Four different implementations of SVMs in Julia (Quadratic optimization of primal/dual problems/gradient descent)

SVM.jl

### A Pluto.jl notebook ###
# v0.18.1

using Markdown
using InteractiveUtils

# ╔═╡ 4bdb73f8-a2c2-11ec-078d-27a1269db616
begin
	import JuMP as J
	import Plots
	import Distributions
	import Ipopt
	import Flux
end

# ╔═╡ 37439e4d-cb89-4591-9a5c-99399b0113e6
function gen_random_data_1(n::Int)
	a, b = (Distributions.MvNormal([1.5, 1.], [.7 .1; .1 .8]), Distributions.MvNormal([1.1, -2.], [.9 .1; .1 .9]))
	ma = fill(-1, 1, n)
	mb = fill(1, 1, n)
	hcat(rand(a, n), rand(b, n)), hcat(ma, mb)
end

# ╔═╡ 07f279aa-a8fa-4c40-93bb-bde5f1645533
function plot_data(d::Matrix, labels::Matrix)
	Plots.plot()
	Plots.scatter!(d[1,:], d[2,:], zcolor=labels', label="data")
	Plots.plot!()
end

# ╔═╡ ef81fb41-fbd4-493a-9b85-235ad13dec5e
data, labels = gen_random_data_1(10)

# ╔═╡ 41441ea5-9785-4036-a92b-c10b12685704
function SVM_1(data::Matrix, labels::Matrix)
	d, n = size(data)

	model = J.Model(Ipopt.Optimizer)
	J.@variable(model, W[1:d, 1:1])
	J.@variable(model, b)

	J.@objective(model, Min, sum(W'*W))
	J.@constraint(model, c_n[i=1:n], labels[:, i] .* (sum(W' * data[:, i]) .+ b) .>= 1)

	J.optimize!(model)
	W, b = J.value.(W), J.value.(b)
	classification = W' * data .+ b
	classification, W, b
end

# ╔═╡ 46d8a454-9d42-4e1a-a228-afd151739b49
begin
	@time class1, W1, b1 = SVM_1(data, labels)
end

# ╔═╡ c89f1739-c34e-43b9-980d-d51a0955e4ca
function SVM_2(data::Matrix, labels::Matrix; slackyes=1)
	# Lagrangian Primal approach
	d, n = size(data)
	td = size(labels, 1)

	model = J.Model(Ipopt.Optimizer)

	J.@variable(model, W[1:d])
	J.@variable(model, b)
	J.@variable(model, 1 >= a[i=1:n] >= 0)
	J.@variable(model, μ[i=1:n] >= 0)
	J.@variable(model, 0 <= slack[i=1:n] <= 1e5 * slackyes)
	J.@objective(model, Min, sum(W.^2)/2 -
	sum(
		a[i] * (
			labels[1,i] * (sum(W' * data[:,i]) + b) + slack[i] - 1)
		for i in 1:n) + 1e6 * sum(slack) - sum(μ .* slack))
	J.@constraint(model, [i=1:n], labels[1,i]*(sum(W' * data[:,i]) + b) + slack[i] - 1 >= 0)
	J.@constraint(model, [i=1:n], a[i] * (labels[1,i]*(sum(W' * data[:,i]) + b) + slack[i] - 1) == 0)
	J.@constraint(model, sum(μ .* slack) == 0)

	J.optimize!(model)
	W, b, a = J.value.(W), J.value.(b), J.value.(a)
	classification = W' * data .+ b
	classification, W, b, a
end

# ╔═╡ 07df25aa-6885-474a-b593-198bc591ec2c
begin
	@time class2, W2, b2, a2 = SVM_2(data, labels; slackyes=0)
end

# ╔═╡ 87dfcfc7-ee88-4def-a8ef-9abced9c5b8d
function SVM_3(data::Matrix, labels::Matrix)
	# Lagrangian Dual approach
	d, n = size(data)
	td = size(labels, 1)

	model = J.Model(Ipopt.Optimizer)
	J.@variable(model, 0 <= a[i=1:n] <= 1)
	J.@objective(model, Max, sum(a) - 0.5 * sum(
		sum(
			a[i]*a[j]*labels[1,i]*labels[1,j]*sum(data[:,i] .* data[:,j])
		for j in 1:n) for i in 1:n
	))
	J.@constraint(model, sum(a .* labels') == 0)

	J.optimize!(model)

	A = J.value.(a)
	W = sum((A' .* labels) .* data; dims=2)
	b = 1/(n) * sum(labels[1,i] - sum(A[j]*labels[1,j]*sum(data[:, j] .* data[:, i]) for j in 1:n) for i in 1:n)
	classification = W' * data .+ b
	classification, W, b, A
end

# ╔═╡ c2ef5b11-5939-4503-b503-ddb82297820c
begin
	@time class3, W3, b3, a3 = SVM_3(data, labels)
end

# ╔═╡ 48ca6f5f-491a-4f2d-b32f-c9632d29e7dc
a3

# ╔═╡ b7b63f0e-0506-4332-a911-c565947bbaa5
function SVM_gd(data::Matrix, labels::Matrix; iter=100, η=0.01, sigmoid=tanh)
	d, n = size(data)

	W = rand(eltype(data), d, 1)
	b = rand(eltype(data))

	C = 1e3
	loss(W, b, i) = sum(W.^2)/2 + C * sum(1 .- labels[1,i] .* sigmoid.(W' * data[:,i] .+ b))

	for i in 1:iter
		# Stochastic gradient descent is more stable.
		for j in 1:n
			v, g = Flux.withgradient(loss, W, b, j)
			#println("$i: loss = $v")
			W -= η .* g[1]/n
			b -= η * g[2]/n
		end
	end

	classification = sigmoid.(W' * data .+ b)
	(classification, W, [b])
end

# ╔═╡ 98636716-7990-47b1-9b10-c29bf7a5c7a1
class4, W4, b4 = SVM_gd(data, labels; η=1e-3, iter=100)

# ╔═╡ 8217d44c-229c-429f-93d5-48a428502859
y(b, W, x) = (-b[1] - W[1]*x)/W[2]

# ╔═╡ 1ed57562-0a89-4da3-9944-7d4af890521d
begin
	plot_data(data, labels)
	Plots.annotate!([(data[1,i], data[2,i], i) for i in 1:size(data, 2)])
	Plots.plot!(LinRange(-1, 3, 3), x -> y(b1, W1, x), label="SVM_1")
	Plots.plot!(LinRange(-1, 3, 3), x -> y(b2, W2, x), label="SVM_2")
	Plots.plot!(LinRange(-1, 3, 3), x -> y(b3, W3, x), label="SVM_3")
	Plots.plot!(LinRange(-1, 3, 3), x -> y(b4, W4, x), label="SVM_gd")
end

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[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.ZipFile]]
deps = ["Libdl", "Printf", "Zlib_jll"]
git-tree-sha1 = "3593e69e469d2111389a9bd06bac1f3d730ac6de"
uuid = "a5390f91-8eb1-5f08-bee0-b1d1ffed6cea"
version = "0.9.4"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.Zygote]]
deps = ["AbstractFFTs", "ChainRules", "ChainRulesCore", "DiffRules", "Distributed", "FillArrays", "ForwardDiff", "IRTools", "InteractiveUtils", "LinearAlgebra", "MacroTools", "NaNMath", "Random", "Requires", "SparseArrays", "SpecialFunctions", "Statistics", "ZygoteRules"]
git-tree-sha1 = "bf526aa30677f1dde58febc67cb9021aab5eb396"
uuid = "e88e6eb3-aa80-5325-afca-941959d7151f"
version = "0.6.36"

[[deps.ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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