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Function minimization: Simulated Annealing led by variance criteria vs Nelder Mead

Most of the datamining problems can be reduced as a minimization/maximization problem.

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Examples
Let's consider  easy scenarios where the function cost is conditionated just by two parameters.

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As you can see the above problems have in common a massive presence of local minimum :).
Let's see how to handle these problems through an approach that I define hybrid, that is, obtained mixing different methods and properties.
Simulated Annealing (SA) "variance based" approach
 

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Variance trick 

Let's consider a sampling in two different region of a function f :

  • sampling_1 in a region having a smooth minimum and evaluate such points
  • sampling_2 in a region having a spiky minimum and evaluate such points
Take now the variance of these evaluated sets.
How does the variance of f(sampling_1)  change respect the variance of f(sampling_2)?  
Here you are the answer:
As you can see the variance can be used as indicator of a minimum region of the cost function.
Instead of explore randomly the solution space, the approach I propose is led by the variance used as  above. 
What happens if the smooth valley is a global minimum?
There are two intuitive factors to avoid that the algorithm jams in a local minimum:
1. The acceptance law admit also pejorative solutions.
2. The sampling procedure moderate by variance (if the variance is chosen high enough) allows explorations of better regions.
The algorithm

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Tests
Let's see how the approach works.
As usual, all my experiments are compared with other techniques. In this case I chose as comparative method the well known "Nelder Mead" method.
I tried to optimize the Nelder Mead method playing around its param setting: shrink ratio, contraction ratio and reflection ratio.

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On the left side it has been represented the solution found by Nelder Mead method.
On the right side the solution found by the SA presented in this post: the blue point depicts the starting point and the red point depicts the solution found.
The last chart shows the space solution explored

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EXPERIMENT 2:

On the left side it has been represented the solution found by Nelder Mead method.
On the right side the solution found by the SA presented in this post: the blue point depicts the starting point and the red point depicts the solution found.

Experiment 3

Experiment 4:

Conclusion
The approach seems promising and works well in different contexts:  I'm quite sure that many improvements can be implemented.
I'm looking forward to receive your comments, ideas and other comparative tests.
Stay tuned!
 
cristian

 

 

 

 

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