Mathias Brandewinder on .NET, F#, VSTO and Excel development, and quantitative analysis / machine learning.
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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24. April 2011 11:44

April 2011’s issue of MSDN Magazine had an interesting piece on Bee Colony Algorithms by Dr. James McCaffrey, explaining the concepts and providing an example, applying the algorithm to the Traveling Salesman Problem. In a nutshell, the algorithm is a meta-heuristic, that is, a method that is not guaranteed to produce an optimal solution, but will search for “decent” solutions in a large space. In a real-life bee hive,  bees scout for areas rich with food, keep visiting them until they are exhausted, and tell other bees about good spots so that more bees come search that area. By analogy, the algorithm uses scout bees, which search for new random solutions, and recruit inactive bees which become active and start searching for improved solutions around their current solution.

I found the algorithm intriguing, and thought it would be a good learning exercise to try and adapt it to F#.

Disclaimer: I am still learning the ropes in F#, so take the code that follows with a grain of salt. I’ll gladly take advice and criticism to make this better – my intent is to share my learning experience with the language, not to teach you best practices.

In the case of the Traveling Salesman Problem, the goal is to find the shortest (or some other cost measure) closed route connecting a list of cities. In order to do this, we need to be able to create random solutions, as well as solutions in the neighborhood of an existing solution.

Assuming we begin with an initial list of Cities (the cities our salesman needs to visit), we can generate random solutions by shuffling that list, using the Fisher-Yates shuffle algorithm. We can generate the sequence of index pairs that need to be swapped with the following

let SwapIndexPairs list =
let random = new Random()
seq {
for i in (List.length list - 1) .. -1 .. 1 do
yield (i, random.Next(i + 1)) }

Running this in the interactive window produces the following:

> open System;;
> let SwapIndexPairs list =
let random = new Random()
seq {
for i in (List.length list - 1) .. -1 .. 1 do
yield (i, random.Next(i + 1)) };;

val SwapIndexPairs : 'a list -> seq<int * int>

> let i = SwapIndexPairs [0;1;2;3;4;5] |> Seq.toList;;

val i : (int * int) list = [(5, 0); (4, 0); (3, 2); (2, 1); (1, 1)]

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