Dropout

Let's think briefly about what we expect from a good predictive model. We want it to peform well on unseen data. Classical generalization theory suggests that to close the gap between train and test performance, we should aim for a simple model. Simplicity can come in the form of a small number of dimensions. We explored this when discussing the monomial basis functions of linear models in :numref:sec_generalization_basics. Additionally, as we saw when discussing weight decay (${\ell }_2$ regularization) in :numref:sec_weight_decay, the (inverse) norm of the parameters also represents a useful measure of simplicity. Another useful notion of simplicity is smoothness, i.e., that the function should not be sensitive to small changes to its inputs. For instance, when we classify images, we would expect that adding some random noise to the pixels should be mostly harmless.

Bishop [71] formalized this idea when he proved that training with input noise is equivalent to Tikhonov regularization. This work drew a clear mathematical connection between the requirement that a function be smooth (and thus simple), and the requirement that it be resilient to perturbations in the input.

Then, Srivastava et al. [70] developed a clever idea for how to apply Bishop's idea to the internal layers of a network, too. Their idea, called dropout, involves injecting noise while computing each internal layer during forward propagation, and it has become a standard technique for training neural networks. The method is called dropout because we literally drop out some neurons during training. Throughout training, on each iteration, standard dropout consists of zeroing out some fraction of the nodes in each layer before calculating the subsequent layer.

To be clear, we are imposing our own narrative with the link to Bishop. The original paper on dropout offers intuition through a surprising analogy to sexual reproduction. The authors argue that neural network overfitting is characterized by a state in which each layer relies on a specific pattern of activations in the previous layer, calling this condition co-adaptation. Dropout, they claim, breaks up co-adaptation just as sexual reproduction is argued to break up co-adapted genes. While such an justification of this theory is certainly up for debate, the dropout technique itself has proved enduring, and various forms of dropout are implemented in most deep learning libraries.

The key challenge is how to inject this noise. One idea is to inject it in an unbiased manner so that the expected value of each layer–-while fixing the others–-equals the value it would have taken absent noise. In Bishop's work, he added Gaussian noise to the inputs to a linear model. At each training iteration, he added noise sampled from a distribution with mean zero $ϵ\sim \mathcal{N}(0,{\sigma }^2)$ to the input $\mathbf{x}$, yielding a perturbed point ${\mathbf{x}}^{\prime }=\mathbf{x}+ϵ$. In expectation, $E[{\mathbf{x}}^{\prime }]=\mathbf{x}$.

In standard dropout regularization, one zeros out some fraction of the nodes in each layer and then debiases each layer by normalizing by the fraction of nodes that were retained (not dropped out). In other words, with dropout probability $p$, each intermediate activation $h$ is replaced by a random variable $h^{\prime }$ as follows:

$

\begin{aligned} h' = \begin{cases} 0 & \textrm{ with probability } p
\frac{h}{1-p} & \textrm{ otherwise} \end{cases} \end{aligned} $

By design, the expectation remains unchanged, i.e., $E[h^{\prime }]=h$.

using Pkg
Pkg.activate("../../d2lai")
using d2lai, Flux, Distributions

  Activating project at `/workspace/workspace/d2l-julia/d2lai`

Dropout in Practice

Recall the MLP with a hidden layer and five hidden units from Figure. When we apply dropout to a hidden layer, zeroing out each hidden unit with probability $p$, the result can be viewed as a network containing only a subset of the original neurons. In Figure, $h_2$ and $h_5$ are removed. Consequently, the calculation of the outputs no longer depends on $h_2$ or $h_5$ and their respective gradient also vanishes when performing backpropagation. In this way, the calculation of the output layer cannot be overly dependent on any one element of $h_1,\dots ,h_5$.

MLP before and after dropout.

Typically, we disable dropout at test time. Given a trained model and a new example, we do not drop out any nodes and thus do not need to normalize. However, there are some exceptions: some researchers use dropout at test time as a heuristic for estimating the uncertainty of neural network predictions: if the predictions agree across many different dropout outputs, then we might say that the network is more confident.

Implementation from Scratch

To implement the dropout function for a single layer, we must draw as many samples from a Bernoulli (binary) random variable as our layer has dimensions, where the random variable takes value $1$ (keep) with probability $1-p$ and $0$ (drop) with probability $p$. One easy way to implement this is to first draw samples from the uniform distribution $U[0,1]$. Then we can keep those nodes for which the corresponding sample is greater than $p$, dropping the rest.

In the following code, we (implement a dropout_layer function that drops out the elements in the tensor input X with probability dropout), rescaling the remainder as described above: dividing the survivors by 1.0-dropout.

function dropout_layer(X::AbstractArray, dropout)
    probs = rand(Bernoulli(dropout), size(X, 1))
    return probs .* X
end

dropout_layer (generic function with 1 method)

We can test out the dropout_layer function on a few examples. In the following lines of code, we pass our input X through the dropout operation, with probabilities 0, 0.5, and 1, respectively.

X = reshape(1:16, 2, 8)
println("dropout with p = 0", dropout(X, 0.))
println("dropout with p = 0.5", dropout(X, 0.5))
println("dropout with p = 1.0", dropout(X, 1.))

dropout with p = 0[1.0 3.0 5.0 7.0 9.0 11.0 13.0 15.0; 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0]
dropout with p = 0.5[2.0 0.0 0.0 14.0 18.0 0.0 0.0 0.0; 0.0 8.0 12.0 0.0 0.0 24.0 28.0 32.0]
dropout with p = 1.0[0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0]

Defining the Model

The model below applies dropout to the output of each hidden layer (following the activation function). We can set dropout probabilities for each layer separately. A common choice is to set a lower dropout probability closer to the input layer. We ensure that dropout is only active during training.

mutable struct DropoutScratchMLP{N, A} <: AbstractClassifier
    net::N 
    args::A 
    train::Bool
end
Flux.@layer DropoutScratchMLP trainable=(net,)
function DropoutScratchMLP(; args...)
    net = Chain(Dense(args[:num_inputs], args[:num_hidden_1]), Dense(args[:num_hidden_1], args[:num_hidden_2]), Dense(args[:num_hidden_2], args[:num_outputs]), Flux.softmax)
    DropoutScratchMLP(net, NamedTuple(args), true)
end

function d2lai.forward(mlp::DropoutScratchMLP, x)
    lin1, lin2, lin3, softmax = mlp.net.layers
    h1 = model.train ? dropout_layer(lin1(x), mlp.args.dropout_1) : lin1(x)
    h2 = model.train ? dropout_layer(lin2(h1), mlp.args.dropout_2) : lin2(h1)
    h3 = lin3(h2)
    return softmax(h3)
end

Training

The following is similar to the training of MLPs described previously.

hparams = (num_inputs = 28*28, num_outputs = 10, num_hidden_1 = 256, num_hidden_2 = 256,
           dropout_1 = 0.5, dropout_2 = 0.5, lr = 0.1)
model = DropoutScratchMLP(; hparams...)

opt = Descent(0.1)
data = d2lai.FashionMNISTData(; batchsize = 256, flatten = true)
trainer = Trainer(model, data, opt; max_epochs = 10)
d2lai.fit(trainer)