Edward is a probabilistic programming language like Stan, PyMC3 and Turing.jl. You write a model out and can perform statistical inference on some data. Edward is a more ‘machine learning’ focused than Stan as you can build neural nets and all the fun, modern techniques that the cool kids are using. Flux.jl takes a similar ‘machine learning’ approach but this time in Julia. Flux can build neural nets and also much more. There is an intersection here between Edward and Flux, so it makes sense to think that you can perform similar tasks. That is the whole point of this blog post, translating this Edward tutorial: http://edwardlib.org/getting-started into Julia code for Flux. So anyone reading this can get an idea of:

  • what Flux is all about,
  • how the Python Edward code can be transformed into Julia,
  • how to build a simple neural nets and infer the parameters in a Bayesian way.

I’ll start off by reproducing the model in Flux before calling on Turing to do the Bayes bit. This is my first time writing about Flux, so hopefully you’ll find this useful if it is also your first foray into doing some machine learning in Julia.

using Flux
using Plots
using Distributions

The original tutorial wants to learn the cos function of which we add some Gaussian noise.

f(x) = cos(x) + rand(Normal(0, 0.1))

xTrain = collect(-3:0.1:3)
yTrain = f.(xTrain)
plot(xTrain, yTrain, seriestype=:scatter, label="Train Data")
plot!(xTrain, cos.(xTrain), label="Truth")

svg

To build the neural net we chain two dense layers with tanh nonlinearities inbetween the layers. We use the mean square error as our loss function which we minimise using gradient descent. This is the most basic way in which you can build and train the neural net. Directly replicating what Edward does comes later, for now we just want to get things working.

With Flux you can specify what type of loss quite easily, with a list of all that comes prepackaged found here https://fluxml.ai/Flux.jl/stable/models/losses. But if none of those take your fancy you can write your own and pass a simple Julia function.

model = Chain(Dense(1, 2, tanh), Dense(2, 1))
loss(x, y) = Flux.Losses.mse(model(x), y)
optimiser = Descent(0.1);

Likewise for the optimiser. I’ve chose gradient descent, but again, you can switch it out for a more advanced approach, such as ADAM. See if any of those listed here https://fluxml.ai/Flux.jl/stable/training/optimisers/ take your fancy.

In short, the neural net takes one value as an input, passes through to two nodes on the hidden layer with a tanh activation function before outputting a single value as the output.

We now set up the training data. We take 100 random normal samples as our inputs, use our f function to generate the outputs and then repeat the dataset 100 times.

x = rand(Normal(), 100)
y = f.(x)
train_data = Iterators.repeated((Array(x'), Array(y')), 100);

We train the model for 10 epochs.

Flux.@epochs 10 Flux.train!(loss, Flux.params(model), train_data, optimiser)

Which takes no time at all to complete. We now want to make sure the results are sensible and it has actually learnt the underlying function.

yOut = zeros(length(xTrain))
for (i, x) in enumerate(xTrain)
    yOut[i] = model([xTrain[i]])[1]
end

plot(xTrain, yOut, label="Predicted")
plot!(xTrain, cos.(xTrain), label="True")
plot!(x, y, seriestype=:scatter, label="Data")

svg

Everything looking good. Our neural net has found something similar to the true function, so hopefully you are convinced that we can now add a layer of complexity. Slight disagreement around the tails, but that is where we are lacking some data, so not too big of a deal.

The Same - But Bayes

All the above captured the spirit, but not the whole point of the Edward tutorial, which is to create a Bayesian neural net. With a Bayesian neural net there is a probability distribution over the weights, rather than just singular values to maximise. In Julia, we have to call upon our old friend Turing.jl. The Turing team have written about Bayesian nets here. I’ve taken some of that code and shuffled it about for this application.

using Turing

Firstly, have the parameters for each of the layers of the neural net. In total there are 6 parameters for the two layer model, so using the unpack function we take a vector of length 6 and break it up into the weights and biases for each layer which we then build in the nn_forward function.

function unpack(nn_params::AbstractVector)
    W₁ = reshape(nn_params[1:2], 2, 1);   
    b₁ = reshape(nn_params[3:4], 2)
    
    W₂ = reshape(nn_params[4:5], 1, 2); 
    b₂ = [nn_params[6]]
    
    return W₁, b₁, W₂, b₂
end

function nn_forward(xs, nn_params::AbstractVector)
    W₁, b₁, W₂, b₂ = unpack(nn_params)
    nn = Chain(Dense(W₁, b₁, tanh), Dense(W₂, b₂))
    return nn(xs)
end

These utility functions mean that we are tearing down the neural net and rebuilding it with the new parameters with each iteration. Turing deals with finding the best parameters by doing the Bayesian sampling (see my previous post on Hamilton MCMC sampling).

We think the 6 parameters of the neural net are drawn from a multivariate normal distribution and the parameters are all uncorrelated with each other. The observations are also from a normal distribution with mean from the neural net and variance \(\sigma\).

alpha = 0.1
sig = sqrt(1.0 / alpha)

@model bayes_nn(xs, ys) = begin
    
    nn_params ~ MvNormal(zeros(6), sig .* ones(6)) #Prior
    
    preds = nn_forward(xs, nn_params) #Build the net
    sigma ~ Gamma(0.01, 1/0.01) # Prior for the variance
    for i = 1:length(ys)
        ys[i] ~ Normal(preds[i], sigma)
    end
end;

Model built, we now want to use the NUTS algorithm to sample from the posterior. Again, like Flux, you can use other sampling methods such as HMC or Metropolis-Hastings.

N = 5000
ch1 = sample(bayes_nn(hcat(x...), y), NUTS(0.65), N);
ch2 = sample(bayes_nn(hcat(x...), y), NUTS(0.65), N);

We sample from posterior 5,000 times with two chains to check the convergence later. When selecting the final parameters we chose those that maximise the log posterior of the model. For each of the datapoints we add the prediction from the neural net.

lp, maxInd = findmax(ch1[:lp])

params, _ = ch1.name_map
bestParams = map(x-> ch1[x].data[maxInd], params[1:6])
plot(x, cos.(x), seriestype=:line, label="True")
plot!(x, Array(nn_forward(hcat(x...), bestParams)'), 
      seriestype=:scatter, label="MAP Estimate")

svg

We can also sample from this posterior distribution by looking at the different parameters from the sampling process.

xPlot = sort(x)

sp = plot()

for i in max(1, (maxInd[1]-100)):min(N, (maxInd[1]+100))
    paramSample = map(x-> ch1[x].data[i], params)
    plot!(sp, xPlot, Array(nn_forward(hcat(xPlot...), paramSample)'), 
        label=:none, colour="blue")
    
end

plot!(sp, x, y, seriestype=:scatter, label="Training Data", colour="red")

sp

svg

Again, looking sensible and representing the true underlying function. Plus as we have done this using Bayes, we have a good visualisation of the uncertainty of the model too.

And then finally we can assess how the chains of the model look and make a judgement on whether they have converged.

lPlot = plot(ch1[:lp], label="Chain 1", title="Log Posterior")
plot!(lPlot, ch2[:lp], label="Chain 2")

sigPlot = plot(ch1[:sigma], label= "Chain 1", title="Variance")
plot!(sigPlot, ch2[:sigma], label="Chain 2")

plot(lPlot, sigPlot)

svg

Both chains are looking like they have converged, so we can trust our sampling process.

Conclusion

I actually surprised myself with how easy it was to build and sample from the neural net in a Bayesian manner. I originally thought that I would only be able to replicate the model using frequentist methods, but not the Bayes steps. This just goes to show what a great job the Turing.jl team have done, making the whole model building and sampling simple. So after reading this you can extended these models to bigger and badder neural nets if that is what your problem needs. Easily swapping out losses, optimisers and posterior sampling methods as needed.

This post also shows how easy it is to translate back and forth between the different machine learning libraries and even programming languages, although there is not too much of a difference between Julia and Python. So if anyone out there is reading this trying to decide on what language to pick up, I’d be biased and say Julia, but really, you’ll be fine with either one.