04_LocalSearch

# 04_LocalSearch - Local Search Scaling Up So far, we have...

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Local Search

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Scaling Up So far, we have considered methods that systematically explore the full search space, possibly using principled pruning (A* etc.). The current best such algorithms (RBFS / SMA*) can handle search spaces of up to 10 100 states → ~ 500 binary valued variables. But search spaces for some real-world problems might be much bigger - e.g. 10 30,000 states. Here, a completely different kind of search is needed. Local Search Methods
Optimization Problems We're interested in the Goal State - not in how to get there. Optimization Problem: - State: vector of variables - Objective Function: f : state - Goal: find state that maximizes or minimizes the objective function Examples: VLSI layout, job scheduling, map coloring, N-Queens

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Representations for 8Q problem
Heuristic for 8Q problem?

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Heuristic for 8Q problem?
Local Search Methods Applicable to optimization problems. Basic idea: - use a single current state - don't save paths followed - generally move only to successors/neighbors of that state Generally require a complete state description .

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Hill-Climbing Search function HILL-CLIMBING ( ) returns a solution state inputs: , a problem static: , a node MAKE-NODE(INITIAL-STATE[ ]) loop do a highest-valued succe problem problem current current problem next ssor of if VALUE[next] < VALUE[current] then return end current current current next
Current State Evaluation

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Hill Climbing Pathologies Objective function Shoulder Global Maximum Local Maximum “flat” local maximum State Space Value of current solution

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Local Maximum Example

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Neutral “Sideways” moves Take new state even if not strictly better (just equal) Allows exploring plateaus …But can get into cycles
Neutral “Sideways” moves 8Q problem without sideways Stuck 86% time 4 steps to succeed 3 to get stuck 8Q problem with sideways Succeeds 94% time 21 steps to succeed 64 to get stuck

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Random restarts Random restarts: Simply restart at a new random state after a pre-defined number of steps. Is it worth it? If probability of success is p, then Expected number of trials to success is 1/p
Neutral “Sideways” moves 8Q problem without sideways Stuck 86% time 4 steps to succeed 3 to get stuck 8Q problem with sideways Succeeds 94% time 21 steps to succeed 64 to get stuck A = With Sideways B = Without Sideways

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Nelder-Mead (Simplex) Method Reflect the point with the highest WSS through centroid (center) of the simplex If this produces the lowest WSS (best point) expand the simplex and reflect further If this is just a good point start at the top and reflect again
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## 04_LocalSearch - Local Search Scaling Up So far, we have...

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