Mathias Brandewinder on .NET, F#, VSTO and Excel development, and quantitative analysis / machine learning.
by Mathias 15. February 2014 12:51

My favorite column in MSDN Magazine is Test Run; it was originally focused on testing, but the author, James McCaffrey, has been focusing lately on topics revolving around numeric optimization and machine learning, presenting a variety of methods and approaches. I quite enjoy his work, with one minor gripe –his examples are all coded in C#, which in my opinion is really too bad, because the algorithms would gain much clarity if written in F# instead.

Back in June 2013, he published a piece on Amoeba Method Optimization using C#. I hadn’t seen that approach before, and found it intriguing. I also found the C# code a bit too hairy for my feeble brain to follow, so I decided to rewrite it in F#.

In a nutshell, the Amoeba approach is a heuristic to find the minimum of a function. Its proper respectable name is the Nelder-Nead method. The reason it is also called the Amoeba method is because of the way the algorithm works: in its simple form, it starts from a triangle, the “Amoeba”; at each step, the Amoeba “probes” the value of 3 points in its neighborhood, and moves based on how much better the new points are. As a result, the triangle is iteratively updated, and behaves a bit like an Amoeba moving on a surface.

Before going into the actual details of the algorithm, here is how my final result looks like. You can find the entire code here on GitHub, with some usage examples in the Sample.fsx script file. Let’s demo the code in action: in a script file, we load the Amoeba code, and use the same function the article does, the Rosenbrock function. We transform the function a bit, so that it takes a Point (an alias for an Array of floats, essentially a vector) as an input, and pass it to the solve function, with the domain where we want to search, in that case, [ –10.0; 10.0 ] for both x and y:

#load "Amoeba.fs"

open Amoeba
open Amoeba.Solver

let g (x:float) y =
    100. * pown (y - x * x) 2 + pown (1. - x) 2

let testFunction (x:Point) =
    g x.[0] x.[1]

solve Default [| (-10.,10.); (-10.,10.) |] testFunction 1000

Running this in the F# interactive window should produce the following:

val it : Solution = (0.0, [|1.0; 1.0|])
>

The algorithm properly identified that the minimum is 0, for a value of x = 1.0 and y = 1.0. Note that results may vary: this is a heuristic, which starts with a random initial amoeba, so each run could produce slightly different results, and might at times epically fail.

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by Mathias 27. June 2012 15:06

This months’ issue of MSDN Magazine has an interesting piece on evolutionary algorithms. The article applies a genetic algorithm to identify the minimum value of a “pathological” continuous function, the Schwefel function.

SchefelFunction

The Schwefel function

For X and Y values between –500 and 500, the “correct” answer is X and Y = 420.9687.

This function is designed to give fits to optimization algorithms. The issue here is that  the function has numerous peaks and valleys. As a result, if the search strategy is to use some form of gradient approach, that is, from a starting point, follow the steepest descent until a minimum is reached, there is a big risk to end up in a place which is a local minimum: the algorithm gets stuck in a valley with no path downwards, but there are other, better solutions around, which can be reached only by “climbing out of the hole” and temporarily following a path which heads in the wrong direction.

Out of curiosity, I checked how the Excel Solver would fare on this problem:

ExcelSchwefel

The result was an abject failure – not even close to the true solution.

I thought it would be interesting to see how Bumblebee, my Artificial Bee Colony framework, would perform. There are some general similarities between the underlying ideas behind the articles’ algorithm and Bumblebee, the main difference being that Bumblebee simply mutates individual solutions, and doesn’t create “crossover solutions”.

Let’s open Visual Studio, create an F# Console project, grab the NuGet package for Bumblebee – and start coding.

As usual, we need 4 elements to leverage Bumblebee – a Type of Solution, and 3 functions: a Generator, which returns a brand-new, random solution, a Mutator, which transforms a known solution into a new, similar solution, and an Evaluator, which evaluates a solution and returns a float, increasing with the quality of the solution.

In this case, the Solution type is fairly straightforward. We are looking for 2 floats x and y, so we’ll go for a Tuple. Similarly, the Evaluation is a given, it is the negative of the Schwefel function. The reason we go for the negative is that Bumblebee will try to maximize the Evaluation, so if we are looking for a minimum, we need to reverse the sign – because the Minimum of a function is the Maximum of its negative.

let schwefel x = 
   -x * Math.Sin(Math.Sqrt(Math.Abs(x)))

let evaluate (x, y) = 
   - schwefel (x) - schwefel (y)

The Generation function is also trivial – we’ll simply pick random pairs of floats in the [ –500.0; 500.0 ] interval:

let min = -500.0
let max = 500.0

let generate (rng: Random) = (
   rng.NextDouble() * (max - min) + min,
   rng.NextDouble() * (max - min) + min)

The Mutation function takes a tiny bit more of effort. The idea I followed is essentially the same as the one used in the article: given a solution defined by a pair of floats, randomly decide if any of the elements will be mutated, and if yes, add a random permutation to that element, scaled to the precision we want to achieve:

let precision = 0.00001
let rate = 0.5

let mutate (rng: Random) solution =
   let (x, y) = solution
   let x =
      if rng.NextDouble() < rate 
      then x + precision * ((max - min) * rng.NextDouble() + min)
      else x
   let y =
      if rng.NextDouble() < rate 
      then y + precision * ((max - min) * rng.NextDouble() + min)
      else y
   (x, y)

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by Mathias 29. July 2009 11:24

Today I came across a post which demonstrates how to use Goal Seek to determine how to save for your retirement. Goal Seek is essentially a simplified solver: point Goal Seek at a cell, tell it how much you want it to be and what cells it can tinker with, and Goal Seek will try to find values that reach that goal. The post is an excellent illustration of what’s great about it: it’s super easy to use, and very practical.

However, there is no such thing as a perfect tool, and Goal Seek can fail miserably at finding the optimal answer to very simple problems. After reading this, I thought it would be a good public service to illustrate what its shortcomings are, especially if you are going to trust it for questions as important as your retirement!

For our illustration, we will use the following setup.

Setup

Now let’s say we want to find a value such that B2 = 250. Following Pointy Haired Dilbert, let’s use Goal Seek:

GoalSeek

Put in a value of 1 in cell B1, and run Goal Seek - here is what happens:

GoalSeekFail

Goal Seek fails to find a value in B1 such that B2 = 250. Complete failure.

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by Mathias 4. December 2008 07:53

Great post on the Excel blog, explaining from start to finish how to set up a shortest path problem using Excel, and how to use the solver to identify the optimal solution.

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