Informed search algorithms Chapter 4Local search algorithms In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution State space = set of "complete" configurations Find configuration satisfying constraints, e.g., n-queens In such cases, we can use local search algorithms keep a single "current" state, try to improve it. Very memory efficient (only remember current state)Example: n-queens Put n queens on an n × n board with no two queens on the same row, column, or diagonal Note that a state cannot be an incomplete configuration with m<n queensHill-climbing search Problem: depending on initial state, can get stuck in local maximaGradient Descent • Assume we have some cost-function: and we want minimize over continuous variables X1,X2,..,Xn 1. Compute the gradient : 2. Take a small step downhill in the direction of the gradient: 3. Check if 4. If true then accept move, if not reject. 5. Repeat.Exercise • Describe the gradient descent algorithm for the cost function:Line Search • In GD you need to choose a step-size. • Line search picks a direction, v, (say the gradient direction) and searches along that direction for the optimal step: • Repeated doubling can be used to effectively search for the optimal step: • There are many methods to pick search direction v. Very good method is “conjugate gradients”. € η* = argmin C(xt+ηvt)€ η→2η→4η→8η (until cost increases)Hill-climbing search: 8-queens problem h = number of pairs of queens that are attacking each other, either directly or indirectly (h = 17 for the above state) Each number indicates h if we move a queen in its corresponding columnHill-climbing search: 8-queens problem A local minimum with h = 1 what can you do to get out of this local minima?)Simulated annealing search Idea: escape local maxima by allowing some "bad" moves but gradually decrease their frequency. This is like smoothing the cost landscape.Properties of simulated annealing search One can prove: If T decreases slowly enough, then simulated annealing search will find a global optimum with probability approaching 1 (however, this may take VERY long) Widely used in VLSI layout, airline scheduling, etc.Local beam search Keep track of k states rather than just one. Start with k randomly generated states. At each iteration, all the successors of all k states are generated. If any one is a goal state, stop; else select the k best successors from the complete list and repeat.Genetic algorithms A successor state is generated by combining two parent states Start with k randomly generated states (population) A state is represented as a string over a finite alphabet (often a string of 0s and 1s) Evaluation function (fitness function). Higher values for better states. Produce the next generation of states by selection, crossover, and mutation Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 × 7/2 = 28) P(child) = 24/(24+23+20+11) = 31% P(child) = 23/(24+23+20+11) = 29% etc fitness: #non-attacking queens probability of being regenerated in next generationGiven N cities and all their distances, find the shortest tour through all cities. Try formulating this as a search problem. Ie what are the states, step-cost, initial state, goal state, successor function. Can you think of ways to try to solve these problems? Travelling Salesman
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