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Optimization stages of Sequential Minimal Optimization (SMO) algorithm in F#
// Copyright 2016 Serge Slipchenko (Serge.Slipchenko@gmail.com)
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
namespace Semagle.MachineLearning.SVM
open LanguagePrimitives
/// Implementation of Sequential Minimal Optimization (SMO) algorithm
module SMO =
[<Literal>]
let private tau = 1e-12
[<Literal>]
let private not_found = -1
type WSSStrategy = MaximalViolatingPair | SecondOrderInformation
type C_SVM = { iterations : int; epsilon : float; C : float; strategy : WSSStrategy }
let C_SVC (X : 'X[]) (Y : float[]) (K : Kernel<'X>) (parameters : C_SVM) =
if Array.length X <> Array.length Y then
invalidArg "X and Y" "have different lengths"
let info = printfn
let N = Array.length X
let C = Array.create N parameters.C
let epsilon = parameters.epsilon
let A = Array.zeroCreate N // optimization variables
let G = Array.create N -1.0 // gradient
// helper functions
let inline _y_gf i = -G.[i]*Y.[i]
let inline Q i j = (K X.[i] X.[j])*Y.[i]*Y.[j]
let I_up () = seq { 0..N-1 } |> Seq.filter (fun i -> (Y.[i] = +1.0 && A.[i] < C.[i]) || (Y.[i] = -1.0 && A.[i] > 0.0))
let I_low () = seq { 0..N-1 } |> Seq.filter (fun i -> (Y.[i] = -1.0 && A.[i] < C.[i]) || (Y.[i] = +1.0 && A.[i] > 0.0))
let inline maxUp () =
let ts = I_up ()
if Seq.isEmpty ts then not_found else Seq.maxBy _y_gf ts
let inline minLow () =
let ts = I_low ()
if Seq.isEmpty ts then not_found else Seq.minBy _y_gf ts
let inline minLowTo s =
let ts = I_low () |> Seq.filter (fun t -> _y_gf t < _y_gf s)
let objective t =
let a_ts = (Q t t) + (Q s s) - 2.0*(Q t s)*Y.[t]*Y.[s]
let b_ts = _y_gf t - _y_gf s
-b_ts*b_ts/(if a_ts > 0.0 then a_ts else tau)
if Seq.isEmpty ts then not_found else Seq.minBy objective ts
/// Maximal violating pair working set selection strategy
let maximalViolatingPair () =
let i = maxUp()
if i = not_found then None else Some (i, minLow())
/// Second order information working set selection strategy
let secondOrderInformation () =
let i = maxUp()
if i = not_found then None else Some (i, minLowTo i)
let selectWorkingSet =
match parameters.strategy with
| MaximalViolatingPair -> maximalViolatingPair
| SecondOrderInformation -> secondOrderInformation
/// Solve an optimization sub-problem
let inline solve i j =
let a_ij = (Q i i) + (Q j j) - 2.0*(Q i j)*Y.[i]*Y.[j]
if Y.[i] <> Y.[j] then
let delta = (-G.[i]-G.[j])/(if a_ij < 0.0 then tau else a_ij)
let diff = A.[i] - A.[j]
match (A.[i] + delta, A.[j] + delta) with
| _, a_j when diff > 0.0 && a_j < 0.0 -> (diff, 0.0)
| a_i, _ when diff <= 0.0 && a_i < 0.0 -> (0.0, diff)
| _, a_j when diff <= C.[i] - C.[j] && a_j > C.[j] -> (C.[j]+diff, C.[j])
| a_i, _ when diff > C.[i] - C.[j] && a_i > C.[i] -> (C.[i], C.[i] - diff)
| a_i, a_j -> a_i, a_j
else
let delta = (G.[i]-G.[j])/(if a_ij < 0.0 then tau else a_ij)
let sum = A.[i] + A.[j]
match (A.[i] - delta, A.[j] + delta) with
| a_i, _ when sum > C.[i] && a_i > C.[i] -> (C.[i], sum - C.[i])
| _, a_j when sum <= C.[i] && a_j < 0.0 -> (sum, 0.0)
| _, a_j when sum > C.[j] && a_j > C.[j] -> (sum - C.[j], C.[j])
| a_i, _ when sum <= C.[j] && a_i < 0.0 -> (0.0, sum)
| a_i, a_j -> a_i, a_j
/// Update gradient
let inline updateG i j a_i a_j =
for t in 0..N-1 do
G.[t] <- G.[t] + (Q t i)*(a_i - A.[i]) + (Q t j)*(a_j - A.[j])
/// Check stop conditions
let stopCriterion () =
let inline m y = I_up () |> Seq.filter (fun i -> Y.[i] = y) |> Seq.map _y_gf |> Seq.max
let inline M y = I_low () |> Seq.filter (fun i -> Y.[i] = y) |> Seq.map _y_gf |> Seq.min
((m +1.0) - (M +1.0) < epsilon) || ((m -1.0) - (M -1.0) < epsilon)
/// Sequential Minimal Optimization (SMO) Algorithm
let rec optimize k =
if k < parameters.iterations then
/// 1. Find a pair of elements that violate the optimality condition
match selectWorkingSet () with
| Some (i, j) ->
/// 2. Solve the optimization sub-problem
let a_i, a_j = solve i j
/// 3. Update the gradient and the solution
updateG i j a_i a_j
A.[i] <- a_i; A.[j] <- a_j
if stopCriterion() then k else optimize (k + 1)
| None -> k
else
failwith "Exceeded iterations limit"
let iterations = optimize 0
info "#iterations = %d" iterations
/// Reconstruction of hyperplane bias
let bias =
let mutable b = 0.0
let mutable M = 0
for i = 0 to N-1 do
if 0.0 < A.[i] && A.[i] < C.[i] then
b <- b + _y_gf i
M <- M + 1
DivideByInt b M
/// Remove support vectors with A.[i] = 0.0 and compute Y.[i]*A.[i]
let N' = Array.sumBy (fun a -> if a <> 0.0 then 1 else 0) A
let X' = Array.zeroCreate N'
let A' = Array.zeroCreate N'
let mutable j = 0
for i = 0 to N-1 do
if A.[i] <> 0.0 then
X'.[j] <- X.[i]
A'.[j] <- Y.[i]*A.[i]
j <- j + 1
info "support vectors = %d" N'
SVM(K,X',A',bias)
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