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Stochastic optimization is the general class of

algorithms and techniques which employ

some degree of randomness to find optimal

(or as optimal as possible) solutions to hardproblems.

Metaheuristics are the most general of these

kinds of algorithms, and are applied to a verywide range of problems.


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Metaheuristics are applied to I know it when I see it

problems.

Theyre algorithms used to findanswers to problems

when you have very little to help you:

you dont know what the optimal solution looks like you dont know how to go about finding it in a principled

way

you have very little heuristic information to go on,

and brute-force search is out of the question because thespace is too large.

But if youre given a candidate solution to your problem,

you can test it and assess howgood it is.

That is, you know a good one when you see it.


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Suppose youre trying to find an optimal set of robotbehaviors for a soccer goalie robot.

What do you have?

You have a simulator for the robot

and can test any given robot behavior set and assign it aquality (you know a good one when you see it).

And youve come up with a definition for what robotbehavior sets look like in general.

What you do not have? You have no idea what the optimal behavior set is

nor even how to go about finding it.


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Simplest way is to Do a Random Search:

just try random behavior sets as long as you have time,

and return the best one you discovered.


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You may consider the following alternative strategy

Start with a random behavior set.

Then make a small, random modification to it and try thenew version.

If the new version is better, throw the old one away. Else throw the new version away.

Now make another small, random modification to yourcurrent version.

If this newest version is better, throw away your current

version, else throw away the newest version. Repeat as long as you can.

This is called Hill Climbing!


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Similar solutions tend to behave similarly (and tend to

have similar quality),

This is usually true for many problems

So small modifications will generally result in small, well-

behaved changes in quality, allowing us to climb the hill

of quality up to good solutions.

This heuristic belief is one of the central defining features

of metaheuristics: indeed, nearly all metaheuristics are essentially elaborate

combinations of hill-climbing and random search.


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This is not a metaheuristic technique!

A traditional mathematical method for finding themaximum of a function

Consider a functionf(x), which we want to maximize

Identify the slope and move up it


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So, very simple method

This method doesnt require us to compute or even know

f(x).

But it does assume we can compute the slope ofx, that is,we havef(x).

Gradient Descent: it is used to find the minimum of a

function.

x


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For one dimension:


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When does the algorithm end???

As soon as we have found the ideal solution

Or, we have run out of time!

How to know when we have found the ideal solution?

Easy! When the slope is zero

But WAIT!!!! Slope is zero for minima as well! However, we

can get around this.

But, Slope is zero also for saddle points!


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Overshooting the Maximum:

As we get close to the maximum

of the function, Gradient Ascent

will overshoot the top and land

on the other side of the hill.

It mayovershoot the top many

times, bouncing back and forth

as it moves closer to the

maximum.


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One reason for Overshooting:

size of the jumps is entirely based on the current slope.

slope is steeper => jump is larger

One Solution: Adjust \alpha

\alpha smaller => no overshooting

\alpha smaller =>Long time to march up the hills

\alpha higher => constant overshooting; high convergencetime

So, we want \alpha to be just right


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Assume that we can computef(x).

Then we can modify as follows:

This dampens \alpha as we approach zero slope.

Multidimensional version is not so easy!!! Multidimensional version off(x) is a complex matrix

called a Hessian.

Also, matrix inversion is involved!!


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Hessian consists ofpartial second

derivatives along each dimension


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For one dimension:


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Because it employs the second derivative,

Newtons Method generally converges faster

than regular Gradient Ascent

it also gives us the information to determine ifwere at the top of a local maximum (as

opposed to a minimum or saddle point)

at a maximum,f(x) is zero and f(x) isnegative.

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