Base implementation of SVGP

Project.toml

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+name = "SparseGPs"
+uuid = "298c2ebc-0411-48ad-af38-99e88101b606"
+authors = ["Ross Viljoen <ross@viljoen.co.uk>"]
+version = "0.1.0"
+
+[deps]
+AbstractGPs = "99985d1d-32ba-4be9-9821-2ec096f28918"
+ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
+FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838"
+FillArrays = "1a297f60-69ca-5386-bcde-b61e274b549b"
+GPLikelihoods = "6031954c-0455-49d7-b3b9-3e1c99afaf40"
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
+Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
+StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"

examples/classification.jl

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+# Recreation of https://gpflow.readthedocs.io/en/master/notebooks/basics/classification.html
+
+# %%
+using SparseGPs
+using AbstractGPs
+using GPLikelihoods
+using StatsFuns
+using FastGaussQuadrature
+using Distributions
+using LinearAlgebra
+using DelimitedFiles
+using IterTools
+
+using Plots
+default(; legend=:outertopright, size=(700, 400))
+
+using Random
+Random.seed!(1234)
+
+# %%
+# Read in the classification data
+data_file = pkgdir(SparseGPs) * "/examples/data/classif_1D.csv"
+x, y = eachcol(readdlm(data_file))
+scatter(x, y)
+
+
+# %%
+# First, create the GP kernel from given parameters k
+function make_kernel(k)
+    return softplus(k[1]) * (SqExponentialKernel() ∘ ScaleTransform(softplus(k[2])))
+end
+
+k = [10, 0.1]
+
+kernel = make_kernel(k)
+f = LatentGP(GP(kernel), BernoulliLikelihood(), 0.1)
+fx = f(x)
+
+
+# %%
+# Then, plot some samples from the prior underlying GP
+x_plot = 0:0.02:6
+prior_f_samples = rand(f.f(x_plot, 1e-6),20)
+
+plt = plot(
+    x_plot,
+    prior_f_samples;
+    seriescolor="red",
+    linealpha=0.2,
+    label=""
+)
+scatter!(plt, x, y; seriescolor="blue", label="Data points")
+
+# %%
+# Plot the same samples, but pushed through a logistic sigmoid to constrain
+# them in (0, 1).
+prior_y_samples = mean.(f.lik.(prior_f_samples))
+
+plt = plot(
+    x_plot,
+    prior_y_samples;
+    seriescolor="red",
+    linealpha=0.2,
+    label=""
+)
+scatter!(plt, x, y; seriescolor="blue", label="Data points")
+
+
+# %%
+# A simple Flux model
+using Flux
+
+struct SVGPModel
+    k # kernel parameters
+    m # variational mean
+    A # variational covariance
+    z # inducing points
+end
+
+@Flux.functor SVGPModel (k, m, A,) # Don't train the inducing inputs
+
+lik = BernoulliLikelihood()
+function (m::SVGPModel)(x)
+    kernel = make_kernel(m.k)
+    f = LatentGP(GP(kernel), BernoulliLikelihood(), 0.1)
+    q = MvNormal(m.m, m.A'm.A)
+    fx = f(x)
+    fu = f(m.z).fx
+    return fx, fu, q
+end
+
+function flux_loss(x, y; n_data=length(y))
+    fx, fu, q = model(x)
+    return -SparseGPs.elbo(fx, y, fu, q; n_data, method=MonteCarlo())
+end
+
+# %%
+M = 15 # number of inducing points
+
+# Initialise the parameters
+k = [10, 0.1]
+m = zeros(M)
+A = Matrix{Float64}(I, M, M)
+z = x[1:M]
+
+model = SVGPModel(k, m, A, z)
+
+opt = ADAM(0.1)
+parameters = Flux.params(model)
+
+# %%
+# Negative ELBO before training
+println(flux_loss(x, y))
+
+# %%
+# Train the model
+Flux.train!(
+    (x, y) -> flux_loss(x, y),
+    parameters,
+    ncycle([(x, y)], 1000), # Train for 1000 epochs
+    opt
+)
+
+# %%
+# Negative ELBO after training
+println(flux_loss(x, y))
+
+# %%
+# After optimisation, plot samples from the underlying posterior GP.
+
+fu = f(z).fx # want the underlying FiniteGP
+post = SparseGPs.approx_posterior(SVGP(), fu, MvNormal(m, A'A))
+l_post = LatentGP(post, BernoulliLikelihood(), 0.1)
+
+post_f_samples = rand(l_post.f(x_plot, 1e-6), 20)
+
+plt = plot(
+    x_plot,
+    post_f_samples;
+    seriescolor="red",
+    linealpha=0.2,
+    legend=false
+)
+
+# %%
+# As above, push these samples through a logistic sigmoid to get posterior predictions.
+post_y_samples = mean.(l_post.lik.(post_f_samples))
+
+plt = plot(
+    x_plot,
+    post_y_samples;
+    seriescolor="red",
+    linealpha=0.2,
+    # legend=false,
+    label=""
+)
+scatter!(plt, x, y; seriescolor="blue", label="Data points")
+vline!(z; label="Pseudo-points")

examples/data/classif_1D.csv

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