Bayesian Neural Network

We borrow this tutorial from the official Turing Docs. We will show how the explicit parameterization of Lux enables first-class composability with packages which expect flattened out parameter vectors.

Note: The tutorial in the official Turing docs is now using Lux instead of Flux.

We will use Turing.jl with Lux.jl to implement implementing a classification algorithm. Lets start by importing the relevant libraries.

# Import libraries

using Lux, Turing, CairoMakie, Random, Tracker, Functors, LinearAlgebra

# Sampling progress
Turing.setprogress!(true);

[ Info: [Turing]: progress logging is enabled globally
[ Info: [AdvancedVI]: global PROGRESS is set as true

Generating data

Our goal here is to use a Bayesian neural network to classify points in an artificial dataset. The code below generates data points arranged in a box-like pattern and displays a graph of the dataset we'll be working with.

# Number of points to generate
N = 80
M = round(Int, N / 4)
rng = Random.default_rng()
Random.seed!(rng, 1234)

# Generate artificial data
x1s = rand(rng, Float32, M) * 4.5f0;
x2s = rand(rng, Float32, M) * 4.5f0;
xt1s = Array([[x1s[i] + 0.5f0; x2s[i] + 0.5f0] for i in 1:M])
x1s = rand(rng, Float32, M) * 4.5f0;
x2s = rand(rng, Float32, M) * 4.5f0;
append!(xt1s, Array([[x1s[i] - 5.0f0; x2s[i] - 5.0f0] for i in 1:M]))

x1s = rand(rng, Float32, M) * 4.5f0;
x2s = rand(rng, Float32, M) * 4.5f0;
xt0s = Array([[x1s[i] + 0.5f0; x2s[i] - 5.0f0] for i in 1:M])
x1s = rand(rng, Float32, M) * 4.5f0;
x2s = rand(rng, Float32, M) * 4.5f0;
append!(xt0s, Array([[x1s[i] - 5.0f0; x2s[i] + 0.5f0] for i in 1:M]))

# Store all the data for later
xs = [xt1s; xt0s]
ts = [ones(2 * M); zeros(2 * M)]

# Plot data points

function plot_data()
    x1 = first.(xt1s)
    y1 = last.(xt1s)
    x2 = first.(xt0s)
    y2 = last.(xt0s)

    fig = Figure()
    ax = CairoMakie.Axis(fig[1, 1]; xlabel="x", ylabel="y")

    scatter!(ax, x1, y1; markersize=16, color=:red, strokecolor=:black, strokewidth=2)
    scatter!(ax, x2, y2; markersize=16, color=:blue, strokecolor=:black, strokewidth=2)

    return fig
end

plot_data()

Building the Neural Network

The next step is to define a feedforward neural network where we express our parameters as distributions, and not single points as with traditional neural networks. For this we will use Dense to define liner layers and compose them via Chain, both are neural network primitives from Lux. The network nn we will create will have two hidden layers with tanh activations and one output layer with sigmoid activation, as shown below.

The nn is an instance that acts as a function and can take data, parameters and current state as inputs and output predictions. We will define distributions on the neural network parameters.

# Construct a neural network using Lux
nn = Chain(Dense(2 => 3, tanh), Dense(3 => 2, tanh), Dense(2 => 1, sigmoid))

# Initialize the model weights and state
ps, st = Lux.setup(rng, nn)

Lux.parameterlength(nn) # number of parameters in NN

20

The probabilistic model specification below creates a parameters variable, which has IID normal variables. The parameters represents all parameters of our neural net (weights and biases).

# Create a regularization term and a Gaussian prior variance term.
alpha = 0.09
sig = sqrt(1.0 / alpha)

3.3333333333333335

Construct named tuple from a sampled parameter vector. We could also use ComponentArrays here and simply broadcast to avoid doing this. But let's do it this way to avoid dependencies.

function vector_to_parameters(ps_new::AbstractVector, ps::NamedTuple)
    @assert length(ps_new) == Lux.parameterlength(ps)
    i = 1
    function get_ps(x)
        z = reshape(view(ps_new, i:(i + length(x) - 1)), size(x))
        i += length(x)
        return z
    end
    return fmap(get_ps, ps)
end

vector_to_parameters (generic function with 1 method)

To interface with external libraries it is often desirable to use the StatefulLuxLayer to automatically handle the neural network states.

const model = StatefulLuxLayer{true}(nn, nothing, st)

# Specify the probabilistic model.
@model function bayes_nn(xs, ts)
    # Sample the parameters
    nparameters = Lux.parameterlength(nn)
    parameters ~ MvNormal(zeros(nparameters), Diagonal(abs2.(sig .* ones(nparameters))))

    # Forward NN to make predictions
    preds = Lux.apply(model, xs, vector_to_parameters(parameters, ps))

    # Observe each prediction.
    for i in eachindex(ts)
        ts[i] ~ Bernoulli(preds[i])
    end
end

bayes_nn (generic function with 2 methods)

Inference can now be performed by calling sample. We use the HMC sampler here.

# Perform inference.
N = 5000
ch = sample(bayes_nn(reduce(hcat, xs), ts), HMC(0.05, 4; adtype=AutoTracker()), N)

Chains MCMC chain (5000×30×1 Array{Float64, 3}):

Iterations        = 1:1:5000
Number of chains  = 1
Samples per chain = 5000
Wall duration     = 21.64 seconds
Compute duration  = 21.64 seconds
parameters        = parameters[1], parameters[2], parameters[3], parameters[4], parameters[5], parameters[6], parameters[7], parameters[8], parameters[9], parameters[10], parameters[11], parameters[12], parameters[13], parameters[14], parameters[15], parameters[16], parameters[17], parameters[18], parameters[19], parameters[20]
internals         = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, numerical_error, step_size, nom_step_size

Summary Statistics
      parameters      mean       std      mcse   ess_bulk   ess_tail      rhat   ess_per_sec
          Symbol   Float64   Float64   Float64    Float64    Float64   Float64       Float64

   parameters[1]    2.0965    4.4644    1.3270    12.0887    23.0787    1.3335        0.5586
   parameters[2]    0.0843    0.3952    0.0510    79.9413    35.2287    1.0240        3.6940
   parameters[3]    4.9916    1.9230    0.4456    19.3935    81.5852    1.2170        0.8961
   parameters[4]    0.3356    2.3096    0.6117    14.3653    24.1405    1.2067        0.6638
   parameters[5]    5.0569    2.3192    0.6141    15.0672    34.9402    1.1058        0.6962
   parameters[6]    0.7127    1.2191    0.2991    21.7200    21.7891    1.1550        1.0037
   parameters[7]    1.7149    3.8638    1.1143    12.4420    21.7560    1.1439        0.5749
   parameters[8]    0.3690    1.2835    0.2555    26.1914    29.6846    1.0246        1.2103
   parameters[9]   -0.4968    2.2271    0.6133    14.1406    24.5983    1.2648        0.6534
  parameters[10]    0.0842    2.1828    0.5840    14.0865    21.8689    1.1832        0.6509
  parameters[11]   -1.0288    1.5663    0.3628    18.3294    27.7789    1.0536        0.8470
  parameters[12]   -4.1763    1.7426    0.3705    23.0782    28.1565    1.0633        1.0664
  parameters[13]    3.1846    1.4791    0.3401    19.4472    53.5564    1.0444        0.8986
  parameters[14]    2.7199    1.9547    0.5178    14.3490    46.8984    1.3048        0.6630
  parameters[15]   -2.0613    1.4937    0.3727    16.3035    39.2852    1.0885        0.7534
  parameters[16]   -2.9853    1.4059    0.2557    31.3669    31.3956    1.0012        1.4494
  parameters[17]   -2.4061    2.6897    0.7370    15.6752    20.0204    1.0733        0.7243
  parameters[18]   -5.3040    1.1943    0.1791    44.9414    68.7121    1.0867        2.0767
  parameters[19]   -5.1706    2.2709    0.5991    17.6788    18.3886    1.0540        0.8169
  parameters[20]   -5.1303    1.3008    0.2366    30.4880    60.9517    1.0163        1.4088

Quantiles
      parameters      2.5%     25.0%     50.0%     75.0%     97.5%
          Symbol   Float64   Float64   Float64   Float64   Float64

   parameters[1]   -7.7024    0.5794    2.9500    5.0242    8.8277
   parameters[2]   -0.6189   -0.1406    0.0435    0.2877    1.0813
   parameters[3]    1.2261    3.6852    4.9109    6.5386    8.2717
   parameters[4]   -3.9045   -0.9307    0.1458    1.1265    5.7834
   parameters[5]    1.5809    3.2130    4.8534    6.4543   10.0149
   parameters[6]   -0.8100   -0.0584    0.3864    1.1121    4.4225
   parameters[7]   -5.8787   -0.8376    1.2050    4.7533    8.9694
   parameters[8]   -2.3155   -0.3960    0.2950    1.1550    2.9515
   parameters[9]   -5.8990   -1.7942   -0.1511    1.0876    2.9460
  parameters[10]   -4.7703   -1.0568    0.0534    1.4673    4.2833
  parameters[11]   -4.5806   -1.9640   -0.8771    0.0209    1.7384
  parameters[12]   -8.2243   -5.3092   -4.0546   -2.7977   -1.3694
  parameters[13]    0.8904    2.0618    2.9913    4.0971    6.5956
  parameters[14]   -0.9166    1.5821    2.5812    3.9071    6.6712
  parameters[15]   -4.9708   -3.1310   -2.0122   -1.0256    0.7273
  parameters[16]   -6.2647   -3.8094   -2.8050   -1.8812   -0.8946
  parameters[17]   -6.2695   -3.8672   -2.6943   -1.7352    5.5834
  parameters[18]   -7.6970   -6.0973   -5.2714   -4.5053   -2.9747
  parameters[19]   -8.3524   -6.4017   -5.5673   -4.6134    2.1958
  parameters[20]   -7.6214   -5.9847   -5.1652   -4.2672   -2.6349

Now we extract the parameter samples from the sampled chain as θ (this is of size 5000 x 20 where 5000 is the number of iterations and 20 is the number of parameters). We'll use these primarily to determine how good our model's classifier is.

# Extract all weight and bias parameters.
θ = MCMCChains.group(ch, :parameters).value;

Prediction Visualization

# A helper to run the nn through data `x` using parameters `θ`
nn_forward(x, θ) = model(x, vector_to_parameters(θ, ps))

# Plot the data we have.
fig = plot_data()

# Find the index that provided the highest log posterior in the chain.
_, i = findmax(ch[:lp])

# Extract the max row value from i.
i = i.I[1]

# Plot the posterior distribution with a contour plot
x1_range = collect(range(-6; stop=6, length=25))
x2_range = collect(range(-6; stop=6, length=25))
Z = [nn_forward([x1, x2], θ[i, :])[1] for x1 in x1_range, x2 in x2_range]
contour!(x1_range, x2_range, Z; linewidth=3, colormap=:seaborn_bright)
fig

The contour plot above shows that the MAP method is not too bad at classifying our data. Now we can visualize our predictions.

$$p(\tilde{x}|X,\alpha )={\int }_{\theta }p(\tilde{x}|\theta )p(\theta |X,\alpha )\approx \sum _{\theta \sim p(\theta |X,\alpha )}{f}_{\theta }(\tilde{x})$$

The nn_predict function takes the average predicted value from a network parameterized by weights drawn from the MCMC chain.

# Return the average predicted value across multiple weights.
nn_predict(x, θ, num) = mean([first(nn_forward(x, view(θ, i, :))) for i in 1:10:num])

nn_predict (generic function with 1 method)

Next, we use the nn_predict function to predict the value at a sample of points where the x1 and x2 coordinates range between -6 and 6. As we can see below, we still have a satisfactory fit to our data, and more importantly, we can also see where the neural network is uncertain about its predictions much easier–-those regions between cluster boundaries.

Plot the average prediction.

fig = plot_data()

n_end = 1500
x1_range = collect(range(-6; stop=6, length=25))
x2_range = collect(range(-6; stop=6, length=25))
Z = [nn_predict([x1, x2], θ, n_end)[1] for x1 in x1_range, x2 in x2_range]
contour!(x1_range, x2_range, Z; linewidth=3, colormap=:seaborn_bright)
fig

Suppose we are interested in how the predictive power of our Bayesian neural network evolved between samples. In that case, the following graph displays an animation of the contour plot generated from the network weights in samples 1 to 5,000.

fig = plot_data()
Z = [first(nn_forward([x1, x2], θ[1, :])) for x1 in x1_range, x2 in x2_range]
c = contour!(x1_range, x2_range, Z; linewidth=3, colormap=:seaborn_bright)
record(fig, "results.gif", 1:250:size(θ, 1)) do i
    fig.current_axis[].title = "Iteration: $i"
    Z = [first(nn_forward([x1, x2], θ[i, :])) for x1 in x1_range, x2 in x2_range]
    c[3] = Z
    return fig
end

"results.gif"

Appendix

using InteractiveUtils
InteractiveUtils.versioninfo()

if @isdefined(MLDataDevices)
    if @isdefined(CUDA) && MLDataDevices.functional(CUDADevice)
        println()
        CUDA.versioninfo()
    end

    if @isdefined(AMDGPU) && MLDataDevices.functional(AMDGPUDevice)
        println()
        AMDGPU.versioninfo()
    end
end

Julia Version 1.10.6
Commit 67dffc4a8ae (2024-10-28 12:23 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 16 default, 0 interactive, 8 GC (on 16 virtual cores)
Environment:
  JULIA_CPU_THREADS = 16
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/01872db4-8c79-43af-ab7d-12abac4f24f6
  JULIA_PKG_SERVER = 
  JULIA_NUM_THREADS = 16
  JULIA_CUDA_HARD_MEMORY_LIMIT = 100%
  JULIA_PKG_PRECOMPILE_AUTO = 0
  JULIA_DEBUG = Literate

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