diff --git a/.github/workflows/Docs.yml b/.github/workflows/Docs.yml index 009fcdef..a032455f 100644 --- a/.github/workflows/Docs.yml +++ b/.github/workflows/Docs.yml @@ -14,7 +14,7 @@ jobs: - uses: actions/checkout@v2 - uses: julia-actions/setup-julia@latest with: - version: '1.6' + version: '1.7' - name: Install dependencies run: julia --project=docs/ -e 'using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()' - name: Build and deploy diff --git a/examples/b-classification/Manifest.toml b/examples/b-classification/Manifest.toml index a8b06699..93de88a1 100644 --- a/examples/b-classification/Manifest.toml +++ b/examples/b-classification/Manifest.toml @@ -1,1094 +1,1117 @@ # This file is machine-generated - editing it directly is not advised -[[AbstractFFTs]] +julia_version = "1.7.0" +manifest_format = "2.0" + +[[deps.AbstractFFTs]] deps = ["LinearAlgebra"] git-tree-sha1 = "485ee0867925449198280d4af84bdb46a2a404d0" uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c" version = 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["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" diff --git a/examples/d-augmented-gp/Project.toml b/examples/d-augmented-gp/Project.toml new file mode 100644 index 00000000..a78be1e3 --- /dev/null +++ b/examples/d-augmented-gp/Project.toml @@ -0,0 +1,11 @@ +[deps] +ApproximateGPs = "298c2ebc-0411-48ad-af38-99e88101b606" +AugmentedGaussianProcesses = "38eea1fd-7d7d-5162-9d08-f89d0f2e271e" +Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Literate = "98b081ad-f1c9-55d3-8b20-4c87d4299306" +Optim = "429524aa-4258-5aef-a3af-852621145aeb" +ParameterHandling = "2412ca09-6db7-441c-8e3a-88d5709968c5" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" +Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f" diff --git a/examples/d-augmented-gp/script.jl b/examples/d-augmented-gp/script.jl new file mode 100644 index 00000000..8aae3c37 --- /dev/null +++ b/examples/d-augmented-gp/script.jl @@ -0,0 +1,223 @@ +# # Augmented Sparse Variational Approximation for Non-Conjugate Likelihoods, +# # a Classification Example +# +# +# This example demonstrates how to use augmentation to improve convergence speed +# for variational Gaussian process with non-conjugate likelihoods. +# inference using the stochastic variational Gaussian process (SVGP) model. For +# a basic introduction to the functionality of this library, please refer to the +# [User Guide](@ref). +# +# ## Setup + +using ApproximateGPs +using ParameterHandling +using Zygote +using Distributions +using LinearAlgebra +using Optim + +using Plots +default(; legend=:outertopright, size=(700, 400)) + +# We import AGP.jl for the Polya-Gamma distribution +using AugmentedGaussianProcesses.ComplementaryDistributions: PolyaGamma + +using Random +Random.seed!(1234); + +# ## Generate some training data +# +# We work in the same setting as the other [classification example](../b-classification). +# For our binary classification model, we will use the standard approach of a +# latent GP with a Bernoulli likelihood. This results in a generative model that +# we can use to produce some training data. +# +# First, we define the underlying latent GP +# ```math +# f \sim \mathcal{GP}(0, k(\cdot, \cdot')) +# ``` +# and sample a function `f`. + +k_true = [30.0, 1.5] +kernel_true = k_true[1] * (Matern32Kernel() ∘ ScaleTransform(k_true[2])) + +jitter = 1e-6 # for numeric stability +lik = BernoulliLikelihood() +lgp = LatentGP(GP(kernel_true), lik, jitter) +x_true = 0:0.02:6 +f_true, y_true = rand(lgp(x_true)) + +plot(x_true, f_true; seriescolor="red", label="") # Plot the sampled function + +# Then, the output of this sampled function is pushed through a logistic sigmoid +# `μ = σ(f)` to constrain the output to `[0, 1]`. + +μ = mean.(lgp.lik.(f_true)) +plot(x_true, μ; seriescolor="red", label="") + +# Finally, the outputs `y` of the process are sampled from a Bernoulli +# distribution with mean `μ`. We're only interested in the outputs at a subset +# of inputs `x`, so we first pick some random input locations and then find the +# corresponding values for `y`. + +N = 200 # The number of training points +mask = sample(1:length(x_true), N; replace=false, ordered=true) # Subsample some input locations +x, y = x_true[mask], y_true[mask] + +scatter(x, y; label="Sampled outputs") +plot!(x_true, mean.(lgp.lik.(f_true)); seriescolor="red", label="True mean") + +# Let's first see how we can obtain the optimal variational parameters +# for our variational distribution q(u) = N(m, S). +# We will for now use the true kernel parameters, +# We define the inducing points locations and the associated `FiniteGP` +M = 15; +z = range(0.1, 5.9; length=M); +fz = GP(kernel_true)(z, jitter); + +# We initialize the variational parameters q(u) = N(m, S) and q(ωᵢ) = p(ωᵢ) = PolyaGamma(1,0) +m = zeros(M); +S = Matrix{Float64}(I(M)); +qω = [PolyaGamma(1, 0.0) for _ in 1:length(x)]; + +# Following the derivations from the paper [Efficient Gaussian Process Classification Using Polya-Gamma Data Augmentation](https://arxiv.org/abs/1802.06383), +# we build the following `cavi!` function which will update the parameters `m` and `S` as well as the variables `ω` + +function cavi!(fz::AbstractGPs.FiniteGP, x, y, m, S, qω; niter=10) + K = ApproximateGPs._chol_cov(fz) + κ = K \ cov(fz.f, fz.x, x) + y_sign = sign.(y .- 0.5) + for _ in 1:niter + postu = posterior(SparseVariationalApproximation(Centered(), fz, MvNormal(m, S))) + postfs = marginals(postu(x)) + marginals_to_aug_posterior!(qω, postfs) + S .= inv(Symmetric(inv(K) + project(Diagonal(mean.(qω)), κ))) + m .= S * (project(y_sign / 2, κ) - K \ mean(fz)) + end + return m, S, qω +end + +function marginals_to_aug_posterior!(qω::AbstractVector, ps::AbstractVector{<:Normal}) + map!(qω, ps) do p + PolyaGamma(1, sqrt(abs2(mean(p)) + var(p))) + end +end + +# where `project` which project the noise to the right dimension +project(x::AbstractVector, κ) = κ * x +project(X::AbstractMatrix, κ) = κ * X * κ' + +# Let's now compare both results +post_svgp_init = posterior(SparseVariationalApproximation(Centered(), fz, MvNormal(m, S))) +cavi!(fz, x, y, m, S, qω) # m, S and qω are going to be modified +post_svgp_opt = posterior(SparseVariationalApproximation(Centered(), fz, MvNormal(m, S))) + +pf = scatter(x, y; legend=false, xlabel="x", ylabel="f") +plot!(x_true, f_true; seriescolor="red", label="True latent") +plot!(x_true, post_svgp_init(x_true); label="Initial Posterior") +plot!(x_true, post_svgp_opt(x_true); label="Final Posterior") +py = scatter(x, y; label="Training data", xlabel="x", ylabel="y") +plot!(x_true, mean.(lik.(f_true)); label="True mean") +plot!(x_true, mean.(lik.(mean(post_svgp_init(x_true)))); label="Initial Posterior") +plot!(x_true, mean.(lik.(mean(post_svgp_opt(x_true)))); label="Final Posterior") +plot(pf, py) +# ## Optimizing the hyperparameters + +# What if like in the [classical optimization case](../b-classification) +# we want to optimize hyperparameters as well? +# Let's use the same approach! +raw_initial_params = ( + k=(var=positive(rand()), precision=positive(rand())), + z=bounded.(range(0.1, 5.9; length=M), 0.0, 6.0), # constrain z to simplify optimisation +); +flat_init_params, unflatten = ParameterHandling.flatten(raw_initial_params) +unpack = ParameterHandling.value ∘ unflatten; + +# We now need to pass our variational parameters and modify the elbo +# to be able to infer correctly the results. +# This means we will have to implicitly pass the variational parameters +# `var_params` and update them after each update of the hyperparameters + +# ### The augmented ELBO +function augmented_elbo( + sva::SparseVariationalApproximation, + lfx::AbstractGPs.LatentFiniteGP, + y::AbstractVector, + qω::AbstractVector; + num_data=length(y), +) + post = posterior(sva) + q_f = marginals(post(lfx.fx.x)) + variational_exp = expected_aug_loglik(y, q_f, lfx.lik, qω) + + n_batch = length(y) + scale = num_data / n_batch + + return sum(variational_exp) * scale - Zygote.@ignore(kl_term(lik, qω)) - + ApproximateGPs.kl_term(sva, post) +end +# We ignored the gradient of the KL divergence with respect to the augmented +# variables as it does not depend on the kernel parameters + +function kl_term(::BernoulliLikelihood{<:LogisticLink}, qω::AbstractVector{<:PolyaGamma}) + sum(qω) do q + c = q.c + -abs2(c) * mean(q) + 2log(cosh(c / 2)) + end +end + +function expected_aug_loglik(y, q_f, ::BernoulliLikelihood, qω) + return map(y, q_f, qω) do y, q, qω + m = mean(q) + sign(y - 0.5) * m / 2 - (abs2(m) + var(q)) * mean(qω) / 2 - log(2) + end +end + +# ### The complete inference process +function loss(params::NamedTuple) + fz, lf = build_fz_lf(params) + fx = lf(x) + Zygote.@ignore cavi!(fz, x, y, m, S, qω; niter=1) + augsvgp = SparseVariationalApproximation(Centered(), fz, MvNormal(m, Symmetric(S))) + return -augmented_elbo(augsvgp, fx, y, qω) +end + +# We ignore the `cavi!` part as it involves inplace operations +# and does not contribute to the final elbo + +function build_fz_lf(params::NamedTuple) + kernel = params.k.var * (Matern32Kernel() ∘ ScaleTransform(params.k.precision)) + lf = LatentGP(GP(kernel), lik, jitter) + fz = GP(kernel)(params.z) + return fz, lf +end + +opt = optimize( + loss ∘ unpack, + θ -> only(Zygote.gradient(loss ∘ unpack, θ)), + flat_init_params, + LBFGS(; + alphaguess=Optim.LineSearches.InitialStatic(; scaled=true), + linesearch=Optim.LineSearches.BackTracking(), + ), + Optim.Options(; iterations=100); + inplace=false, +) + +# And we can now compare with the optimal result we obtained earlier +θ_opt = unpack(opt.minimizer) +fz_opt, lf_opt = build_fz_lf(θ_opt) +post_svgp_hpopt = posterior( + SparseVariationalApproximation(Centered(), fz_opt, MvNormal(m, S)) +) + +pf = scatter(x, y; legend=false, xlabel="x", ylabel="f") +plot!(x_true, f_true; seriescolor="red", label="True latent") +plot!(x_true, post_svgp_opt(x_true); label="Posterior") +plot!(x_true, post_svgp_hpopt(x_true); label="Posterior (hp opt)") +py = scatter(x, y; label="Training data", xlabel="x", ylabel="y") +plot!(x_true, mean.(lik.(f_true)); label="True mean") +plot!(x_true, mean.(lik.(mean(post_svgp_opt(x_true)))); label="Posterior") +plot!(x_true, mean.(lik.(mean(post_svgp_hpopt(x_true)))); label="Posterior (hp opt)") +plot(pf, py) diff --git a/src/expected_loglik.jl b/src/expected_loglik.jl index c29ff3bb..dd96108d 100644 --- a/src/expected_loglik.jl +++ b/src/expected_loglik.jl @@ -72,10 +72,9 @@ function expected_loglik( # using a reparameterisation by change of variable # (see e.g. en.wikipedia.org/wiki/Gauss%E2%80%93Hermite_quadrature) xs, ws = gausshermite(gh.n_points) - # size(fs): (length(y), n_points) - fs = √2 * std.(q_f) .* xs' .+ mean.(q_f) + fs = sqrt(2) * std.(q_f) .* xs' .+ mean.(q_f) lls = loglikelihood.(lik.(fs), y) - return sum((1 / √π) * lls * ws) + return sum(lls * ws) / sqrt(π) end ChainRulesCore.@non_differentiable gausshermite(n)