From 125e09769ab8e2ea95bf7ccf1c708f9ed474008a Mon Sep 17 00:00:00 2001 From: James Foster Date: Fri, 10 Nov 2023 14:56:42 +1100 Subject: [PATCH 01/11] Initial version --- docs/make.jl | 1 + docs/src/tutorials/nonlinear/classifiers.jl | 185 ++++++++++++++++++++ docs/styles/Vocab/JuMP-Vocab/accept.txt | 1 + 3 files changed, 187 insertions(+) create mode 100644 docs/src/tutorials/nonlinear/classifiers.jl diff --git a/docs/make.jl b/docs/make.jl index 67fdfd181cb..54f353c66f3 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -317,6 +317,7 @@ const _PAGES = [ "tutorials/nonlinear/nested_problems.md", "tutorials/nonlinear/querying_hessians.md", "tutorials/nonlinear/complementarity.md", + "tutorials/nonlinear/classifiers.md", ], "Conic programs" => [ "tutorials/conic/introduction.md", diff --git a/docs/src/tutorials/nonlinear/classifiers.jl b/docs/src/tutorials/nonlinear/classifiers.jl new file mode 100644 index 00000000000..1f2728e593a --- /dev/null +++ b/docs/src/tutorials/nonlinear/classifiers.jl @@ -0,0 +1,185 @@ +# Copyright 2023 James Foster and contributors #src +# This Source Code Form is subject to the terms of the Mozilla Public License #src +# v.2.0. If a copy of the MPL was not distributed with this file, You can #src +# obtain one at https://mozilla.org/MPL/2.0/. #src + +# # Classifiers + +# Classification problems deal with constructing functions, called *classifiers*, +# that can efficiently classify data into two or more distinct sets. +# A common application is classifying previously unseen data points +# after training a classifier on known data. + +# The theory and models in this tutorial come from +# Section 9.4 "Classification Problems" of the book +# [*Linear Programming with MATLAB*](https://doi.org/10.1137/1.9780898718775) (2007) +# by M. C. Ferris, O. L. Mangasarian, and S. J. Wright. + +# ## Required packages + +# This tutorial uses the following packages + +using JuMP +import DataFrames +import HiGHS +import Ipopt +import LinearAlgebra +import Plots +import Random +import SparseArrays +import Test #src + +# ## Data and visualisation + +# To start, let's generate some points to test with. +# The argument ``m`` is the number of 2-dimensional points: + +function generate_test_points(m; random_seed = 1) + rng = Random.MersenneTwister(random_seed) + return 2.0 .* rand(rng, Float64, m, 2) +end + +# For the sake of the example, let's take ``m = 100``: +P = generate_test_points(100); + +# Note that the points are represented row-wise in the generated array. +# Let's visualise the points using the `Plots` package: + +r = 1.01 * maximum(abs.(P)) +plot1 = Plots.scatter( + P[:, 1], + P[:, 2]; + xlim = (0, r), + ylim = (0, r), + label = nothing, + color = :white, + shape = :circle, + size = (600, 600), + legend = false, +) + +# We want to split the points into two distinct sets on either side of a dividing line. +# We'll then label each point depending on which side of the line it happens to fall. +# Based on the labels of the point, we'll show how to create a classifier using a JuMP model. +# We can then test how well our classifier reproduces the original labels and the boundary between them. + +# Let's make a line to divide the point into two sets by defining a gradient and constant: +w0 = [5, 3]; +g0 = 8; +line(v::AbstractArray; w = w0, g = g0) = LinearAlgebra.dot(w, v) - g +line(x::Real; w = w0, g = g0) = -(w[1] * x - g) / w[2]; + +# Julia's multiple dispatch feature allows us to define the vector and single-variable form +# of the `line` function under the same name. + +# Let's add this to the plot: + +Plots.plot!( + plot1, + x -> line(x), + 0.0:0.01:2.0; + seriestype = :line, + linewidth = 5, + lineopacity = 0.6, + c = :darkcyan, +) + +# Now we label the points relative to which side of the line they are. +P_pos = hcat(filter(v -> line(v) > 0, eachrow(P))...)' +P_neg = hcat(filter(v -> line(v) < 0, eachrow(P))...)' + +@assert size(P_pos, 1) + size(P_neg, 1) == size(P, 1) #src + +Plots.scatter!( + P_pos[:, 1], + P_pos[:, 2]; + shape = :cross, + markercolor = :blue, + markersize = 8, +); +Plots.scatter!( + P_neg[:, 1], + P_neg[:, 2]; + shape = :xcross, + markercolor = :crimson, + markersize = 8, +); +plot1 + +# The goal is to show we can reconstruct the line from *just* the points and labels. + +# ## Formulation: linear support vector machine + +# Firstly, we will put the point set back together row-wise as a matrix, with the labelled points group together: + +A = [P_pos; P_neg]; +m, n = size(A) + +# To keep track of the labels, we'll use a diagonal matrix where entry ``i`` of the diagonal is the +# label for point ``i`` (row ``i`` of the matrix). + +D = LinearAlgebra.Diagonal([ones(size(P_pos, 1)); -ones(size(P_neg, 1))]) + +# It is numerically useful to have the labels +1 for `S_pos` and -1 for `S_neg`. + +# A classifier known as the linear *support vector machine* looks for the line function data +# ``w`` and ``g`` such that the function ``L(v) = w^T v - g`` satisfies +# ``L(p) < 0`` for a point ``p`` in `S_neg` and ``L(p) > 0`` on `S_pos`. +# The linearly constrained quadratic program that implements this as follows: + +# ```math +# \begin{aligned} +# & \min_{w \in \mathbb{R}^n, \; g \in \mathbb{R}, \; y \in \mathbb{R}^m} & \frac{1}{2} w^T w + C \; \sum_{i=1}^m y_i \\ +# & \text{subject to} & D_{ii}( A_{i :} w - g ) + y_i & \geq 1, & i = 1 \ldots m \\ +# \end{aligned} +# ``` + +# We need a value for the positive penalty parameter ``C``: +C = 1e3; + +# ## JuMP formulation + +# Now let's build and the JuMP model. We'll get ``w`` and ``g`` from the optimal solution after the +# solve. + +m, n = size(A) + +model = Model(Ipopt.Optimizer) +# set_silent(model) +@variable(model, w[1:n]) +@variable(model, g) +@variable(model, y[1:m] >= 0) +@constraint(model, [i in 1:m], D[i, i] * (A[i, :]' * w - g) + y[i] >= 1) +@objective(model, Min, 0.5 * w' * w + C * sum(y)) +optimize!(model) +Test.@test termination_status(model) == LOCALLY_SOLVED #src +Test.@test primal_status(model) == FEASIBLE_POINT #src +solution_summary(model) + +# ## Results + +# We recover the solution values +w_sol = value.(w) +g_sol = value(g) +y_sol = value.(y); +println("Minimum slack: ", minimum(y_sol), "\nMaximum slack: ", maximum(y_sol)) + +# With the solution, we can ask: was the value of the penalty constant "sufficiently large" +# for this data set? This can be judged in part by the range of the slack variables. + +# Let's add this to the plot as well and check how we did: + +Plots.plot!( + plot1, + x -> line(x; w = w_sol, g = g_sol), + 0.0:0.01:2.0; + seriestype = :line, + linewidth = 5, + linestyle = :dashdotdot, + lineopacity = 0.9, + c = :darkblue, +); +plot1 + +# We find that we have recovered the dividing line from just the information of the points +# and their labels. diff --git a/docs/styles/Vocab/JuMP-Vocab/accept.txt b/docs/styles/Vocab/JuMP-Vocab/accept.txt index d188dcc1ea1..c91bde97415 100644 --- a/docs/styles/Vocab/JuMP-Vocab/accept.txt +++ b/docs/styles/Vocab/JuMP-Vocab/accept.txt @@ -205,6 +205,7 @@ Linial Luangkesorn Lubin Madani +Mangasarian Marek Markowitz Mathieu From eb9a23b81cacdbfe7154ac0fea886212315a8d29 Mon Sep 17 00:00:00 2001 From: Oscar Dowson Date: Sun, 12 Nov 2023 11:57:07 +1300 Subject: [PATCH 02/11] Apply suggestions from code review --- docs/src/tutorials/nonlinear/classifiers.jl | 32 +++++++++------------ 1 file changed, 14 insertions(+), 18 deletions(-) diff --git a/docs/src/tutorials/nonlinear/classifiers.jl b/docs/src/tutorials/nonlinear/classifiers.jl index 1f2728e593a..04a71b26eda 100644 --- a/docs/src/tutorials/nonlinear/classifiers.jl +++ b/docs/src/tutorials/nonlinear/classifiers.jl @@ -46,7 +46,7 @@ P = generate_test_points(100); # Let's visualise the points using the `Plots` package: r = 1.01 * maximum(abs.(P)) -plot1 = Plots.scatter( +plot = Plots.scatter( P[:, 1], P[:, 2]; xlim = (0, r), @@ -64,8 +64,8 @@ plot1 = Plots.scatter( # We can then test how well our classifier reproduces the original labels and the boundary between them. # Let's make a line to divide the point into two sets by defining a gradient and constant: -w0 = [5, 3]; -g0 = 8; +w0 = [5, 3] +g0 = 8 line(v::AbstractArray; w = w0, g = g0) = LinearAlgebra.dot(w, v) - g line(x::Real; w = w0, g = g0) = -(w[1] * x - g) / w[2]; @@ -75,7 +75,7 @@ line(x::Real; w = w0, g = g0) = -(w[1] * x - g) / w[2]; # Let's add this to the plot: Plots.plot!( - plot1, + plot, x -> line(x), 0.0:0.01:2.0; seriestype = :line, @@ -87,24 +87,23 @@ Plots.plot!( # Now we label the points relative to which side of the line they are. P_pos = hcat(filter(v -> line(v) > 0, eachrow(P))...)' P_neg = hcat(filter(v -> line(v) < 0, eachrow(P))...)' - @assert size(P_pos, 1) + size(P_neg, 1) == size(P, 1) #src - Plots.scatter!( + plot, P_pos[:, 1], P_pos[:, 2]; shape = :cross, markercolor = :blue, markersize = 8, -); +) Plots.scatter!( + plot, P_neg[:, 1], P_neg[:, 2]; shape = :xcross, markercolor = :crimson, markersize = 8, -); -plot1 +) # The goal is to show we can reconstruct the line from *just* the points and labels. @@ -112,7 +111,7 @@ plot1 # Firstly, we will put the point set back together row-wise as a matrix, with the labelled points group together: -A = [P_pos; P_neg]; +A = [P_pos; P_neg] m, n = size(A) # To keep track of the labels, we'll use a diagonal matrix where entry ``i`` of the diagonal is the @@ -131,6 +130,7 @@ D = LinearAlgebra.Diagonal([ones(size(P_pos, 1)); -ones(size(P_neg, 1))]) # \begin{aligned} # & \min_{w \in \mathbb{R}^n, \; g \in \mathbb{R}, \; y \in \mathbb{R}^m} & \frac{1}{2} w^T w + C \; \sum_{i=1}^m y_i \\ # & \text{subject to} & D_{ii}( A_{i :} w - g ) + y_i & \geq 1, & i = 1 \ldots m \\ +# & & y \ge 0. # \end{aligned} # ``` @@ -143,9 +143,8 @@ C = 1e3; # solve. m, n = size(A) - model = Model(Ipopt.Optimizer) -# set_silent(model) +set_silent(model) @variable(model, w[1:n]) @variable(model, g) @variable(model, y[1:m] >= 0) @@ -159,9 +158,7 @@ solution_summary(model) # ## Results # We recover the solution values -w_sol = value.(w) -g_sol = value(g) -y_sol = value.(y); +w_sol, g_sol, y_sol = value.(w), value(g), value.(y) println("Minimum slack: ", minimum(y_sol), "\nMaximum slack: ", maximum(y_sol)) # With the solution, we can ask: was the value of the penalty constant "sufficiently large" @@ -170,7 +167,7 @@ println("Minimum slack: ", minimum(y_sol), "\nMaximum slack: ", maximum(y_sol)) # Let's add this to the plot as well and check how we did: Plots.plot!( - plot1, + plot, x -> line(x; w = w_sol, g = g_sol), 0.0:0.01:2.0; seriestype = :line, @@ -178,8 +175,7 @@ Plots.plot!( linestyle = :dashdotdot, lineopacity = 0.9, c = :darkblue, -); -plot1 +) # We find that we have recovered the dividing line from just the information of the points # and their labels. From dfba120649c9d2a1a644cb0fcf2eff7faf75e3b3 Mon Sep 17 00:00:00 2001 From: Oscar Dowson Date: Sun, 12 Nov 2023 11:59:57 +1300 Subject: [PATCH 03/11] Apply suggestions from code review --- docs/src/tutorials/nonlinear/classifiers.jl | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/docs/src/tutorials/nonlinear/classifiers.jl b/docs/src/tutorials/nonlinear/classifiers.jl index 04a71b26eda..0f3726aa795 100644 --- a/docs/src/tutorials/nonlinear/classifiers.jl +++ b/docs/src/tutorials/nonlinear/classifiers.jl @@ -20,13 +20,10 @@ # This tutorial uses the following packages using JuMP -import DataFrames -import HiGHS import Ipopt import LinearAlgebra import Plots import Random -import SparseArrays import Test #src # ## Data and visualisation @@ -40,6 +37,7 @@ function generate_test_points(m; random_seed = 1) end # For the sake of the example, let's take ``m = 100``: + P = generate_test_points(100); # Note that the points are represented row-wise in the generated array. @@ -64,6 +62,7 @@ plot = Plots.scatter( # We can then test how well our classifier reproduces the original labels and the boundary between them. # Let's make a line to divide the point into two sets by defining a gradient and constant: + w0 = [5, 3] g0 = 8 line(v::AbstractArray; w = w0, g = g0) = LinearAlgebra.dot(w, v) - g @@ -85,6 +84,7 @@ Plots.plot!( ) # Now we label the points relative to which side of the line they are. + P_pos = hcat(filter(v -> line(v) > 0, eachrow(P))...)' P_neg = hcat(filter(v -> line(v) < 0, eachrow(P))...)' @assert size(P_pos, 1) + size(P_neg, 1) == size(P, 1) #src @@ -135,6 +135,7 @@ D = LinearAlgebra.Diagonal([ones(size(P_pos, 1)); -ones(size(P_neg, 1))]) # ``` # We need a value for the positive penalty parameter ``C``: + C = 1e3; # ## JuMP formulation @@ -158,6 +159,7 @@ solution_summary(model) # ## Results # We recover the solution values + w_sol, g_sol, y_sol = value.(w), value(g), value.(y) println("Minimum slack: ", minimum(y_sol), "\nMaximum slack: ", maximum(y_sol)) From 50a403ba514d03b60bd9ca51cc0c551c0a13e8d7 Mon Sep 17 00:00:00 2001 From: odow Date: Sun, 12 Nov 2023 17:49:38 +1300 Subject: [PATCH 04/11] Oscar's updates --- docs/src/tutorials/nonlinear/classifiers.jl | 101 ++++++++++---------- 1 file changed, 48 insertions(+), 53 deletions(-) diff --git a/docs/src/tutorials/nonlinear/classifiers.jl b/docs/src/tutorials/nonlinear/classifiers.jl index 0f3726aa795..275926ef274 100644 --- a/docs/src/tutorials/nonlinear/classifiers.jl +++ b/docs/src/tutorials/nonlinear/classifiers.jl @@ -6,14 +6,14 @@ # # Classifiers # Classification problems deal with constructing functions, called *classifiers*, -# that can efficiently classify data into two or more distinct sets. +# that can efficiently classify data into two or more distinct sets. # A common application is classifying previously unseen data points # after training a classifier on known data. -# The theory and models in this tutorial come from -# Section 9.4 "Classification Problems" of the book -# [*Linear Programming with MATLAB*](https://doi.org/10.1137/1.9780898718775) (2007) -# by M. C. Ferris, O. L. Mangasarian, and S. J. Wright. +# The theory and models in this tutorial come from Section 9.4 +# "Classification Problems" of the book +# [_Linear Programming with MATLAB_](https://doi.org/10.1137/1.9780898718775) +# (2007), by M. C. Ferris, O. L. Mangasarian, and S. J. Wright. # ## Required packages @@ -28,8 +28,8 @@ import Test #src # ## Data and visualisation -# To start, let's generate some points to test with. -# The argument ``m`` is the number of 2-dimensional points: +# To start, let's generate some points to test with. The argument ``m`` is the +# number of 2-dimensional points: function generate_test_points(m; random_seed = 1) rng = Random.MersenneTwister(random_seed) @@ -40,8 +40,8 @@ end P = generate_test_points(100); -# Note that the points are represented row-wise in the generated array. -# Let's visualise the points using the `Plots` package: +# Note that the points are represented row-wise in the generated array. Let's +# visualise the points using the `Plots` package: r = 1.01 * maximum(abs.(P)) plot = Plots.scatter( @@ -56,20 +56,21 @@ plot = Plots.scatter( legend = false, ) -# We want to split the points into two distinct sets on either side of a dividing line. -# We'll then label each point depending on which side of the line it happens to fall. -# Based on the labels of the point, we'll show how to create a classifier using a JuMP model. -# We can then test how well our classifier reproduces the original labels and the boundary between them. +# We want to split the points into two distinct sets on either side of a +# dividing line. We'll then label each point depending on which side of the line +# it happens to fall. Based on the labels of the point, we'll show how to create +# a classifier using a JuMP model. We can then test how well our classifier +# reproduces the original labels and the boundary between them. -# Let's make a line to divide the point into two sets by defining a gradient and constant: +# Let's make a line to divide the point into two sets by defining a gradient and +# constant: -w0 = [5, 3] -g0 = 8 -line(v::AbstractArray; w = w0, g = g0) = LinearAlgebra.dot(w, v) - g -line(x::Real; w = w0, g = g0) = -(w[1] * x - g) / w[2]; +w_0, g_0 = [5, 3], 8 +line(v::AbstractArray; w = w_0, g = g_0) = w' * v - g +line(x::Real; w = w_0, g = g_0) = -(w[1] * x - g) / w[2]; -# Julia's multiple dispatch feature allows us to define the vector and single-variable form -# of the `line` function under the same name. +# Julia's multiple dispatch feature allows us to define the vector and +# single-variable form of the `line` function under the same name. # Let's add this to the plot: @@ -83,10 +84,12 @@ Plots.plot!( c = :darkcyan, ) -# Now we label the points relative to which side of the line they are. +# Now we label the points relative to which side of the line they are. It is +# numerically useful to have the labels +1 for `P_pos` and -1 for `P_neg`. -P_pos = hcat(filter(v -> line(v) > 0, eachrow(P))...)' -P_neg = hcat(filter(v -> line(v) < 0, eachrow(P))...)' +labels = ifelse.(line.(eachrow(P)) .>= 0, 1, -1) +P_pos = P[labels .== 1, :] +P_neg = P[labels .== -1, :] @assert size(P_pos, 1) + size(P_neg, 1) == size(P, 1) #src Plots.scatter!( plot, @@ -105,34 +108,24 @@ Plots.scatter!( markersize = 8, ) -# The goal is to show we can reconstruct the line from *just* the points and labels. +# The goal is to show we can reconstruct the line from *just* the points and +# the labels. # ## Formulation: linear support vector machine -# Firstly, we will put the point set back together row-wise as a matrix, with the labelled points group together: - -A = [P_pos; P_neg] -m, n = size(A) - -# To keep track of the labels, we'll use a diagonal matrix where entry ``i`` of the diagonal is the -# label for point ``i`` (row ``i`` of the matrix). - -D = LinearAlgebra.Diagonal([ones(size(P_pos, 1)); -ones(size(P_neg, 1))]) - -# It is numerically useful to have the labels +1 for `S_pos` and -1 for `S_neg`. - -# A classifier known as the linear *support vector machine* looks for the line function data -# ``w`` and ``g`` such that the function ``L(v) = w^T v - g`` satisfies -# ``L(p) < 0`` for a point ``p`` in `S_neg` and ``L(p) > 0`` on `S_pos`. -# The linearly constrained quadratic program that implements this as follows: +# A classifier known as the linear _support vector machine_ looks for the line +# function ``L(p) = w^\top p - g`` that satisfies ``L(p) < 0`` for a point ``p`` +# in `S_neg` and ``L(p) > 0`` on `S_pos`. +# The linearly constrained quadratic program that implements this is: # ```math # \begin{aligned} -# & \min_{w \in \mathbb{R}^n, \; g \in \mathbb{R}, \; y \in \mathbb{R}^m} & \frac{1}{2} w^T w + C \; \sum_{i=1}^m y_i \\ -# & \text{subject to} & D_{ii}( A_{i :} w - g ) + y_i & \geq 1, & i = 1 \ldots m \\ -# & & y \ge 0. +# \min_{w \in \mathbb{R}^n, \; g \in \mathbb{R}, \; y \in \mathbb{R}^m} & \frac{1}{2} w^T w + C \; \sum_{i=1}^m y_i \\ +# \text{subject to} & D \cdot (P w - g) + y & \geq \mathbf{1} \\ +# & y \ge 0. # \end{aligned} # ``` +# where ``D`` is a diagonal matrix of the labels. # We need a value for the positive penalty parameter ``C``: @@ -140,17 +133,17 @@ C = 1e3; # ## JuMP formulation -# Now let's build and the JuMP model. We'll get ``w`` and ``g`` from the optimal solution after the -# solve. +# Here is the JuMP model: -m, n = size(A) +m, n = size(P) model = Model(Ipopt.Optimizer) set_silent(model) @variable(model, w[1:n]) @variable(model, g) @variable(model, y[1:m] >= 0) -@constraint(model, [i in 1:m], D[i, i] * (A[i, :]' * w - g) + y[i] >= 1) -@objective(model, Min, 0.5 * w' * w + C * sum(y)) +D = LinearAlgebra.Diagonal(labels) +@constraint(model, D * (P * w .- g) .+ y .>= 1) +@objective(model, Min, 1 / 2 * w' * w + C * sum(y)) optimize!(model) Test.@test termination_status(model) == LOCALLY_SOLVED #src Test.@test primal_status(model) == FEASIBLE_POINT #src @@ -163,10 +156,12 @@ solution_summary(model) w_sol, g_sol, y_sol = value.(w), value(g), value.(y) println("Minimum slack: ", minimum(y_sol), "\nMaximum slack: ", maximum(y_sol)) -# With the solution, we can ask: was the value of the penalty constant "sufficiently large" -# for this data set? This can be judged in part by the range of the slack variables. +# With the solution, we can ask: was the value of the penalty constant +# "sufficiently large" for this data set? This can be judged in part by the +# range of the slack variables. If the slack is too large, then we need to +# increase the penalty constant. -# Let's add this to the plot as well and check how we did: +# Let's plot the solution and check how we did: Plots.plot!( plot, @@ -179,5 +174,5 @@ Plots.plot!( c = :darkblue, ) -# We find that we have recovered the dividing line from just the information of the points -# and their labels. +# We find that we have recovered the dividing line from just the information of +# the points and their labels. From d75f12212bf8577e49b463a8c106fb01e4cc2b44 Mon Sep 17 00:00:00 2001 From: James Foster Date: Sun, 12 Nov 2023 20:55:42 +1100 Subject: [PATCH 05/11] Tweaks: Initial commment, typos --- docs/src/tutorials/nonlinear/classifiers.jl | 17 ++++++++++------- 1 file changed, 10 insertions(+), 7 deletions(-) diff --git a/docs/src/tutorials/nonlinear/classifiers.jl b/docs/src/tutorials/nonlinear/classifiers.jl index 275926ef274..d3c74c3104e 100644 --- a/docs/src/tutorials/nonlinear/classifiers.jl +++ b/docs/src/tutorials/nonlinear/classifiers.jl @@ -5,6 +5,9 @@ # # Classifiers +# The purpose of this tutorial is to show how JuMP can be used to easily +# formulate classification problems. + # Classification problems deal with constructing functions, called *classifiers*, # that can efficiently classify data into two or more distinct sets. # A common application is classifying previously unseen data points @@ -88,8 +91,8 @@ Plots.plot!( # numerically useful to have the labels +1 for `P_pos` and -1 for `P_neg`. labels = ifelse.(line.(eachrow(P)) .>= 0, 1, -1) -P_pos = P[labels .== 1, :] -P_neg = P[labels .== -1, :] +P_pos = P[labels.==1, :] +P_neg = P[labels.==-1, :] @assert size(P_pos, 1) + size(P_neg, 1) == size(P, 1) #src Plots.scatter!( plot, @@ -113,15 +116,15 @@ Plots.scatter!( # ## Formulation: linear support vector machine -# A classifier known as the linear _support vector machine_ looks for the line -# function ``L(p) = w^\top p - g`` that satisfies ``L(p) < 0`` for a point ``p`` -# in `S_neg` and ``L(p) > 0`` on `S_pos`. +# A classifier known as the linear _support vector machine_ looks for the affine +# function ``L(p) = w^\top p - g`` that satisfies ``L(p) < 0`` for all points ``p`` +# in `P_neg` and ``L(p) > 0`` for all points ``p`` in `P_pos`. # The linearly constrained quadratic program that implements this is: # ```math # \begin{aligned} -# \min_{w \in \mathbb{R}^n, \; g \in \mathbb{R}, \; y \in \mathbb{R}^m} & \frac{1}{2} w^T w + C \; \sum_{i=1}^m y_i \\ -# \text{subject to} & D \cdot (P w - g) + y & \geq \mathbf{1} \\ +# \min_{w \in \mathbb{R}^n, \; g \in \mathbb{R}, \; y \in \mathbb{R}^m} \quad & \frac{1}{2} w^T w + C \; \sum_{i=1}^m y_i \\ +# \text{subject to } \quad & D \cdot (P w - g) + y \geq \mathbf{1} \\ # & y \ge 0. # \end{aligned} # ``` From 997c36c78186d10d7b38bd7a49b52335ba6fcda0 Mon Sep 17 00:00:00 2001 From: James Foster Date: Mon, 13 Nov 2023 00:13:19 +1100 Subject: [PATCH 06/11] More advance sections (draft) --- docs/src/assets/checker/README | 103 + docs/src/assets/checker/checker.mat | Bin 0 -> 2248 bytes docs/src/assets/checker/checker.txt | 1000 + docs/src/assets/checker/checkerboard.eps | 1451 ++ docs/src/assets/checker/checkerboard.pdf | Bin 0 -> 114028 bytes docs/src/assets/checker/poly6.eps | 20528 +++++++++++++++++ docs/src/assets/checker/poly6.pdf | Bin 0 -> 661712 bytes docs/src/assets/checker/sin.eps | 20679 ++++++++++++++++++ docs/src/assets/checker/sin.pdf | Bin 0 -> 665330 bytes docs/src/tutorials/conic/ellipse_approx.jl | 2 +- docs/src/tutorials/nonlinear/classifiers.jl | 125 +- docs/styles/Vocab/JuMP-Vocab/accept.txt | 1 + 12 files changed, 43872 insertions(+), 17 deletions(-) create mode 100644 docs/src/assets/checker/README create mode 100644 docs/src/assets/checker/checker.mat create mode 100755 docs/src/assets/checker/checker.txt create mode 100644 docs/src/assets/checker/checkerboard.eps create mode 100644 docs/src/assets/checker/checkerboard.pdf create mode 100644 docs/src/assets/checker/poly6.eps create mode 100644 docs/src/assets/checker/poly6.pdf create mode 100644 docs/src/assets/checker/sin.eps create mode 100644 docs/src/assets/checker/sin.pdf diff --git a/docs/src/assets/checker/README b/docs/src/assets/checker/README new file mode 100644 index 00000000000..956828f2df9 --- /dev/null +++ b/docs/src/assets/checker/README @@ -0,0 +1,103 @@ +File: /afs/cs.wisc.edu/p/ftp/math-prog/cpo-dataset/machine-learn/checker/README +This directory contains six files: + +README (this file), checker.mat and checker.txt, checkerboard.eps, poly6.eps and sin.eps. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +This dataset is made public with the permission of its originators: + +@inproceedings{hk:96, + author = "Tin Kam Ho and Eugene M. Kleinberg", + title = "Building Projectable Classifiers of Arbitrary Complexity", + booktitle={Proceedings of the 13th International Conference on Pattern R + ecognition}, + editor = "", + publisher = "", + note = {http://cm.bell-labs.com/who/tkh/pubs.html}, + address = {Vienna, Austria }, + pages = {880-885}, + date = {August 25--30}, + year = 1996} + +This dataset has been used in the following papers: + +@incollection{Kaufman98a, + author = {L. Kaufman}, + title={Solving the Quadratic Programming problem arising in support + vector classification}, + editor = {Bernhard {Sch\"olkopf} and Christopher J. C. Burges and + Alexander J. Smola}, + booktitle = {Advances in Kernel Methods {-} Support Vector Learning}, + publisher = {MIT Press}, + pages = {147-167}, + year= {1999} } + + +@techreport{mm:99, + author = "O. L. Mangasarian and David R. Musicant", + title = "Data Discrimination via Nonlinear Generalized Support + Vector Machines", + institution = uwcs, + month = {March}, + year = 1999, + number = {99-03}, + address = "Madison, Wisconsin", + note={ftp://ftp.cs.wisc.edu/math-prog/tech-reports/99-03.ps}} + + +@techreport{lm:99, + author = "Yuh-Jye Lee and O. L. Mangasarian", + title = "{SSVM}: A Smooth Support Vector Machine", + institution = "Data Mining Institute, Computer Sciences Department, University of Wisconsin", + month = {September}, + year = 1999, + number = {99-03}, + address = "Madison, Wisconsin", + note={ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/99-03.ps}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +checker.mat +---------- + +This file can be loaded into MATLAB directly and contains two matrices, A and B. + +A is 486 x 2 matrix which represents 486 points of class I in R^2 space. + +B is 514 x 2 matrix which represents 514 points of class II in R^2 space. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +checker.txt +---------- + +This text file contains three columns which represent 1000 points in R^2, +the two dimensional real space. + +The first column is the indicator of classes. + +The second and third column are the coordinates of the data points. + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +checkerboard.eps +-------------- + +This eps file contains the figure of checkerboard dataset. + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +poly6.eps +-------- + +This eps file depicts the result of checkerboard problem using SSVM with sixth drgree polynomial kernel. +SSVM stands for Smooth Support Vector Machine. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +sin.eps + +This eps file depicts the result of checkerboard problem using SSVM with sinusoidal kernel. \ No newline at end of file diff --git a/docs/src/assets/checker/checker.mat b/docs/src/assets/checker/checker.mat new file mode 100644 index 0000000000000000000000000000000000000000..c44aed1ccfed2da734712328942fceb15e1356e3 GIT binary patch literal 2248 zcmb7^={G8j7RE!7A$53>gM`=IiINaSnG=aJ#EGI~%2b&S88T%mI!YXKB1u$29kb}m z`*+;G<2v`;ulK{@|hJe8la5y9;D=Q26|Cbw}yc7Tc_yYjI 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{pop StandardEncoding} ifelse +exch +dup 3 1 roll +findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def + currentdict +end +definefont pop +} bdef + +/isroman { +findfont /CharStrings get +/Agrave known +} bdef + +/FMSR { +3 1 roll 1 index +dup isroman +{reencode} {pop pop} ifelse +exch FMS +} bdef + +/csm { + 1 dpi2point div -1 dpi2point div scale + neg translate + landscapeMode eq {90 rotate} if + } bdef + +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef + +% macros for lines and objects +/L { + lineto + stroke + } bdef +/MP { + 3 1 roll moveto + 1 sub {rlineto} repeat + } bdef +/AP { + {rlineto} repeat + } bdef +/PP { + closepath eofill + } bdef +/DP { + closepath stroke + } bdef +/MR { + 4 -2 roll moveto + dup 0 exch rlineto + exch 0 rlineto + neg 0 exch rlineto + closepath + } bdef +/FR { + MR stroke + } bdef +/PR { + MR fill + } bdef +/L1i { + { currentfile picstr readhexstring pop } image + } bdef + +/tMatrix matrix def +/MakeOval { + newpath + tMatrix currentmatrix pop + translate scale + 0 0 1 0 360 arc + tMatrix setmatrix + } bdef +/FO { + MakeOval + stroke + } bdef +/PO { + MakeOval + fill + } bdef + +/PD { + currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap + } bdef + +/FA { + newpath + tMatrix currentmatrix pop + translate scale + 0 0 1 5 -2 roll arc + tMatrix setmatrix + stroke + } bdef +/PA { + newpath + tMatrix currentmatrix pop + translate 0 0 moveto scale + 0 0 1 5 -2 roll arc + closepath + tMatrix setmatrix + fill + } bdef + +/FAn { + newpath + tMatrix currentmatrix pop + translate scale + 0 0 1 5 -2 roll arcn + tMatrix setmatrix + stroke + } bdef +/PAn { + newpath + tMatrix currentmatrix pop + translate 0 0 moveto scale + 0 0 1 5 -2 roll arcn + closepath + 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zzy9N^(?32uejF$GI7Xk{p8ok)Uw(S`z5bb+= 0) -D = LinearAlgebra.Diagonal(labels) -@constraint(model, D * (P * w .- g) .+ y .>= 1) -@objective(model, Min, 1 / 2 * w' * w + C * sum(y)) -optimize!(model) -Test.@test termination_status(model) == LOCALLY_SOLVED #src -Test.@test primal_status(model) == FEASIBLE_POINT #src -solution_summary(model) +function solve_SVM_classifier(P, labels; optimizer = Ipopt.Optimizer, C = C_0) + m, n = size(P) + model = Model(optimizer) + set_silent(model) + @variable(model, w[1:n]) + @variable(model, g) + @variable(model, y[1:m] >= 0) + D = LinearAlgebra.Diagonal(labels) + @constraint(model, D * (P * w .- g) .+ y .>= 1) + @objective(model, Min, 1 / 2 * w' * w + C * sum(y)) + optimize!(model) + return model, value.(w), value(g), value.(y) +end # ## Results # We recover the solution values -w_sol, g_sol, y_sol = value.(w), value(g), value.(y) +model, w_sol, g_sol, y_sol = solve_SVM_classifier(P, labels) +Test.@test termination_status(model) == LOCALLY_SOLVED #src +Test.@test primal_status(model) == FEASIBLE_POINT #src println("Minimum slack: ", minimum(y_sol), "\nMaximum slack: ", maximum(y_sol)) # With the solution, we can ask: was the value of the penalty constant @@ -179,3 +185,90 @@ Plots.plot!( # We find that we have recovered the dividing line from just the information of # the points and their labels. + +# ## Nonseparable classes of points + +# Now, what if the point sets are not cleanly separable by a line +# (or a hyperplane in higher dimensions)? Does this still work? +# Let's repeat the above process but this time we will simulate nonseparable +# classes of points by intermingling a few nearby points across the previously used line. + +nearby_indices = (abs.(line.(eachrow(P))) .< 1.1) +labels_new = ifelse.(nearby_indices, -labels, labels) + +P_pos = P[labels_new.==1, :] +P_neg = P[labels_new.==-1, :] +@assert size(P_pos, 1) + size(P_neg, 1) == size(P, 1) #src +Plots.scatter!( + plot_copy, + P_pos[:, 1], + P_pos[:, 2]; + shape = :cross, + markercolor = :blue, + markersize = 8, +) +Plots.scatter!( + plot_copy, + P_neg[:, 1], + P_neg[:, 2]; + shape = :xcross, + markercolor = :crimson, + markersize = 8, +) + +model, w_sol, g_sol, y_sol = solve_SVM_classifier(P, labels_new) +Test.@test termination_status(model) == LOCALLY_SOLVED #src +Test.@test primal_status(model) == FEASIBLE_POINT #src +println("Minimum slack: ", minimum(y_sol), "\nMaximum slack: ", maximum(y_sol)) + +Plots.plot!( + plot_copy, + x -> line(x; w = w_sol, g = g_sol), + 0.0:0.01:2.0; + seriestype = :line, + linewidth = 5, + linestyle = :dashdotdot, + lineopacity = 0.9, + c = :darkblue, +) + +# So it still works as a classifier, with allowance for the nonseparable points. +# +# We can find out which points are contributing to the shape of the line by +# looking at the dual values of the affine constraints and comparing them +# to the penalty constant ``C``: + +affine_cons = all_constraints(model, AffExpr, MOI.GreaterThan{Float64}) +active_cons = findall(isapprox.(dual.(affine_cons), C_0; atol = 0.001)) + +findall(nearby_indices) ⊆ active_cons + +# The last statement tells us that our nonseparable points are actively contributing +# to how the classifier is defined. The remaining points are of interest and +# are highlighted: + +P_active = P[setdiff(active_cons, findall(nearby_indices)), :] + +Plots.scatter!( + plot_copy, + P_active[:, 1], + P_active[:, 2]; + shape = :hexagon, + markercolor = :green, + markersize = 8, + markeropacity = 0.5, +) + +# ## Advanced: Duality and the kernel method + +# Linear SVM techniques are not limited to finding separating hyperplanes +# in the original space of the dataset. We can first transform the +# training data under a nonlinear mapping, apply our method, then +# map the hyperplane back into original space. + +# ### Checkerboard dataset +# ![Checkerboard](../../assets/checker/checkerboard.pdf) + +# ### Classifier using a sixth degree polynomial kernel + +# ![Sine Kernel](../../assets/checker/poly6.pdf) diff --git a/docs/styles/Vocab/JuMP-Vocab/accept.txt b/docs/styles/Vocab/JuMP-Vocab/accept.txt index c91bde97415..5c23db83ea6 100644 --- a/docs/styles/Vocab/JuMP-Vocab/accept.txt +++ b/docs/styles/Vocab/JuMP-Vocab/accept.txt @@ -73,6 +73,7 @@ minimizers nl|NL [Nn]onconvex [Nn]on(posi|nega)tiv(e|es|ity) +[Nn]onseparable [Nn]onsmooth [Oo]ptigraph(?s) [Oo]ptinode(?s) From 55223a05167ac646d045d84293f6446cb6750fdd Mon Sep 17 00:00:00 2001 From: James Foster Date: Mon, 13 Nov 2023 22:20:58 +1100 Subject: [PATCH 07/11] Add kernel method on checkerboards --- docs/src/assets/checker/checker.txt | 1000 ----------------- docs/src/assets/checker/checkerboard.pdf | Bin 114028 -> 0 bytes docs/src/assets/checker/poly6.pdf | Bin 661712 -> 0 bytes docs/src/assets/checker/sin.pdf | Bin 665330 -> 0 bytes docs/src/tutorials/nonlinear/classifiers.jl | 214 +++- .../nonlinear/data}/checker/README | 0 .../nonlinear/data}/checker/checker.mat | Bin .../nonlinear/data/checker/checker.txt | 1000 +++++++++++++++++ .../nonlinear/data}/checker/checkerboard.eps | 0 .../nonlinear/data/checker/checkerboard.png | Bin 0 -> 447300 bytes .../nonlinear/data}/checker/poly6.eps | 0 .../nonlinear/data}/checker/sin.eps | 0 12 files changed, 1205 insertions(+), 1009 deletions(-) delete mode 100755 docs/src/assets/checker/checker.txt delete mode 100644 docs/src/assets/checker/checkerboard.pdf delete mode 100644 docs/src/assets/checker/poly6.pdf delete mode 100644 docs/src/assets/checker/sin.pdf rename docs/src/{assets => tutorials/nonlinear/data}/checker/README (100%) rename docs/src/{assets => tutorials/nonlinear/data}/checker/checker.mat (100%) create mode 100755 docs/src/tutorials/nonlinear/data/checker/checker.txt rename docs/src/{assets => tutorials/nonlinear/data}/checker/checkerboard.eps (100%) create mode 100644 docs/src/tutorials/nonlinear/data/checker/checkerboard.png rename docs/src/{assets => tutorials/nonlinear/data}/checker/poly6.eps (100%) rename docs/src/{assets => tutorials/nonlinear/data}/checker/sin.eps (100%) diff --git a/docs/src/assets/checker/checker.txt b/docs/src/assets/checker/checker.txt deleted file mode 100755 index 9d449ab1f51..00000000000 --- a/docs/src/assets/checker/checker.txt +++ /dev/null @@ -1,1000 +0,0 @@ -0 174 172 -1 53 18 -0 182 80 -1 127 180 -0 81 67 -1 146 150 -0 123 126 -0 16 42 -0 9 137 -1 161 30 -0 151 198 -1 163 12 -1 103 52 -1 34 174 -1 30 77 -1 13 85 -0 85 72 -0 102 34 -1 162 131 -1 189 146 -1 42 150 -1 132 158 -0 9 104 -1 9 181 -0 165 54 -0 176 91 -1 39 81 -1 90 101 -1 58 34 -0 198 76 -1 175 32 -0 33 7 -1 51 146 -1 158 114 -0 160 188 -1 133 163 -0 104 115 -1 84 116 -0 150 156 -0 114 3 -1 45 197 -1 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zzy9N^(?32uejF$GI7Xk{p8ok)Uw(S`z5bb+ 0`` for all points ``p`` in `P_pos`. # The linearly constrained quadratic program that implements this is: # ```math # \begin{aligned} -# \min_{w \in \mathbb{R}^n, \; g \in \mathbb{R}, \; y \in \mathbb{R}^m} \quad & \frac{1}{2} w^T w + C \; \sum_{i=1}^m y_i \\ +# \min_{w \in \mathbb{R}^n, \; g \in \mathbb{R}, \; y \in \mathbb{R}^m} \quad & \frac{1}{2} w^\top w + C \; \sum_{i=1}^m y_i \\ # \text{subject to } \quad & D \cdot (P w - g) + y \geq \mathbf{1} \\ # & y \ge 0. # \end{aligned} @@ -158,7 +159,7 @@ end # ## Results -# We recover the solution values +# Let's recover the values that define the classifier by solving the model: model, w_sol, g_sol, y_sol = solve_SVM_classifier(P, labels) Test.@test termination_status(model) == LOCALLY_SOLVED #src @@ -221,6 +222,8 @@ Test.@test termination_status(model) == LOCALLY_SOLVED #src Test.@test primal_status(model) == FEASIBLE_POINT #src println("Minimum slack: ", minimum(y_sol), "\nMaximum slack: ", maximum(y_sol)) +# + Plots.plot!( plot_copy, x -> line(x; w = w_sol, g = g_sol), @@ -258,17 +261,210 @@ Plots.scatter!( markersize = 8, markeropacity = 0.5, ) - # ## Advanced: Duality and the kernel method +# ### The dual program + +# The dual of the linear SVM program is also a linearly constrained quadratic program: +# ```math +# \begin{aligned} +# \min_{u \in \mathbb{R}^m} \quad & \frac{1}{2} u^\top D P P^\top D u - \; \mathbf{1}^\top u \\ +# \text{subject to } \quad & \mathbf{1}^\top D u = 0 \\ +# & 0 \leq u \leq C\mathbf{1}. +# \end{aligned} +# ``` +# This is the JuMP model: +function solve_dual_SVM_classifier( + P, + labels; + optimizer = Ipopt.Optimizer, + C = C_0, +) + m, n = size(P) + model = Model(optimizer) + set_silent(model) + @variable(model, 0 <= u[1:m] <= C) + D = LinearAlgebra.Diagonal(labels) + con = @constraint(model, sum(D * u) >= 0) + @objective(model, Min, 1 / 2 * u' * D * P * P' * D * u - C * sum(u)) + optimize!(model) + u_sol = value.(u) + w_sol = P' * D * u_sol + g_sol = dual(con) + return model, u_sol, w_sol, g_sol +end + +# We recover the line gradient vector ``w`` through setting ``w = P^\top D u`` +# and the line constant ``g`` as the dual value of the single affine constraint. +# In many cases it may be simpler to solve the dual form. We can check that this +# has recovered the classifier: + +model_dual, u_d, w_d, g_d = solve_dual_SVM_classifier(P, labels) +Test.@test termination_status(model_dual) == LOCALLY_SOLVED #src +Test.@test primal_status(model_dual) == FEASIBLE_POINT #src +Plots.plot!( + plot_copy, + x -> line(x; w = w_d, g = g_d), + 0.0:0.01:2.0; + seriestype = :line, + linewidth = 2, + linestyle = :dash, + lineopacity = 0.9, + c = :red, +) + +# ## The kernel method + # Linear SVM techniques are not limited to finding separating hyperplanes -# in the original space of the dataset. We can first transform the +# in the original space of the dataset. One could first transform the # training data under a nonlinear mapping, apply our method, then # map the hyperplane back into original space. +# The actual data describing the point set is held in a matrix ``P``, +# but looking at the dual program we see that what actually matters is the +# [Gram matrix](https://en.wikipedia.org/wiki/Gram_matrix) ``P P^\top``, +# expressing a pairwise comparison (an inner-product) between each point vector. +# It follows that any mapping of the point set only needs +# to be defined at the level of _pairwise_ maps between points. +# Such maps are known as _kernel functions_ +# ```math +# k \; : \; \mathbb{R}^n \times \mathbb{R}^n \; \rightarrow \mathbb{R}, \qquad +# (s, t) \mapsto \left< \Phi(s), \Phi(t) \right> +# ``` +# where the right-hand side applies some transformation ``\Phi : \mathbb{R}^n \rightarrow \mathbb{R}^{n'}`` +# followed by an inner-product in that image space. In practice, we can avoid having +# ``\Phi`` explicitly given but instead define a kernel function directly between pairs of vectors. +# This move to using a kernel function without knowing the map is called the _kernel method_ +# (or sometimes, the _kernel trick_). + +# ### Classifier using a Gaussian kernel + +# We will demonstrate the application of a _Gaussian_ or _radial basis function_ kernel +# ```math +# k(s, t) = \exp\left( -\mu \lVert s - t \rVert^2_2 \right) +# ``` +# for some positive parameter ``\mu``. + +k_gauss(s::Vector, t::Vector; μ = 0.5) = exp(-μ * LinearAlgebra.norm(s - t)^2) + +# Given a matrix of points expressed row-wise and a kernel, let's return +# the transformed matrix ``K`` that replaces ``P P^\top``: + +function pairwise_transform(kernel_func, P) + m, n = size(P) + K = zeros(eltype(P), m, m) + + for j in 1:m + for i in 1:j + K[i, j] = kernel_func(P[i, :], P[j, :]) + end + end + + return LinearAlgebra.Symmetric(K, :U) +end + +K_0 = pairwise_transform(k_gauss, P) + +function solve_kernel_SVM_classifier( + kernel_func, + P, + labels; + optimizer = Ipopt.Optimizer, + C = C_0, + kwargs..., +) + m, n = size(P) + K = pairwise_transform(kernel_func, P) + model = Model(optimizer) + set_silent(model) + @variable(model, 0 <= u[1:m] <= C) + D = LinearAlgebra.Diagonal(labels) + con = @constraint(model, sum(D * u) >= 0) + @objective(model, Min, 1 / 2 * u' * D * K * D * u - C * sum(u)) + optimize!(model) + u_sol = value.(u) + g_sol = dual(con) + + function kernel_classifier(v::Vector) + return sum( + D[i, i] * u_sol[i] * kernel_func(P[i, :], v; kwargs...) for i in 1:m + ) + end + + return model, u_sol, g_sol, kernel_classifier +end + +# This time, we don't recover the line gradient vector ``w`` directly. +# Instead, we compute the classifier ``f`` using the dual values and the +# kernel function: +# ```math +# f(v) = \sum_{i=1}^m D_{ii} u_i \; k(p_i, v ) - g +# ``` +# where ``p_i`` is row vector ``i`` of ``P``. + # ### Checkerboard dataset -# ![Checkerboard](../../assets/checker/checkerboard.pdf) -# ### Classifier using a sixth degree polynomial kernel +# To demonstrate this nonlinear technique, we'll use the [checkerboard dataset](data/checker/README). + +checkerboard = DelimitedFiles.readdlm( + joinpath(@__DIR__, "data", "checker", "checker.txt"), + ' ', + Int64, +) +labels_checker = ifelse.(checkerboard[:, 1] .== 0, -1, 1) +B = checkerboard[:, 2:3] ./ 100.0 # rescale to [0,2] x [0,2] square. +B_pos = B[labels_checker.==1, :] +B_neg = B[labels_checker.==-1, :] +@assert size(B_pos, 1) + size(B_neg, 1) == size(B, 1) #src +r = 1.01 * maximum(abs.(B)) +plot_checkers = Plots.scatter( + B_pos[:, 1], + B_pos[:, 2]; + xlim = (0, r), + ylim = (0, r), + label = nothing, + color = :white, + shape = :circle, + size = (600, 600), + legend = false, +) +Plots.scatter!( + plot_checkers, + B_neg[:, 1], + B_neg[:, 2]; + color = :black, + shape = :circle, + markersize = 2, +) + +# Is the technique capable of generating a distinctly nonlinear surface? +# The result has a fairly strong dependence on the choice of parameters: +C_1 = 100_000 +μ_1 = 20 +println("C = ", C_1, ", μ = ", μ_1) + +# Let's solve the Gaussian kernel based quadratic problem with these parameters: + +model_kernel, u_k, g_k, f_gauss = + solve_kernel_SVM_classifier(k_gauss, B, labels_checker; C = C_1, μ = μ_1) +Test.@test termination_status(model_kernel) == LOCALLY_SOLVED #src +Test.@test primal_status(model_kernel) == FEASIBLE_POINT #src + +grid = [[x, y] for x in 0:0.01:2, y in 0:0.01:2] +grid_pos = grid[f_gauss.(grid).>0] + +plot_board = Plots.scatter( + plot_checkers, + Tuple.(grid_pos); + shape = :circle, + markersize = 0.5, + markeropacity = 0.8, + xlim = (0, r), + ylim = (0, r), + size = (600, 600), + legend = false, +) -# ![Sine Kernel](../../assets/checker/poly6.pdf) +# We find that the kernel method can perform well as a nonlinear classifier. +# Determining a better performing kernel function or choice of parameters is left as +# a further challenge. diff --git a/docs/src/assets/checker/README b/docs/src/tutorials/nonlinear/data/checker/README similarity index 100% rename from docs/src/assets/checker/README rename to docs/src/tutorials/nonlinear/data/checker/README diff --git a/docs/src/assets/checker/checker.mat b/docs/src/tutorials/nonlinear/data/checker/checker.mat similarity index 100% rename from docs/src/assets/checker/checker.mat rename to docs/src/tutorials/nonlinear/data/checker/checker.mat diff --git a/docs/src/tutorials/nonlinear/data/checker/checker.txt b/docs/src/tutorials/nonlinear/data/checker/checker.txt new file mode 100755 index 00000000000..18da90070fd --- /dev/null +++ b/docs/src/tutorials/nonlinear/data/checker/checker.txt @@ -0,0 +1,1000 @@ +0 174 172 +1 53 18 +0 182 80 +1 127 180 +0 81 67 +1 146 150 +0 123 126 +0 16 42 +0 9 137 +1 161 30 +0 151 198 +1 163 12 +1 103 52 +1 34 174 +1 30 77 +1 13 85 +0 85 72 +0 102 34 +1 162 131 +1 189 146 +1 42 150 +1 132 158 +0 9 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z$S{UWzhu%aYkUK==A0S`UJl4g-B1gk++qiySA80`qSmADA2&gX-%)V!>a@b{U)C`4 zc`eu^(}P0ONtTk`brN*o5)6)#%|mGX1kZJyi=uGasE@Z4PEuHV^AdYLf<5T%Wy|?i z4P!Q02|@jNg0DO8GjGc`ZMF>k*G%8#rzOaK?^0vCgSXom`>T_yBduo(DJBT#UeVdn zK}qN*kc86lyRc7NPd|{rRXJc)an7JDXNx=|Z9G~^tsJ`s@&>kGyn?21)v*-{X!*Qx8Xg@!~``%sPwmfs%? zGgaSC-F|dtc@SP=;veu4Yu)prm{_}=WOJkRdObEHO%HnuL68|H#67Qgch~Qlbmw-J zcU&?L{%%pGIXfCujXkzI4)o97?|;`5j*+Fn%%xsuCvVjIN4NXG5D)(Gx7tp?u`zSH zm@0#^De)@~GVZy8HhB iALjqB%>V1#Y)htW8Jo-!=iq_||EMTxDiq0C-1 Date: Tue, 14 Nov 2023 12:11:10 +1100 Subject: [PATCH 08/11] Fix formulation, improve text --- docs/src/tutorials/nonlinear/classifiers.jl | 36 +++++++++++++-------- 1 file changed, 22 insertions(+), 14 deletions(-) diff --git a/docs/src/tutorials/nonlinear/classifiers.jl b/docs/src/tutorials/nonlinear/classifiers.jl index 94fb9e6c4b5..c5d00e48831 100644 --- a/docs/src/tutorials/nonlinear/classifiers.jl +++ b/docs/src/tutorials/nonlinear/classifiers.jl @@ -121,8 +121,9 @@ Plots.scatter!( # ## Formulation: linear support vector machine -# A classifier known as the linear _support vector machine_ (SVM) looks for the affine -# function ``L(p) = w^\top p - g`` that satisfies ``L(p) < 0`` for all points ``p`` +# A classifier known as the linear [_support vector machine_](https://en.wikipedia.org/wiki/Support_vector_machine) +# (SVM) looks for the affine function ``L(p) = w^\top p - g`` +# that satisfies ``L(p) < 0`` for all points ``p`` # in `P_neg` and ``L(p) > 0`` for all points ``p`` in `P_pos`. # The linearly constrained quadratic program that implements this is: @@ -285,8 +286,8 @@ function solve_dual_SVM_classifier( set_silent(model) @variable(model, 0 <= u[1:m] <= C) D = LinearAlgebra.Diagonal(labels) - con = @constraint(model, sum(D * u) >= 0) - @objective(model, Min, 1 / 2 * u' * D * P * P' * D * u - C * sum(u)) + con = @constraint(model, sum(D * u) == 0) + @objective(model, Min, 1 / 2 * u' * D * P * P' * D * u - sum(u)) optimize!(model) u_sol = value.(u) w_sol = P' * D * u_sol @@ -363,7 +364,9 @@ function pairwise_transform(kernel_func, P) return LinearAlgebra.Symmetric(K, :U) end -K_0 = pairwise_transform(k_gauss, P) +# Now we're ready to define our optimization problem. We need to provide the +# kernel function to be used in the problem. Note that any extra keyword +# arguments here (like parameter values) are passed through to the kernel function. function solve_kernel_SVM_classifier( kernel_func, @@ -379,8 +382,8 @@ function solve_kernel_SVM_classifier( set_silent(model) @variable(model, 0 <= u[1:m] <= C) D = LinearAlgebra.Diagonal(labels) - con = @constraint(model, sum(D * u) >= 0) - @objective(model, Min, 1 / 2 * u' * D * K * D * u - C * sum(u)) + con = @constraint(model, sum(D * u) == 0) + @objective(model, Min, 1 / 2 * u' * D * K * D * u - sum(u)) optimize!(model) u_sol = value.(u) g_sol = dual(con) @@ -438,12 +441,13 @@ Plots.scatter!( ) # Is the technique capable of generating a distinctly nonlinear surface? -# The result has a fairly strong dependence on the choice of parameters: +# Let's solve the Gaussian kernel based quadratic problem with these parameters: + C_1 = 100_000 -μ_1 = 20 +μ_1 = 10 println("C = ", C_1, ", μ = ", μ_1) -# Let's solve the Gaussian kernel based quadratic problem with these parameters: +# model_kernel, u_k, g_k, f_gauss = solve_kernel_SVM_classifier(k_gauss, B, labels_checker; C = C_1, μ = μ_1) @@ -457,8 +461,7 @@ plot_board = Plots.scatter( plot_checkers, Tuple.(grid_pos); shape = :circle, - markersize = 0.5, - markeropacity = 0.8, + markersize = 0.2, xlim = (0, r), ylim = (0, r), size = (600, 600), @@ -466,5 +469,10 @@ plot_board = Plots.scatter( ) # We find that the kernel method can perform well as a nonlinear classifier. -# Determining a better performing kernel function or choice of parameters is left as -# a further challenge. +# The result has a fairly strong dependence on the choice of parameters, +# with larger values of ``\mu`` allowing for a more complex boundary while +# smaller values lead to a smoother boundary for the classifier. +# Determining a better performing kernel function and choice of parameters is +# covered by the process of _cross-validation_ with respect to the dataset, +# where different testing, training and tuning sets are used to validate +# the best choice of parameters against a statistical measure of error. From c99c8022ad778bcab0bf70c022e9048fdd811399 Mon Sep 17 00:00:00 2001 From: odow Date: Wed, 15 Nov 2023 14:52:59 +1300 Subject: [PATCH 09/11] Take a pass and simplify a lot of the code --- docs/src/tutorials/nonlinear/classifiers.jl | 384 ++++++++------------ 1 file changed, 154 insertions(+), 230 deletions(-) diff --git a/docs/src/tutorials/nonlinear/classifiers.jl b/docs/src/tutorials/nonlinear/classifiers.jl index c5d00e48831..e57ca90b433 100644 --- a/docs/src/tutorials/nonlinear/classifiers.jl +++ b/docs/src/tutorials/nonlinear/classifiers.jl @@ -5,13 +5,13 @@ # # Classifiers -# The purpose of this tutorial is to show how JuMP can be used to easily -# formulate classification problems. +# The purpose of this tutorial is to show how JuMP can be used to formulate +# classification problems. # Classification problems deal with constructing functions, called *classifiers*, -# that can efficiently classify data into two or more distinct sets. -# A common application is classifying previously unseen data points -# after training a classifier on known data. +# that can efficiently classify data into two or more distinct sets. A common +# application is classifying previously unseen data points after training a +# classifier on known data. # The theory and models in this tutorial come from Section 9.4 # "Classification Problems" of the book @@ -44,34 +44,27 @@ end P = generate_test_points(100); -# Note that the points are represented row-wise in the generated array. Let's -# visualise the points using the `Plots` package: +# The points are represented row-wise in the matrix `P`. Let's visualise the +# points using the `Plots` package: -r = 1.01 * maximum(abs.(P)) plot = Plots.scatter( P[:, 1], P[:, 2]; - xlim = (0, r), - ylim = (0, r), - label = nothing, + xlim = (0, 2.02), + ylim = (0, 2.02), color = :white, - shape = :circle, size = (600, 600), legend = false, ) -# We'll need this for later comparison: - -plot_copy = deepcopy(plot); - # We want to split the points into two distinct sets on either side of a # dividing line. We'll then label each point depending on which side of the line # it happens to fall. Based on the labels of the point, we'll show how to create # a classifier using a JuMP model. We can then test how well our classifier # reproduces the original labels and the boundary between them. -# Let's make a line to divide the points into two sets by defining a gradient and -# constant: +# Let's make a line to divide the points into two sets by defining a gradient +# and a constant: w_0, g_0 = [5, 3], 8 line(v::AbstractArray; w = w_0, g = g_0) = w' * v - g @@ -82,49 +75,31 @@ line(x::Real; w = w_0, g = g_0) = -(w[1] * x - g) / w[2]; # Let's add this to the plot: -Plots.plot!( - plot, - x -> line(x), - 0.0:0.01:2.0; - seriestype = :line, - linewidth = 5, - lineopacity = 0.6, - c = :darkcyan, -) +Plots.plot!(plot, line, 0.0:0.01:2.0; linewidth = 5) # Now we label the points relative to which side of the line they are. It is -# numerically useful to have the labels +1 for `P_pos` and -1 for `P_neg`. +# numerically useful to have the labels +1 and -1 for the upcoming JuMP +# formulation. labels = ifelse.(line.(eachrow(P)) .>= 0, 1, -1) -P_pos = P[labels.==1, :] -P_neg = P[labels.==-1, :] -@assert size(P_pos, 1) + size(P_neg, 1) == size(P, 1) #src Plots.scatter!( plot, - P_pos[:, 1], - P_pos[:, 2]; - shape = :cross, - markercolor = :blue, - markersize = 8, -) -Plots.scatter!( - plot, - P_neg[:, 1], - P_neg[:, 2]; - shape = :xcross, - markercolor = :crimson, + P[:, 1], + P[:, 2]; + shape = ifelse.(labels .== 1, :cross, :xcross), + markercolor = ifelse.(labels .== 1, :blue, :crimson), markersize = 8, ) -# The goal is to show we can reconstruct the line from *just* the points and +# Our goal is to show we can reconstruct the line from *just* the points and # the labels. # ## Formulation: linear support vector machine # A classifier known as the linear [_support vector machine_](https://en.wikipedia.org/wiki/Support_vector_machine) -# (SVM) looks for the affine function ``L(p) = w^\top p - g`` -# that satisfies ``L(p) < 0`` for all points ``p`` -# in `P_neg` and ``L(p) > 0`` for all points ``p`` in `P_pos`. +# (SVM) looks for the affine function ``L(p) = w^\top p - g`` +# that satisfies ``L(p) < 0`` for all points ``p`` with a label `-1` and +# ``L(p) \ge 0`` for all points ``p`` with a label `+1`. # The linearly constrained quadratic program that implements this is: # ```math @@ -140,32 +115,34 @@ Plots.scatter!( C_0 = 100.0; -# ## JuMP formulation +# ### JuMP formulation # Here is the JuMP model: -function solve_SVM_classifier(P, labels; optimizer = Ipopt.Optimizer, C = C_0) +function solve_SVM_classifier(P::Matrix, labels::Vector; C::Float64 = C_0) m, n = size(P) - model = Model(optimizer) + model = Model(Ipopt.Optimizer) set_silent(model) @variable(model, w[1:n]) @variable(model, g) @variable(model, y[1:m] >= 0) + @objective(model, Min, 1 / 2 * w' * w + C * sum(y)) D = LinearAlgebra.Diagonal(labels) @constraint(model, D * (P * w .- g) .+ y .>= 1) - @objective(model, Min, 1 / 2 * w' * w + C * sum(y)) optimize!(model) - return model, value.(w), value(g), value.(y) + Test.@test termination_status(model) == LOCALLY_SOLVED #src + Test.@test primal_status(model) == FEASIBLE_POINT #src + slack = extrema(value.(y)) + println("Minimum slack: ", slack[1], "\nMaximum slack: ", slack[2]) + classifier(x) = line(x; w = value.(w), g = value(g)) + return model, classifier end -# ## Results +# ### Results # Let's recover the values that define the classifier by solving the model: -model, w_sol, g_sol, y_sol = solve_SVM_classifier(P, labels) -Test.@test termination_status(model) == LOCALLY_SOLVED #src -Test.@test primal_status(model) == FEASIBLE_POINT #src -println("Minimum slack: ", minimum(y_sol), "\nMaximum slack: ", maximum(y_sol)) +_, classifier = solve_SVM_classifier(P, labels) # With the solution, we can ask: was the value of the penalty constant # "sufficiently large" for this data set? This can be judged in part by the @@ -176,12 +153,10 @@ println("Minimum slack: ", minimum(y_sol), "\nMaximum slack: ", maximum(y_sol)) Plots.plot!( plot, - x -> line(x; w = w_sol, g = g_sol), + classifier, 0.0:0.01:2.0; - seriestype = :line, linewidth = 5, linestyle = :dashdotdot, - lineopacity = 0.9, c = :darkblue, ) @@ -190,71 +165,58 @@ Plots.plot!( # ## Nonseparable classes of points -# Now, what if the point sets are not cleanly separable by a line -# (or a hyperplane in higher dimensions)? Does this still work? -# Let's repeat the above process but this time we will simulate nonseparable -# classes of points by intermingling a few nearby points across the previously used line. +# Now, what if the point sets are not cleanly separable by a line (or a +# hyperplane in higher dimensions)? Does this still work? Let's repeat the +# process, but this time we will simulate nonseparable classes of points by +# intermingling a few nearby points across the previously used line. -nearby_indices = (abs.(line.(eachrow(P))) .< 1.1) +nearby_indices = abs.(line.(eachrow(P))) .< 1.1 labels_new = ifelse.(nearby_indices, -labels, labels) - -P_pos = P[labels_new.==1, :] -P_neg = P[labels_new.==-1, :] -@assert size(P_pos, 1) + size(P_neg, 1) == size(P, 1) #src -Plots.scatter!( - plot_copy, - P_pos[:, 1], - P_pos[:, 2]; - shape = :cross, - markercolor = :blue, - markersize = 8, +model, classifier = solve_SVM_classifier(P, labels_new) +plot = Plots.scatter( + P[:, 1], + P[:, 2]; + xlim = (0, 2.02), + ylim = (0, 2.02), + color = :white, + size = (600, 600), + legend = false, ) Plots.scatter!( - plot_copy, - P_neg[:, 1], - P_neg[:, 2]; - shape = :xcross, - markercolor = :crimson, + plot, + P[:, 1], + P[:, 2]; + shape = ifelse.(labels_new .== 1, :cross, :xcross), + markercolor = ifelse.(labels_new .== 1, :blue, :crimson), markersize = 8, ) - -model, w_sol, g_sol, y_sol = solve_SVM_classifier(P, labels_new) -Test.@test termination_status(model) == LOCALLY_SOLVED #src -Test.@test primal_status(model) == FEASIBLE_POINT #src -println("Minimum slack: ", minimum(y_sol), "\nMaximum slack: ", maximum(y_sol)) - -# - Plots.plot!( - plot_copy, - x -> line(x; w = w_sol, g = g_sol), + plot, + classifier, 0.0:0.01:2.0; - seriestype = :line, linewidth = 5, linestyle = :dashdotdot, - lineopacity = 0.9, c = :darkblue, ) -# So it still works as a classifier, with allowance for the nonseparable points. +# So our JuMP formulation still produces a classifier, but it mis-classifies +# some of the nonseparable points. # # We can find out which points are contributing to the shape of the line by -# looking at the dual values of the affine constraints and comparing them -# to the penalty constant ``C``: +# looking at the dual values of the affine constraints and comparing them to the +# penalty constant ``C``: affine_cons = all_constraints(model, AffExpr, MOI.GreaterThan{Float64}) active_cons = findall(isapprox.(dual.(affine_cons), C_0; atol = 0.001)) - findall(nearby_indices) ⊆ active_cons -# The last statement tells us that our nonseparable points are actively contributing -# to how the classifier is defined. The remaining points are of interest and -# are highlighted: +# The last statement tells us that our nonseparable points are actively +# contributing to how the classifier is defined. The remaining points are of +# interest and are highlighted: P_active = P[setdiff(active_cons, findall(nearby_indices)), :] - Plots.scatter!( - plot_copy, + plot, P_active[:, 1], P_active[:, 2]; shape = :hexagon, @@ -262,11 +224,16 @@ Plots.scatter!( markersize = 8, markeropacity = 0.5, ) -# ## Advanced: Duality and the kernel method + +# ## Advanced: duality and the kernel method + +# We now consider an alternative formulation for a linear SVM by solving the +# dual problem. # ### The dual program -# The dual of the linear SVM program is also a linearly constrained quadratic program: +# The dual of the linear SVM program is also a linearly constrained quadratic +# program: # ```math # \begin{aligned} # \min_{u \in \mathbb{R}^m} \quad & \frac{1}{2} u^\top D P P^\top D u - \; \mathbf{1}^\top u \\ @@ -275,72 +242,72 @@ Plots.scatter!( # \end{aligned} # ``` # This is the JuMP model: -function solve_dual_SVM_classifier( - P, - labels; - optimizer = Ipopt.Optimizer, - C = C_0, -) + +function solve_dual_SVM_classifier(P::Matrix, labels::Vector; C::Float64 = C_0) m, n = size(P) - model = Model(optimizer) + model = Model(Ipopt.Optimizer) set_silent(model) @variable(model, 0 <= u[1:m] <= C) D = LinearAlgebra.Diagonal(labels) - con = @constraint(model, sum(D * u) == 0) @objective(model, Min, 1 / 2 * u' * D * P * P' * D * u - sum(u)) + @constraint(model, con, sum(D * u) == 0) optimize!(model) - u_sol = value.(u) - w_sol = P' * D * u_sol - g_sol = dual(con) - return model, u_sol, w_sol, g_sol + Test.@test termination_status(model) == LOCALLY_SOLVED #src + Test.@test primal_status(model) == FEASIBLE_POINT #src + w = P' * D * value.(u) + g = dual(con) + classifier(x) = line(x; w = w, g = g) + return classifier end -# We recover the line gradient vector ``w`` through setting ``w = P^\top D u`` +# We recover the line gradient vector ``w`` through setting ``w = P^\top D u``, # and the line constant ``g`` as the dual value of the single affine constraint. -# In many cases it may be simpler to solve the dual form. We can check that this -# has recovered the classifier: -model_dual, u_d, w_d, g_d = solve_dual_SVM_classifier(P, labels) -Test.@test termination_status(model_dual) == LOCALLY_SOLVED #src -Test.@test primal_status(model_dual) == FEASIBLE_POINT #src +# The dual problem has fewer variables and fewer constraints, so in many cases +# it may be simpler to solve the dual form. + +# We can check that the dual form has recovered a classifier: + +classifier = solve_dual_SVM_classifier(P, labels) Plots.plot!( - plot_copy, - x -> line(x; w = w_d, g = g_d), + plot, + classifier, 0.0:0.01:2.0; - seriestype = :line, linewidth = 2, linestyle = :dash, - lineopacity = 0.9, c = :red, ) # ## The kernel method -# Linear SVM techniques are not limited to finding separating hyperplanes -# in the original space of the dataset. One could first transform the -# training data under a nonlinear mapping, apply our method, then -# map the hyperplane back into original space. +# Linear SVM techniques are not limited to finding separating hyperplanes in the +# original space of the dataset. One could first transform the training data +# under a nonlinear mapping, apply our method, then map the hyperplane back into +# original space. -# The actual data describing the point set is held in a matrix ``P``, -# but looking at the dual program we see that what actually matters is the +# The actual data describing the point set is held in a matrix ``P``, but +# looking at the dual program we see that what actually matters is the # [Gram matrix](https://en.wikipedia.org/wiki/Gram_matrix) ``P P^\top``, # expressing a pairwise comparison (an inner-product) between each point vector. -# It follows that any mapping of the point set only needs -# to be defined at the level of _pairwise_ maps between points. -# Such maps are known as _kernel functions_ +# It follows that any mapping of the point set only needs to be defined at the +# level of _pairwise_ maps between points. Such maps are known as _kernel functions_: # ```math # k \; : \; \mathbb{R}^n \times \mathbb{R}^n \; \rightarrow \mathbb{R}, \qquad # (s, t) \mapsto \left< \Phi(s), \Phi(t) \right> # ``` -# where the right-hand side applies some transformation ``\Phi : \mathbb{R}^n \rightarrow \mathbb{R}^{n'}`` -# followed by an inner-product in that image space. In practice, we can avoid having -# ``\Phi`` explicitly given but instead define a kernel function directly between pairs of vectors. -# This move to using a kernel function without knowing the map is called the _kernel method_ -# (or sometimes, the _kernel trick_). +# where the right-hand side applies some transformation +# ``\Phi : \mathbb{R}^n \rightarrow \mathbb{R}^{n'}`` followed by an +# inner-product in that image space. + +# In practice, we can avoid having ``\Phi`` explicitly given but instead define +# a kernel function directly between pairs of vectors. This change to using a +# kernel function without knowing the map is called the _kernel method_ (or +# sometimes, the _kernel trick_). # ### Classifier using a Gaussian kernel -# We will demonstrate the application of a _Gaussian_ or _radial basis function_ kernel +# We will demonstrate the application of a _Gaussian_ or _radial basis function_ +# kernel: # ```math # k(s, t) = \exp\left( -\mu \lVert s - t \rVert^2_2 \right) # ``` @@ -348,58 +315,51 @@ Plots.plot!( k_gauss(s::Vector, t::Vector; μ = 0.5) = exp(-μ * LinearAlgebra.norm(s - t)^2) -# Given a matrix of points expressed row-wise and a kernel, let's return -# the transformed matrix ``K`` that replaces ``P P^\top``: +# Given a matrix of points expressed row-wise and a kernel, the next function +# returns the transformed matrix ``K`` that replaces ``P P^\top``: -function pairwise_transform(kernel_func, P) +function pairwise_transform(kernel::Function, P::Matrix{T}) where {T} m, n = size(P) - K = zeros(eltype(P), m, m) - - for j in 1:m - for i in 1:j - K[i, j] = kernel_func(P[i, :], P[j, :]) - end + K = zeros(T, m, m) + for j in 1:m, i in 1:j + K[i, j] = K[j, i] = kernel(P[i, :], P[j, :]) end - - return LinearAlgebra.Symmetric(K, :U) + return LinearAlgebra.Symmetric(K) end # Now we're ready to define our optimization problem. We need to provide the # kernel function to be used in the problem. Note that any extra keyword -# arguments here (like parameter values) are passed through to the kernel function. +# arguments here (like parameter values) are passed through to the kernel. function solve_kernel_SVM_classifier( - kernel_func, - P, - labels; - optimizer = Ipopt.Optimizer, - C = C_0, + kernel::Function, + P::Matrix, + labels::Vector; + C::Float64 = C_0, kwargs..., ) m, n = size(P) - K = pairwise_transform(kernel_func, P) - model = Model(optimizer) + K = pairwise_transform(kernel, P) + model = Model(Ipopt.Optimizer) set_silent(model) @variable(model, 0 <= u[1:m] <= C) D = LinearAlgebra.Diagonal(labels) con = @constraint(model, sum(D * u) == 0) @objective(model, Min, 1 / 2 * u' * D * K * D * u - sum(u)) optimize!(model) - u_sol = value.(u) - g_sol = dual(con) - - function kernel_classifier(v::Vector) + Test.@test termination_status(model) == LOCALLY_SOLVED #src + Test.@test primal_status(model) == FEASIBLE_POINT #src + u_sol, g_sol = value.(u), dual(con) + function classifier(v::Vector) return sum( - D[i, i] * u_sol[i] * kernel_func(P[i, :], v; kwargs...) for i in 1:m - ) + D[i, i] * u_sol[i] * kernel(P[i, :], v; kwargs...) for i in 1:m + ) - g_sol end - - return model, u_sol, g_sol, kernel_classifier + return classifier end -# This time, we don't recover the line gradient vector ``w`` directly. -# Instead, we compute the classifier ``f`` using the dual values and the -# kernel function: +# This time, we don't recover the line gradient vector ``w`` directly. Instead, +# we compute the classifier ``f`` using the function: # ```math # f(v) = \sum_{i=1}^m D_{ii} u_i \; k(p_i, v ) - g # ``` @@ -407,72 +367,36 @@ end # ### Checkerboard dataset -# To demonstrate this nonlinear technique, we'll use the [checkerboard dataset](data/checker/README). +# To demonstrate this nonlinear technique, we'll use the +# [checkerboard dataset](data/checker/README). -checkerboard = DelimitedFiles.readdlm( - joinpath(@__DIR__, "data", "checker", "checker.txt"), - ' ', - Int64, -) -labels_checker = ifelse.(checkerboard[:, 1] .== 0, -1, 1) +filename = joinpath(@__DIR__, "data", "checker", "checker.txt") +checkerboard = DelimitedFiles.readdlm(filename, ' ', Int) +labels = ifelse.(iszero.(checkerboard[:, 1]), -1, 1) B = checkerboard[:, 2:3] ./ 100.0 # rescale to [0,2] x [0,2] square. -B_pos = B[labels_checker.==1, :] -B_neg = B[labels_checker.==-1, :] -@assert size(B_pos, 1) + size(B_neg, 1) == size(B, 1) #src -r = 1.01 * maximum(abs.(B)) -plot_checkers = Plots.scatter( - B_pos[:, 1], - B_pos[:, 2]; - xlim = (0, r), - ylim = (0, r), - label = nothing, - color = :white, - shape = :circle, +plot = Plots.scatter( + B[:, 1], + B[:, 2]; + color = ifelse.(labels .== 1, :white, :black), + markersize = ifelse.(labels .== 1, 4, 2), size = (600, 600), legend = false, ) -Plots.scatter!( - plot_checkers, - B_neg[:, 1], - B_neg[:, 2]; - color = :black, - shape = :circle, - markersize = 2, -) -# Is the technique capable of generating a distinctly nonlinear surface? -# Let's solve the Gaussian kernel based quadratic problem with these parameters: - -C_1 = 100_000 -μ_1 = 10 -println("C = ", C_1, ", μ = ", μ_1) - -# - -model_kernel, u_k, g_k, f_gauss = - solve_kernel_SVM_classifier(k_gauss, B, labels_checker; C = C_1, μ = μ_1) -Test.@test termination_status(model_kernel) == LOCALLY_SOLVED #src -Test.@test primal_status(model_kernel) == FEASIBLE_POINT #src +# Is the technique capable of generating a distinctly nonlinear surface? Let's +# solve the Gaussian kernel based quadratic problem with these parameters: +classifier = solve_kernel_SVM_classifier(k_gauss, B, labels; C = 1e5, μ = 10.0) grid = [[x, y] for x in 0:0.01:2, y in 0:0.01:2] -grid_pos = grid[f_gauss.(grid).>0] - -plot_board = Plots.scatter( - plot_checkers, - Tuple.(grid_pos); - shape = :circle, - markersize = 0.2, - xlim = (0, r), - ylim = (0, r), - size = (600, 600), - legend = false, -) +grid_pos = [Tuple(g) for g in grid if classifier(g) >= 0] +Plots.scatter!(plot, grid_pos; markersize = 0.2) # We find that the kernel method can perform well as a nonlinear classifier. -# The result has a fairly strong dependence on the choice of parameters, -# with larger values of ``\mu`` allowing for a more complex boundary while -# smaller values lead to a smoother boundary for the classifier. -# Determining a better performing kernel function and choice of parameters is -# covered by the process of _cross-validation_ with respect to the dataset, -# where different testing, training and tuning sets are used to validate -# the best choice of parameters against a statistical measure of error. + +# The result has a fairly strong dependence on the choice of parameters, with +# larger values of ``\mu`` allowing for a more complex boundary while smaller +# values lead to a smoother boundary for the classifier. Determining a better +# performing kernel function and choice of parameters is covered by the process +# of _cross-validation_ with respect to the dataset, where different testing, +# training and tuning sets are used to validate the best choice of parameters +# against a statistical measure of error. From b30229d2b72d3e7bd66d3dbe3f7d250ae698962d Mon Sep 17 00:00:00 2001 From: odow Date: Wed, 15 Nov 2023 14:59:38 +1300 Subject: [PATCH 10/11] Further simplify plotting calls --- docs/src/tutorials/nonlinear/classifiers.jl | 12 ++---------- 1 file changed, 2 insertions(+), 10 deletions(-) diff --git a/docs/src/tutorials/nonlinear/classifiers.jl b/docs/src/tutorials/nonlinear/classifiers.jl index e57ca90b433..9527a8036d0 100644 --- a/docs/src/tutorials/nonlinear/classifiers.jl +++ b/docs/src/tutorials/nonlinear/classifiers.jl @@ -75,7 +75,7 @@ line(x::Real; w = w_0, g = g_0) = -(w[1] * x - g) / w[2]; # Let's add this to the plot: -Plots.plot!(plot, line, 0.0:0.01:2.0; linewidth = 5) +Plots.plot!(plot, line; linewidth = 5) # Now we label the points relative to which side of the line they are. It is # numerically useful to have the labels +1 and -1 for the upcoming JuMP @@ -151,14 +151,7 @@ _, classifier = solve_SVM_classifier(P, labels) # Let's plot the solution and check how we did: -Plots.plot!( - plot, - classifier, - 0.0:0.01:2.0; - linewidth = 5, - linestyle = :dashdotdot, - c = :darkblue, -) +Plots.plot!(plot, classifier; linewidth = 5, linestyle = :dashdotdot) # We find that we have recovered the dividing line from just the information of # the points and their labels. @@ -275,7 +268,6 @@ Plots.plot!( 0.0:0.01:2.0; linewidth = 2, linestyle = :dash, - c = :red, ) # ## The kernel method From 48d6d0e332a6a0efb3d94564a61163bb4e547f05 Mon Sep 17 00:00:00 2001 From: odow Date: Wed, 15 Nov 2023 15:02:49 +1300 Subject: [PATCH 11/11] More updates --- docs/src/tutorials/nonlinear/classifiers.jl | 18 ++---------------- 1 file changed, 2 insertions(+), 16 deletions(-) diff --git a/docs/src/tutorials/nonlinear/classifiers.jl b/docs/src/tutorials/nonlinear/classifiers.jl index 9527a8036d0..ee0f21c568a 100644 --- a/docs/src/tutorials/nonlinear/classifiers.jl +++ b/docs/src/tutorials/nonlinear/classifiers.jl @@ -183,14 +183,7 @@ Plots.scatter!( markercolor = ifelse.(labels_new .== 1, :blue, :crimson), markersize = 8, ) -Plots.plot!( - plot, - classifier, - 0.0:0.01:2.0; - linewidth = 5, - linestyle = :dashdotdot, - c = :darkblue, -) +Plots.plot!(plot, classifier; linewidth = 5, linestyle = :dashdotdot) # So our JuMP formulation still produces a classifier, but it mis-classifies # some of the nonseparable points. @@ -213,7 +206,6 @@ Plots.scatter!( P_active[:, 1], P_active[:, 2]; shape = :hexagon, - markercolor = :green, markersize = 8, markeropacity = 0.5, ) @@ -262,13 +254,7 @@ end # We can check that the dual form has recovered a classifier: classifier = solve_dual_SVM_classifier(P, labels) -Plots.plot!( - plot, - classifier, - 0.0:0.01:2.0; - linewidth = 2, - linestyle = :dash, -) +Plots.plot!(plot, classifier; linewidth = 5, linestyle = :dash) # ## The kernel method