Local search algorithms
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1 Local search algorithms Chapter 4, Sections 3 4 Chapter 4, Sections 3 4 1
2 Outline Hill-climbing Simulated annealing Genetic algorithms Local search in continuous spaces (briefly) Chapter 4, Sections 3 4 2
3 Iterative improvement algorithms In many optimization problems, path is irrelevant; the goal state itself is the solution; e.g. N-queen problem. In such cases, we can use iterative improvement algorithms; keep a single current state, try to improve it by making small changes it in each iteration, until no further improvement is possible. Local search in the state space which is the set of complete configurations. find optimal configuration, e.g., TSP or, find configuration satisfying constraints, e.g.,, N-queen, timetablee. Constant space, suitable for online as well as offline search Chapter 4, Sections 3 4 3
4 Example: n-queens Problem: Put n queens on an n n board with no two queens on the same row, column, or diagonal Move a queen to reduce number of conflicts (number of attacking queens). h = 5 h = 2 h = 0 Almost always solves n-queens problems almost instantaneously for very large n, e.g., n = 1 million Chapter 4, Sections 3 4 4
5 Example: Travelling Salesperson Problem Short definition: Find the shortest path connecting N given cities. Start with any complete tour, perform pairwise exchanges: Variants of this approach get within 1% of optimal very quickly with thousands of cities Chapter 4, Sections 3 4 5
6 Iterative Improvement Algorithms Iterative improvement algorithms can be applied to minimize a cost (e.g. TSP or reducing number of conflicts) or to maximize a benefit. At each iteration, whatever measure we use, we will select the configuration that improves that measure. Chapter 4, Sections 3 4 6
7 Hill-climbing (or gradient ascent/descent) function Hill-Climbing( problem) returns a solution state inputs: problem, a problem local variables: current, a node next, a node current Make-Node(Initial-State[problem]) loop do next a highest-valued successor of current if Value[next] < Value[current] then return current current next end Note that this is a maximization problem, but Hill Climbing can be used for minimizing a cost, equally well (use negative of the cost as a benefit). Chapter 4, Sections 3 4 7
8 Hill-climbing contd. Problem: depending on initial state, can get stuck on local maxima value global maximum local maximum states Chapter 4, Sections 3 4 8
9 Continuous state spaces Suppose we want to find 3 locations to build three airports in Romania: 6-D state space defined by (x 1, y 2 ), (x 2, y 2 ), (x 3, y 3 ) objective function f(x 1, y 2, x 2, y 2, x 3, y 3 ) = sum of squared distances from each city to nearest airport Chapter 4, Sections 3 4 9
10 Continuous state spaces In continuous spaces, we have two options: Discretize the allowed values of the variables (modify by fixed amount) Use the gradient Chapter 4, Sections
11 Continuous state spaces - Discretization Discretization methods turn continuous space into discrete space, e.g., empirical gradient considers ±δ change in each coordinate Chapter 4, Sections
12 Continuous state spaces - Gradient Approach Use the gradient The standard way to solving continuous problems. Gradient methods compute f = f, f, f, f, f, f x 1 y 1 x 2 y 2 x 3 y 3 to increase/reduce f, e.g., by x x + α f(x) Problems w/ choosing step size, slow convergence Chapter 4, Sections
13 Continuous state spaces - IGNORE Sometimes can solve for f(x) = 0 exactly (e.g., with one city). Newton Raphson (1664, 1690) iterates x x H 1 f (x) f(x) to solve f(x) = 0, where H ij = 2 f/ x i x j Chapter 4, Sections
14 Stochastic hill-climbing Hill-climbing variations Choose at random from among the uphill moves. The probability of selection can vary with the steepness of the uphill move. Convergence: usually slower than hill climbing Solutions: Sometimes better Chapter 4, Sections
15 First choice hill-climbing Hill-climbing variations A variation of stochastic hill-climbing. Instead of finding all the possible moves (successor states) and picking one at random, it randomly generates a next state until it finds one which is better. Useful when there are many successor states (thousands). Chapter 4, Sections
16 Hill-climbing variations All hill-climbing algorithms so far are incomplete. Random-start-hill-climbing Search a goal state, starting from randomly generated initial states. This variation is complete with probability approaching to 1 (as the number of searches increase). In fact, it can solve a 3-million queen problem in under a minute (for 8-queen, the probability of success is roughly 0.14, hence 7 random starts is expected to find the solution (n = 1/p)) Chapter 4, Sections
17 Simulated annealing Hill-climbing algorithms never makes downhill moves, hence they are guaranteed to be incomplete since they can get stuck in local maxima. Solution: simulated annealing (again, complete only probabilistically) Idea: escape local maxima by allowing some bad moves but gradually decrease their size and frequency E.g. shaking the surface where a ping pong ball is rolling to get it to the global minimum. Devised by Metropolis et al., 1953, for physical process modelling. Widely used in VLSI layout, airline scheduling, etc. Chapter 4, Sections
18 Simulated annealing function Simulated-Annealing( problem, schedule) returns a solution state inputs: problem, a maximization problem schedule, a mapping from time to temperature local variables: current, a node next, a node T, a temperature controlling the probability of downward steps current Make-Node(Initial-State[problem]) for t 1 to do T schedule[t] if T=0 then return current next a randomly selected successor of current E Value[next] Value[current] if E > 0 then current next else current next only with probability e E/T Chapter 4, Sections
19 Local beam search Idea: Keeping only one node in memory is an extreme reaction to memory problems. Local beam search: keep k states instead of just one Loop: Start from k randomly generated states Generate all successors of all k states If goal is found, stop Otherwise, keep the top k of the successor states Chapter 4, Sections
20 Local beam search Not the same as k random-start searches run in parallel! Searches that find good states recruit other searches to join them Problem: quite often, all k states end up on same local hill Idea: Stochastic beam search Choose k successors randomly, biased towards good ones Observe the close analogy to natural selection! Chapter 4, Sections
21 Genetic algorithms = stochastic local beam search + generate successors from pairs of states % % % % (a) Initial Population (b) Fitness Function (c) Selection (d) Cross Over (e) Mutation Chapter 4, Sections
22 Genetic algorithms Population Fitness function Crossover Mutation Selection Chapter 4, Sections
23 Genetic algorithms contd. GAs require states encoded as strings (GPs use programs) Crossover helps iff substrings are meaningful components + = Chapter 4, Sections
24 Genetic algorithms Start with a population of k randomly selected states selection w.r.t the fitness function crossover (mating) mutation Idea: Try to take solutions to different subproblems from different nearsolutions. Chapter 4, Sections
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