CSE 4705 Artificial Intelligence Jinbo Bi Department of Computer Science & Engineering http://www.engr.uconn.edu/~jinbo

Informed search strategies

Dominance

Iterative deepening A* and beyond

Iterative deepening A* and beyond

Informed search II When A* fails – hill climbing, simulated annealing Genetic algorithms

Outline

Local search and optimization

Hill climbing

Hill climbing on a surface of states

Hill climbing search

Hill climbing search example 4

Hill climbing search example: n-queens

Hill climbing search example: 8-queens

Hill climbing search space features

Hill climbing (variant) If we only know the function value at a state x, Hill Climbing ( or downhill) If besides the function value, we can compute gradient, Gradient ascent ( or descent)

Hill climbing (variant) If we only know the function value at a state x, and the state space is finite Hill Climbing ( or downhill) If besides the function value, we can compute gradient, and the state space is continuous Gradient ascent ( or descent)

Gradient ascent/descent Requires line search to decide a step size: (1) exact line search, solve a one-dimensional optimization problem (2)inexact line search (Armijo-Goldstein)

Gradient ascent/descent example min f(x,y) = (x-2)2 + (y-3)2 Initial guess is (1,1), find the optimal solution. Compute gradient f’(x,y) = ( 2(x-2), 2(y-3) ) f’(1,1) = (-2, -4) Set x = (1,1) – s * (-2,-4) Do a line search to find s g(s) = (2s-1)2 + (4s-2)2 = 20s2 – 20s + 5 g’(s) = 40s – 20 = 0 s = 0.5 Hence, x = (2, 3) Check if x is the fixed point

Gradient ascent/descent example From wikipedia

Gradient ascent/descent example

Gradient methods vs. Newton’s method

Drawbacks of hill climbing Local Maxima: peaks that are not the highest point in the space Plateaus: the space has a broad flat region that gives the search algorithm no direction (random walk) Ridges: drop-offs to the sides; steps to the North, East, South and West may go down, but a step to the NW may go up.

Example of a local maximum

The Shape of an Easy Problem (Convex)

The Shape of a Harder Problem

The Shape of an even Harder Problem

Random restart – a remedy to the drawbacks

Local beam search

Simulated annealing

Simulated annealing (SA)

Simulated annealing (SA)

Simulated annealing algorithm Example of simulated annealing https://www.youtube.com/watch?v=VWtYLv-4oP0 Example codes http://www.theprojectspot.com/tutorial-post/simulated-annealing-algorithm-for-beginners/6

Simulated annealing example h = 1, t=1, T = T0/2 = 50 Because T ~= 0, so Next = a random move Δ E = E1 – E0 = 1 If Δ E < 0, Current = Next Else compute p = e (- Δ E/T) =0.98 Generate a random number r from a uniform [0,1] Assume r = 0.8 Then, Current = Next Initial state T0 = 100 Annealing schedule is T T/2

Simulated annealing example h = 1, t=1, T = T0/2 = 50 Because T ~= 0, so Next = a random move Δ E = E1 – E0 = 1 If Δ E < 0, Current = Next Else compute p = e (- Δ E/T) =0.98 Generate a random number r from a uniform [0,1] Assume r = 0.8 Then, Current = Next Initial state T0 = 100 Annealing schedule is T T/2

Simulated annealing (variant) Another annealing scheduling

Genetic algorithms

Genetic algorithms

Representation: strings of genes

Encoding of a chromosome

Example: genetic algorithms for drive train

Operations: crossover

Operations: mutation

The basic genetic algorithm A short video explains GA in 3 minutes https://www.youtube.com/watch?v=ejxfTy4lI6I

Genetic algorithm: 8-queens

Questions?