that algorithm can escape oneed to place a limit on the possible number of sideways moves qfor 8-queens oallow sideways moves with a limit of 100 oraises percentage of problem instances solved from 14-94% ØHowever, 1 steps for every successful solution and 64 for each failure. qtabu Search okeep track of states already visited Local Search CSL30 - ARTIFICIAL INTELLIGENCE 10

11 Simulated Annealing qidea: Escape local maxima by allowing some bad moves, but gradually decrease their size and frequency. ohill climbing is efficient, but incomplete orandom walk is complete but inefficient qinspired by metallurgical process of annealing othe required low energy state is attained by first heating the material and then gradually cooling it qtemperature Schedule mapping time to temperature. qwhen temperature is high allow random moves qwhen temperature is low reduce the probability of random moves o More like hill climbing Local Search CSL30 - ARTIFICIAL INTELLIGENCE 11

12 Simulated Annealing Local Search CSL30 - ARTIFICIAL INTELLIGENCE 1

13 Local Beam Search qidea: Keep k states instead of 1; choose top of all their successors qnot same as searches run in parallel osearches that find good states recruit other searches to join them qproblem: all states end up on the same hill qmodification: Stochastic local beam search opick successors randomly with bias towards good ones qanalogous to natural selection. Local Search CSL30 - ARTIFICIAL INTELLIGENCE 13

14 Genetic Algorithms qidea: Combine stochastic local beam search and generating successors from pairs of states qa state is represented as a string over a finite alphabet o 8-queens position of the queens in each column qpopulation - Start with k randomly generated states qevaluation Function (fitness function) o Higher values for better states o Opposite to heuristic functions qproduce the next generation of states by simulated evolution o Random selection o Crossover o Random mutation Local Search CSL30 - ARTIFICIAL INTELLIGENCE 14

15 Genetic Algorithms 4 starting states Fitness Function Number of non-attacking pairs of queens Probability of selection proportional to the Fitness function Local Search CSL30 - ARTIFICIAL INTELLIGENCE 15

16 General Comments on GA qpros orandom exploration via crossovers can find solutions that local search cannot oconnection to theory of human evolution qcons olarge number of tunable parameters ØDifficult to replicate the performance from one problem to another (too much of randomization) olack of good empirical studies comparing to other methods ouseful on some problems, but no convincing evidence of its superiority over hill-climbing with random restarts. Local Search CSL30 - ARTIFICIAL INTELLIGENCE 16

17 Local Search in Continuous Spaces Local Search CSL30 - ARTIFICIAL INTELLIGENCE 17

18 Plot of the Objective function f(x)

19 Contours of the Objective Function

20 Gradient f (x)

21 Figure 6: The method of Steepest Descent. (a) Starting at, take a step in the direction of steepest descent of. (b) Find the point on the intersection of these two surfaces that minimizes. (c) This parabola is the intersection of surfaces. The bottommost point is our target. (d) The gradient at the bottommost point Steepest Descent 4 (a) (b) (c) (d)

22 Gradient f (x) along the Search Line Figure 6: The method of Steepest Descent. (a) Starting at, take a step in the direction of steepest descent of. (b) Find the point on the intersection of these two surfaces that minimizes. (c) This parabola is the intersection of surfaces. The bottommost point is our target. (d) The gradient at the bottommost point is orthogonal to the gradient of the previous step Figure 7: The gradient is shown at several locations along the search line (solid arrows). Each gradient s projection onto the line is also shown (dotted arrows). The gradient vectors represent the direction of

23 Steepest Descent Convergence

24 Searching with Partial Observations qstate becomes belief state oagent s current belief about the possible physical states it might be in, given the sequence of actions and percepts up to that point qsearching with no observation conformant problem odoctors giving a prescription without a blood test Local Search CSL30 - ARTIFICIAL INTELLIGENCE 4

25 Vacuum world example Local Search CSL30 - ARTIFICIAL INTELLIGENCE 5