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Advanced Computer Graphics Optimization Part 2 Spring 2002 Professor Brogan
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Simulated Annealing Review of Metropolis Procedure –Evaluation function –Current state –Temperature (and annealing schedule) –Change in state
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Computing New States Lacking derivatives (and when data isn’t smooth) –Doesn’t matter With derivatives –If you know which way is downhill (better) Then why risk losing that improvement by picking a new direction randomly
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Going Downhill If you always use derivative to go downhill –Metropolis step is ineffective (you’re always going to take the better answer) Process turns into a simple gradient descent algorithm
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Gradient Descent Follow derivatives downhill Performs poorly in narrow valleys And performs worse as the solution is reached
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Best of Both Worlds Current state = Simplex of N+1 points –Not just one point Envision simplex with lines connecting N+1 points –In three dimensions it is a tetrahedron The goal is for minimum to be spanned by points At each iteration, replace one point with a lower-valued one
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Simplex Method Diameter of simplex gets smaller with each iteration Stop when diameter reaches tolerance
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Example Find minimum of f(x,y) Simplex is a triangle –(N+1 = 3) Start with –A = (x 1, y 1 ), B = (x 2, y 2 ), C = (x 3, y 3 ) Reflection –Compute a new point, D = (x 4, y 4 ) that is reflection of highest point (A in this example) through midpoint of other two points D = B + C - A
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Example Expansion –If D is smaller than A –Then move was good –Try going further in same direction –E = 2D – (B + C) / 2
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Example Contraction –If D has same value as A, then find F and G F = (2A + B + C) / 4 G = (2D +B + C) / 4 –The smallest of F and G is kept
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Example Shrinkage If neither F nor G are smaller than A Then the side connecting A and C must move towards B in order to shrink the simplex –H = (A + B) / 2 –I = (B + C) / 2
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Back to Simulated Annealing Remember our state is N+1 points We add a positive, logarithmically distributed random variable, proportional to the temperature, T, to each point in state We subtract a similar random variable from the function value of each new point that is tried as a replacement point
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Simulated Annealing As temp goes to 0, this becomes Simplex At other T values –Brownian motion of simplex shape and sampling new, random points
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Want to read more? Numerical Recipes in C http://www.nr.com/nronline_switcher.ht mlhttp://www.nr.com/nronline_switcher.ht ml Source code can be found online too
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Golden Search in 1D The root of a function (where it equals 0) can be found when two points bracket it –One point has value > 0 –Other point has value < 0 –Somewhere in middle, function has value = 0
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Finding Min is Different 3 points are needed to find min –f(a) > f(b) –f(c) > f(b) –Min is certainly between a and c Similar to root finding algorithms like Bisection method –Choose a new point x that is between a and b or b and c
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Updating Points Try a point x between b and c If f(x) < f(b) –Points = (b, x, c) Else –Points = (a, b, x) Continue until distance between outer points is small
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Tolerance What is small enough span? (1-e)b < b < (1+e)b –Where e = 3x10 -8 (float) or 3x10 -15 (double) –Shape of f(x) near b is given by Taylor series –Thus, second term is tiny compared to the first and will act just like 0 when added to it –So keep e large (1x10 -4 (float))
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Golden Mean Where should new intermediate point be for (a, b, c)? Let b be fraction w from between a and c Suppose the next trial point were fraction z beyond b: (x – b) / (c – a) Next segment will either be of length w+z or 1- w To minimize worst case possibility, select z so these two are equal: z = 1 – 2w
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Golden Mean Resulting point is –|b-a| = |x-c| z will be in second portion if w < ½ Because w was actually selected using this same algorithm in previous step –x (at dist z) should be same fraction of way from b to c (if that was bigger seg) as was b from a to c (=w) –z / (1-w) = w Solve for w =.38197 (Golden mean)
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