Thursday, February 24, 2011

Support vector machines (SVMs) in F# using Microsoft Solver Foundation

Support vector machines are a super star in machine learning and data mining in the past decade. It reheats statistical learning in machine learning community. It is also one of the classifiers that work practically with applications to other disciplines such as bioinformatics.

In this post, I will give a brief review of the theory of this classifier and show an F# implementation using the quadratic programming (QP) solver in Microsoft Solver Foundation (MSF).

It may seem dull to start with mathematics. But the math in SVMs is actually quite intuitive and would be quite easy if one has some training in linear algebra and calculus. Also understanding the intuition behind the formulae is more important in the formulation itself. There are quite a lot of good SVM tutorials written by researchers. Thus I only write the important formulas that occur in the implementation. The reader could link the F# code with formulas directly.

My note is based on the machine learning course by Prof. Jaakkola [Jaakkola] and the Matlab implementation therein. The course material is highly comprehensive yet accurate.

Background review for SVMs

Given a binary labeled datasetclip_image002, where clip_image004 and clip_image006 is 1 or -1. Think about the data points clip_image008 plotted in a d dimensional (Euclidean) space, the linear SVM classifier is a hyperplane clip_image010 in the space and it “best” separates the two kinds of data points. Here “best” means that the hyperplane should have the largest margin, which is the distance from the plane to the sides of labeled points. Intuitively the larger the margin is, the more robust and confident the classifier is. As if the hyperplane shakes a little, it still classifies well for its being far from both sides.

Let’s take d=2 as the discussing example: data points are plotted in a 2-D planeclip_image012. The aim is to draw a line clip_image014 to separate the two kinds of points such that the margin of the separation is maximized. In the following Figure, the line with maximum margin is shown. After some geometry work, the margin is calculated as clip_image016, thus minimizing clip_image018 with the constraints that the two kinds of points are well separated (clip_image020) gives the max-margin hyperplane.

clip_image022

(Wikipedia picture)

Two important extensions to the above simplest SVM are:

1. Allowing imperfect separation, that is when a hyperplane could not separate the points perfectly, it is allowed to mis-separate some data points. This introduces a slack variable clip_image024 for each data point clip_image008[1], when the point is on the correct side of the plane, the slack variable is 0, otherwise it is measures the distance it goes from the plane minus 1, if the values goes to below zero set it zero. (Please read the Note and reference section for more explanation.)

2. Kernel. The simplest SVM is a linear classifier in the original space. However, there are situations where linear classifier is not sufficient (even the best), one strategy is to do a non-linear transformation or mapping clip_image026 that maps a data point clip_image004[1] to another space, so that a linear classifier in that space, which is non-linear in the original clip_image028 space, does a better separation of the data points. The kernel trick is that we don’t need to define clip_image026[1] explicitly; only the definition of inner product of that space is required: clip_image030.

With the two extensions, the new maximum margin objective becomes:

clip_image032

subject to

clip_image034

The dual form:

clip_image036

subject to

clip_image038

clip_image040

The data points with its alpha value greater than 0 are called support vectors.

With the clip_image042in the dual problem solved, the SVM classification hyperplane is recovered by:

clip_image044

The threshold parameter clip_image046could be calculated by substituting back to the support vectors:

clip_image048

Any clip_image050 and clip_image052 (So that make sure data point clip_image008[2] is on the margin boundary, not in-between) would give a clip_image054, to stabilize the solution, it is common practice to take the median of off the possible clip_image054[1].

The implementation

The solver is actually an Interior Point Solver, which could solve linear and quadratic programming with linear constraints.

Page 14-17 of MSF-SolverProgrammingPrimer.pdf show an example usage of the QP solver in Microsoft Solver Foundation. But the example is not complete and it does not demonstrate one very subtle part: how to set the coefficients for the quadratic terms.

The following code example shows a simple example with the the comment regarding the coefficient setting:

Example usage of the QP solver
  1. #r @"C:\Program Files (x86)\Sho 2.0 for .NET 4\packages\Optimizer\Microsoft.Solver.Foundation.dll"
  2. open Microsoft.SolverFoundation.Common
  3. open Microsoft.SolverFoundation.Solvers
  4. open Microsoft.SolverFoundation.Services
  5. // test QP
  6. module TEST =
  7. (* minimize x^2 + y^2 + 3xy + 2x + y *)
  8. (* notice that in the InteriorPointSolver,
  9. the coefficients for xy & yx should be the same (so only set ONCE!)
  10. if we set both 3xy and 0yx, the solver takes the later coef.
  11. *)
  12. let solver = new InteriorPointSolver()
  13. let _, goal = solver.AddRow("dual objective value")
  14. solver.AddGoal(goal, 0, true)
  15. let _, x = solver.AddVariable("x")
  16. let _, y = solver.AddVariable("y")
  17. solver.SetCoefficient(goal, x, Rational.op_Implicit(2))
  18. solver.SetCoefficient(goal, y, Rational.op_Implicit(1))
  19. // for terms like x-y (where x != y), set its coef for only one time!
  20. solver.SetCoefficient(goal, Rational.op_Implicit(3), x, y)
  21. //solver.SetCoefficient(goal, Rational.Zero, y, x)
  22. solver.SetCoefficient(goal, Rational.op_Implicit(1), x, x)
  23. solver.SetCoefficient(goal, Rational.op_Implicit(1), y, y)
  24. let param = new InteriorPointSolverParams()
  25. solver.Solve(param) |> ignore
  26. //solver.Result
  27. let objectiveValue = solver.GetValue(0).ToDouble()
  28. let x0 = solver.GetValue(1).ToDouble()
  29. let y0 = solver.GetValue(2).ToDouble()
  30. x0*x0 + y0*y0 + 3.0 * x0 * y0 + 2. * x0 + y0
  31. //x0*x0 + y0*y0 + 0.0 * x0 * y0 + 2. * x0 + y0

The implementation of SVM is a straightforward translation of equations (1) to (5). The following shows the svm learning (bulidSvm) and testing (svmClassify) functions:

SVM Implementation
  1. open System
  2. type dataset =
  3. { features: float array array; // (instance = float array) array
  4. mutable labels: int array; //
  5. }
  6. with
  7. member x.NumSamples = x.features.Length
  8. module Array =
  9. let median (a:'a array) =
  10. let sorted = Array.sort a
  11. sorted.[sorted.Length / 2]
  12. module Kernel =
  13. let linear a b =
  14. Array.fold2 (fun acc p q -> acc + p * q) 0.0 a b
  15. let polynomial k a b =
  16. let dot = linear a b
  17. Math.Pow(1.0 + dot, k |> float)
  18. let gaussian beta a b =
  19. let diff = Array.fold2 (fun acc p q -> acc + (p-q)*(p-q)) 0.0 a b
  20. exp (-0.5 * beta * diff)
  21. module SVM =
  22. type svmmodel = {
  23. SVs:dataset;
  24. alpha:float array;
  25. kernel: float[] -> float[] -> float;
  26. w0:float;
  27. }
  28. with
  29. member x.NumSupporVectors = x.SVs.features.Length
  30. let buildSVM (ds:dataset) (C:float) (kernel:float[] -> float[] -> float) =
  31. let n = ds.features.Length
  32. let C = Rational.op_Implicit(C)
  33. let zero = Rational.Zero
  34. // create a interior point solver, which solves the QP problem
  35. let solver = new InteriorPointSolver()
  36. // set the objective value / goal
  37. let _, goal = solver.AddRow("dual objective value")
  38. // false == maximizing the objective value
  39. // the value of goal is (1)
  40. solver.AddGoal(goal, 0, false) |> ignore
  41. // add the Lagangian variables \alpha_i and set their bounds (0 <= \alpha_i <= C)
  42. let alpha = Array.create n 0
  43. for i=0 to n-1 do
  44. let _, out = solver.AddVariable("alpha_"+i.ToString())
  45. alpha.[i] <- out
  46. solver.SetBounds(out, zero, C)
  47. // add contraint: \sum_i \alpha_i * y_i = 0
  48. // equation (2)
  49. let _, sumConstraint = solver.AddRow("SumConstraint")
  50. solver.SetBounds(sumConstraint, zero, zero);
  51. for i=0 to n-1 do
  52. // set the coefs for the sum constraint
  53. // equation (2)
  54. solver.SetCoefficient(sumConstraint, alpha.[i], Rational.op_Implicit(ds.labels.[i]))
  55. // add the \alpha_i terms into the objective
  56. solver.SetCoefficient(goal, alpha.[i], Rational.One)
  57. // add the qudratic terms
  58. for j=0 to i do
  59. // coef = y_i * y_j * K(x_i, x_j)
  60. let coef = float(ds.labels.[i] * ds.labels.[j]) * (kernel ds.features.[i] ds.features.[j])
  61. if i=j then
  62. solver.SetCoefficient(goal, Rational.op_Implicit(-0.5 * coef), alpha.[i], alpha.[j])
  63. else
  64. solver.SetCoefficient(goal, Rational.op_Implicit(-coef), alpha.[i], alpha.[j])
  65. // use the default parameters
  66. let param = new InteriorPointSolverParams()
  67. solver.Solve(param) |> ignore
  68. // get the alpha values out
  69. let alphaValue = Array.init n (fun i -> solver.GetValue(i+1))
  70. (* print optimization result
  71. printfn "goal value = %A" (solver.GetValue(0).ToDouble())
  72. for i=1 to n do
  73. printfn "%A" (solver.GetValue(i).ToDouble())
  74. *)
  75. let alphaNew = new ResizeArray<Rational>()
  76. // extract the non-zero alpha values out and their corresponding support vectors
  77. let SVs =
  78. let feats = new ResizeArray<float[]>()
  79. let labs = new ResizeArray<int>()
  80. let maxAlpha = Array.max alphaValue
  81. let threshold = maxAlpha * Rational.op_Implicit(1e-8)
  82. for i=0 to n-1 do
  83. if alphaValue.[i] > threshold then
  84. feats.Add(ds.features.[i])
  85. labs.Add(ds.labels.[i])
  86. alphaNew.Add(alphaValue.[i])
  87. { features = feats |> Seq.toArray;
  88. labels = labs |> Seq.toArray;
  89. }
  90. // solve w_0 in the primal form
  91. let alphaNZ = alphaNew |> Seq.toArray
  92. // equation (5)
  93. let w0 =
  94. alphaNZ
  95. |> Array.mapi (fun i a ->
  96. if a = C then
  97. None
  98. else
  99. let mutable tmp = 0.0
  100. for j=0 to SVs.NumSamples-1 do
  101. tmp <- tmp + alphaNZ.[j].ToDouble() * (SVs.labels.[j] |> float) * (kernel SVs.features.[i] SVs.features.[j])
  102. Some ((float SVs.labels.[i]) - tmp)
  103. )
  104. |> Array.filter (fun v -> match v with None -> false | _ -> true)
  105. |> Array.map (fun v -> match v with Some v -> v | _ -> 0.0)
  106. |> Array.median
  107. // construct an svm record
  108. {
  109. SVs = SVs;
  110. alpha = alphaNZ |> Array.map (fun v -> v.ToDouble());
  111. kernel = kernel;
  112. w0 = w0;
  113. }
  114. let svmClassify (model:svmmodel) (ds:dataset) =
  115. // equation (4)
  116. let vals =
  117. ds.features
  118. |> Array.map (fun x ->
  119. let mutable sum = 0.0
  120. for i=0 to model.NumSupporVectors-1 do
  121. sum <- sum + model.alpha.[i] * (float model.SVs.labels.[i]) * (model.kernel model.SVs.features.[i] x)
  122. sum + model.w0
  123. )
  124. let nCorrect =
  125. Array.map2 (fun value label -> if (value > 0.0) && (label = 1) || (value < 0.0) && (label = -1) then 1 else 0) vals ds.labels
  126. |> Array.sum
  127. (float nCorrect) / (float ds.NumSamples), vals

Try on the iris data set we used in the Logistic Regression post:

let svm = buildSVM iris 10.0 Kernel.linear
let classifyResult = svmClassify svm iris

val svm : svm =
{SVs =
{features =
[|[|1.0; 6.2; 2.2; 4.5; 1.5|]; [|1.0; 5.9; 3.2; 4.8; 1.8|];
[|1.0; 6.3; 2.5; 4.9; 1.5|]; [|1.0; 6.7; 3.0; 5.0; 1.7|];
[|1.0; 6.0; 2.7; 5.1; 1.6|]; [|1.0; 5.4; 3.0; 4.5; 1.5|];
[|1.0; 4.9; 2.5; 4.5; 1.7|]; [|1.0; 6.0; 2.2; 5.0; 1.5|];
[|1.0; 6.3; 2.7; 4.9; 1.8|]; [|1.0; 6.2; 2.8; 4.8; 1.8|];
[|1.0; 6.1; 3.0; 4.9; 1.8|]; [|1.0; 6.3; 2.8; 5.1; 1.5|];
[|1.0; 6.0; 3.0; 4.8; 1.8|]|];
labels = [|-1; -1; -1; -1; -1; -1; 1; 1; 1; 1; 1; 1; 1|];};
alpha =
[|6.72796421; 10.0; 10.0; 10.0; 10.0; 6.475497298; 1.490719712;
8.547262902; 3.165478894; 10.0; 10.0; 10.0; 10.0|];
kernel = <fun:svm@226-32>;
w0 = -13.63716815;}

>
Real: 00:00:00.002, CPU: 00:00:00.000, GC gen0: 0, gen1: 0, gen2: 0

val classifyResult : float * float [] =
(0.97,
[|-2.787610618; -2.380530973; -1.42477876; -2.929203541; -1.681415929;
-1.96460177; -1.24778761; -6.106194692; -2.761061946; -2.973451328;
…. 4.203539824; 1.82300885|])

Notes and references

The optimization in SVMs could be treated as a general QP problem as shown in our implementation. However, when the number of the data points is big, the QP grows too big to handle by the QP solver.

As a special QP problem, SVM has a lot of novel solutions proposed in the machine learning research area, e.g. the SMO approach [Platt] in LibSVM implementation and the Ball Vector Machine approach [Tsang] transforming the special QP into a (discrete) computational geometry problem, minimum enclosing ball. For practical usage of SVM, one usually uses these dedicated approaches.

Besides the maximum margin interpretation, SVMs also have theory roots in functional regularization theory, in which the optimization clip_image002[5]

is viewed as minimizing the error term clip_image004[7] while controlling the complexity of the function clip_image006[5] and C is a tradeoff parameter between these two. The reader is referred to the teaching notes and papers by Prof. Poggio [Poggio].

[Burges] A Tutorial on Support Vector Machines for Pattern Recognition. http://research.microsoft.com/en-us/um/people/cburges/papers/svmtutorial.pdf

[Jaakkola] http://courses.csail.mit.edu/6.867/.

[Platt] Sequential Minimal Optimization: A Fast Algorithm for Training Support Vector Machines. ftp://ftp.research.microsoft.com/pub/tr/tr-98-14.pdf

[Poggio] http://www.mit.edu/~9.520/spring10/.

[Tsang] http://www.cs.ust.hk/~ivor/cvm.html.

9 comments:

  1. This comment has been removed by the author.

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  2. This comment has been removed by the author.

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  3. Great post Yin!

    I was looking for info on how to use Microsoft Solver Foundation and I stumbled on your great blog again :)

    ReplyDelete
  4. "one _tragedy_ is to do a non-linear transformation" ? I think the spell correction got the better of your text ;-)

    "from the plane minus 1" - I hadn't seen that in my previous reading about SVMs. Isn't the slack just the distance past the boundary (without the -1)?

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  5. The SVM paper I was looking at was http://scribblethink.org/Work/Notes/svmtutorial.pdf, which on second reading does have the step that changes from a 0 to 1 slack excess.

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  6. @philipoakley: Thanks for the correction on the spelling. "from the plane minus 1" -- please search "hinge loss function" to get more details of it:)

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  7. see this link:
    http://research.microsoft.com/en-us/um/people/manik/projects/trade-off/hinge.html

    The blue line is the hinge loss, which is 0 when the difference is below 1.

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  8. This comment has been removed by the author.

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  9. March 2013
    Hi Yin.
    Thanks for your blogs on F# MSF.

    What is the latest status of F# and MSF?

    Thanks, Art Scott

    ReplyDelete