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mdn_julia.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"id": "xVT8RtsWDmAC"
},
"source": [
"# Mixture Density Networks with Julia\n",
"\n",
"[This post is also available as a Jupyter notebook.](https://github.com/jklaise/personal_website/tree/master/notebooks/mdn_julia.ipynb)\n",
"\n",
"Related posts:\n",
"\n",
" - JavaScript [implementation](http://blog.otoro.net/2015/06/14/mixture-density-networks/).\n",
" - TensorFlow [implementation](http://blog.otoro.net/2015/11/24/mixture-density-networks-with-tensorflow/).\n",
" - PyTorch [implementation](https://github.com/hardmaru/pytorch_notebooks/blob/master/mixture_density_networks.ipynb)"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "gJRR0mR2DmAG"
},
"source": [
"This post is a result of me trying to understand how to do deep learning in Julia using the excellent [Flux](https://fluxml.ai/) package as well as getting a better understanding of conditional density estimation using a simple but effective technique—[Mixture Density Networks](https://publications.aston.ac.uk/id/eprint/373/1/NCRG_94_004.pdf) (Bishop, 1994). This post follows very closely the PyTorch implementation (including paraphrasing some statements) listed above, which itself was adapted from the original TensorFlow and JavaScript implementations."
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "JC9FUq63DmAI"
},
"source": [
"## Prelude: Why Julia?"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "CftXbow_DmAJ"
},
"source": [
"If we want to do generic training of models via backpropagation we need an automatic differentiation (AD) solution—nobody has time to write their gradients by hand, you just want to define your forward pass and let the AD handle the backward pass. In Python I might use one of the big frameworks like PyTorch or TensorFlow to do this (if I'm feeling adventurous then maybe [autograd](https://github.com/HIPS/autograd) or [jax](https://github.com/google/jax) at the cost of forgoing higher-level interfaces for neural network layers, optimizers etc.). The problem with these big frameworks is that if you have some very custom model operations on the forward pass, it might be impossible for the AD to take gradients. You may be able to write your own, but it might involve hacking the underlying framework (which is most likely C++ rather than Python) without guarantees of working. A concrete example is [backpropagating through an ODE solver](https://fluxml.ai/2019/03/05/dp-vs-rl.html) to do model based reinforcement learning.\n",
"\n",
"Julia has the promise of native [differentiable programming](https://fluxml.ai/2019/02/07/what-is-differentiable-programming.html) to allow just the kind of use cases. [Swift for Tensorflow](https://www.tensorflow.org/swift) is a similar endeavour, but I think Julia has the upper hand for several reasons: it has an extensive scientific package ecosystem, a flexible compiler enabling things like [source-to-source automatic differentiation](https://github.com/FluxML/Zygote.jl), and it's a solution to the [two language problem](https://thebottomline.as.ucsb.edu/2018/10/julia-a-solution-to-the-two-language-programming-problem) which can plague Python frameworks such as PyTorch and TensorFlow when it comes to writing very custom models."
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "eZPnNxlnDmAJ"
},
"source": [
"## Warm-up: 1-D regression\n",
"Let's warm up by doing some very simple data fitting in 1D by using a neural network with one hidden layer. We will try to fit the sinusoidal function\n",
"\n",
"$$\n",
"y(x)=7 \\sin( 0.75 x ) + 0.5 x + \\epsilon,\n",
"$$\n",
"\n",
"where\n",
"\n",
"- $7 \\sin( 0.75 x)$: a large periodic sine wave.\n",
"- $0.5 x$: add a slight upward slope.\n",
"- $\\epsilon$: add some standard Gaussian noise\n",
"\n",
"First we import the libraries we need."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"id": "IXTOs1vbDmAK"
},
"outputs": [],
"source": [
"# using Distributions\n",
"using Flux\n",
"using Plots\n",
"using Random\n",
"Random.seed!(12345); # for reproducibility"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "OXMmrtHhDmAN"
},
"source": [
"Next we define generate a sample dataset of points $(x, y(x))$ and visualize it:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"id": "kk42CtMSDmAN"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1×1000 Matrix{Float64}\n",
"1×1000 Matrix{Float64}\n"
]
}
],
"source": [
"# Flux expects batch dimension to be last\n",
"function generate_data(n_samples)\n",
" ϵ = randn(1, n_samples) # Normal(μ = 0, σ = 1)\n",
" x = -10.5 .+ ((10.5 + 10.5) .* rand(1, n_samples)) # Uniform(a = -10.5, b = 10.5)\n",
" y = 7sin.(0.75x) + 0.5x + ϵ\n",
" return x, y\n",
"end\n",
"\n",
"n_samples = 1000\n",
"x, y = generate_data(n_samples)\n",
"foreach(println ∘ summary, (x, y))"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"id": "o35-ja8pDmAO",
"outputId": "b6818202-8e77-413a-bdd2-c22256ca867d"
},
"outputs": [
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},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"scatter(x', y', alpha=0.2) # transpose for plotting"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "JCeEP7GYDmAQ"
},
"source": [
"We will now use this data to train a neural network with one hidden layer. The network can be described by the following equation:\n",
"\n",
"$$\n",
"\\hat{y}(x) = W_2\\tanh(W_1x + b_1) +b_2,\n",
"$$\n",
"\n",
"where\n",
"\n",
" - $W_1$, $W_2$ are the weights for the input and output layers\n",
" - $b_1$, $b_2$ are the biases for the input and output layers\n",
" - $\\tanh$ is a the hyperbolic tangent activation function\n",
" \n",
"Let's create this network using 20 hidden nodes. Note that both the input and output are 1-dimensional. We will use Flux's `Dense` layer constructs and the `Chain` model (equivalent to a Sequential layer in PyTorch/Tensorflow) to put these together:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"id": "IJ2__Es2DmAR"
},
"outputs": [],
"source": [
"n_input = 1\n",
"n_hidden = 20\n",
"n_output = 1\n",
"\n",
"model = Chain(\n",
" Dense(n_input, n_hidden, tanh),\n",
" Dense(n_hidden, n_output));"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "jzTAKHsADmAS"
},
"source": [
"To train the network we also need to define a loss function, we will use the standard mean squared error loss for regression models:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"id": "z-u4stDVDmAS"
},
"outputs": [],
"source": [
"loss(x, y) = Flux.mse(model(x), y);"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "G6ffa4eJDmAT"
},
"source": [
"We also need an optimizer which will try to minimize the loss function at training time. We will use [RMSprop](http://ruder.io/optimizing-gradient-descent/index.html#rmsprop), included in Flux:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"id": "xSK_Bs0ZDmAT"
},
"outputs": [],
"source": [
"opt = RMSProp(0.01, 0.99);"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "wINU1ypmDmAT"
},
"source": [
"We need to get the parameters from the model that are to be optimized, this can be done easily within Flux as the parameters (their values and gradients) are tracked when the model was created:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"id": "rEtOIgg-DmAU"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"20-element Vector{Float32}\n",
"20×1 Matrix{Float32}\n",
"1×20 Matrix{Float32}\n",
"1-element Vector{Float32}\n"
]
}
],
"source": [
"pars = Flux.params(model)\n",
"foreach(println ∘ summary, pars.params)"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "VwjOO0ZVDmAU"
},
"source": [
"Finally we define the number of epochs to train for as well as a simple callback function which will allow us to monitor the training loss:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"id": "Vcly3J1UDmAV"
},
"outputs": [],
"source": [
"n_epochs = 3000\n",
"evalcb() = @show(loss(x, y));"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"id": "HlPTt_ZIDmAV"
},
"outputs": [],
"source": [
"data = Iterators.repeated((x, y), n_epochs);"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "SCUDeENFDmAW"
},
"source": [
"We are now ready to train the network using Flux's high-level `train!` API:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"id": "21xQ9fwyDmAW",
"outputId": "99e2163c-22d7-40d3-9fbb-3459b838c817"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"loss(x, y) = 27.298287408520643\n",
"loss(x, y) = 1.1932880406359176\n",
"loss(x, y) = 1.1510033394549746\n"
]
}
],
"source": [
"# high-level\n",
"Flux.train!(loss, pars, data, opt, cb=Flux.throttle(evalcb, 1))"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "jTjiZ0NUDmAX"
},
"source": [
"Note that we created an iterator over the data, repeating it for the number of epochs we wanted to train. Alternatively, instead of using the iterator over the data and the callback, you can write the training loop more explicitly like this:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"id": "9a4kckEcDmAX",
"outputId": "417883a9-b9c6-4306-828c-6e35c6db75f5"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Epoch: 300 loss: 1.1150549085054622\n",
"Epoch: 600 loss: 1.1720569275068586\n",
"Epoch: 900 loss: 1.122003730736572\n",
"Epoch: 1200 loss: 1.0998033504048332\n",
"Epoch: 1500 loss: 1.1733619605717536\n",
"Epoch: 1800 loss: 1.107484668922506\n",
"Epoch: 2100 loss: 1.1120659458041775\n",
"Epoch: 2400 loss: 1.0867373767598607\n",
"Epoch: 2700 loss: 1.1369364727020979\n",
"Epoch: 3000 loss: 1.093817393281882\n"
]
}
],
"source": [
"# lower-level\n",
"data = [(x, y)]\n",
"\n",
"for epoch = 1:n_epochs\n",
" Flux.train!(loss, pars, data, opt)\n",
" if epoch % 300 ==0\n",
" l = loss(x, y)\n",
" println(\"Epoch: \", epoch, \" loss: \", l)\n",
" end\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "RxbeHPmoDmAY"
},
"source": [
"With Flux you can go down to the level of doing the gradient updates manually too, similar how you would do it in PyTorch, we will use this in the next section to update our custom loss function. At the time of writing Flux is [undergoing a change](https://github.com/FluxML/Flux.jl/pull/669) in its automatic differentiation backend which might result in some changes to the low-level API."
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "6cjlkaLVDmAZ"
},
"source": [
"Now that we have trained the model, we can evaluate its performance on some test data:"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"id": "nTi02_l1DmAZ",
"outputId": "f9828832-223a-41e5-fb71-ccd1faed9807"
},
"outputs": [
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},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x_test = range(-10, stop=10, length=n_samples)\n",
"y_test = model(transpose(x_test))\n",
"\n",
"scatter(transpose(x), transpose(y), alpha=0.2)\n",
"scatter!(collect(x_test), transpose(y_test), msc=nothing)"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "qCw_gLxwDmAa"
},
"source": [
"As we can see the network can fit this sinusoidal data quite well as expected. However, this type of fitting works well only if we are modelling a one-to-one or a many-to-one function. Suppose we inverted the training data so that the prediction task would become $x(y)$:"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"id": "2y_Hy1R0DmAa",
"outputId": "ef6347eb-d64b-4bbf-a444-7135fa33072e"
},
"outputs": [
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},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"scatter(transpose(y), transpose(x), alpha=0.2)"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "iI1AeG2NDmAa"
},
"source": [
"Let's use the same network to try and with this inverted data:"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"id": "vmMjfa8nDmAa",
"outputId": "915b063c-1d56-4a25-e076-4c73ee201391"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"loss(x, y) = 2.6277055209793185\n",
"loss(x, y) = 34.376074514701926\n",
"loss(x, y) = NaN\n"
]
}
],
"source": [
"n_epochs = 3000\n",
"data = Iterators.repeated((y,x), n_epochs)\n",
"Flux.train!(loss, pars, data, opt, cb=Flux.throttle(evalcb, 1))"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"id": "nZpU3a6tDmAb",
"outputId": "477a9749-1239-4804-c1b0-4a2cade7615d"
},
"outputs": [
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" \"/>\n",
"<polyline clip-path=\"url(#clip862)\" style=\"stroke:#000000; stroke-linecap:butt; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n",
" 178.867,766.462 2352.76,766.462 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip862)\" style=\"stroke:#000000; stroke-linecap:butt; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n",
" 178.867,442.328 2352.76,442.328 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip862)\" style=\"stroke:#000000; stroke-linecap:butt; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n",
" 178.867,118.193 2352.76,118.193 \n",
" \"/>\n",
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