# 5.6. Dropout¶ Open the notebook in Colab Open the notebook in Colab Open the notebook in Colab Open the notebook in Colab Open the notebook in SageMaker Studio Lab

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 Section 3.6. Additionally, as we saw when discussing weight decay ($$\ell_2$$ regularization) in Section 3.7, 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 (1995) 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. (2014) 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 $$\epsilon \sim \mathcal{N}(0,\sigma^2)$$ to the input $$\mathbf{x}$$, yielding a perturbed point $$\mathbf{x}' = \mathbf{x} + \epsilon$$. In expectation, $$E[\mathbf{x}'] = \mathbf{x}$$.

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 $$h'$$ as follows:

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

By design, the expectation remains unchanged, i.e., $$E[h'] = h$$.

import torch
from torch import nn
from d2l import torch as d2l

from mxnet import autograd, gluon, init, np, npx
from mxnet.gluon import nn
from d2l import mxnet as d2l

npx.set_np()

from functools import partial
import jax
import optax
from flax import linen as nn
from jax import numpy as jnp
from d2l import jax as d2l

No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)

import tensorflow as tf
from d2l import tensorflow as d2l


## 5.6.1. Dropout in Practice¶

Recall the MLP with a hidden layer and five hidden units from Fig. 5.1.1. 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 Fig. 5.6.1, $$h_2$$ 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 $$h_1, \ldots, h_5$$. Fig. 5.6.1 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.

## 5.6.2. 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.

def dropout_layer(X, dropout):
assert 0 <= dropout <= 1
return mask * X / (1.0 - dropout)

def dropout_layer(X, dropout):
assert 0 <= dropout <= 1
if dropout == 1: return np.zeros_like(X)
mask = np.random.uniform(0, 1, X.shape) > dropout
return mask.astype(np.float32) * X / (1.0 - dropout)

def dropout_layer(X, dropout, key=d2l.get_key()):
assert 0 <= dropout <= 1
if dropout == 1: return jnp.zeros_like(X)
mask = jax.random.uniform(key, X.shape) > dropout
return jnp.asarray(mask, dtype=jnp.float32) * X / (1.0 - dropout)

def dropout_layer(X, dropout):
assert 0 <= dropout <= 1
if dropout == 1: return tf.zeros_like(X)
shape=tf.shape(X), minval=0, maxval=1) < 1 - dropout
return tf.cast(mask, dtype=tf.float32) * X / (1.0 - dropout)


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 = torch.arange(16, dtype = torch.float32).reshape((2, 8))
print('dropout_p = 0:', dropout_layer(X, 0))
print('dropout_p = 0.5:', dropout_layer(X, 0.5))
print('dropout_p = 1:', dropout_layer(X, 1))

dropout_p = 0: tensor([[ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.],
[ 8.,  9., 10., 11., 12., 13., 14., 15.]])
dropout_p = 0.5: tensor([[ 0.,  2.,  0.,  6.,  8.,  0.,  0.,  0.],
[16., 18., 20., 22., 24., 26., 28., 30.]])
dropout_p = 1: tensor([[0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0.]])

X = np.arange(16).reshape(2, 8)
print('dropout_p = 0:', dropout_layer(X, 0))
print('dropout_p = 0.5:', dropout_layer(X, 0.5))
print('dropout_p = 1:', dropout_layer(X, 1))

dropout_p = 0: [[ 0.  1.  2.  3.  4.  5.  6.  7.]
[ 8.  9. 10. 11. 12. 13. 14. 15.]]
dropout_p = 0.5: [[ 0.  0.  0.  0.  8. 10. 12.  0.]
[16.  0. 20. 22.  0.  0.  0. 30.]]
dropout_p = 1: [[0. 0. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0.]]
[21:50:21] ../src/storage/storage.cc:196: Using Pooled (Naive) StorageManager for CPU

X = jnp.arange(16, dtype=jnp.float32).reshape(2, 8)
print('dropout_p = 0:', dropout_layer(X, 0))
print('dropout_p = 0.5:', dropout_layer(X, 0.5))
print('dropout_p = 1:', dropout_layer(X, 1))

dropout_p = 0: [[ 0.  1.  2.  3.  4.  5.  6.  7.]
[ 8.  9. 10. 11. 12. 13. 14. 15.]]
dropout_p = 0.5: [[ 0.  0.  0.  6.  0.  0. 12.  0.]
[ 0. 18. 20. 22.  0.  0. 28.  0.]]
dropout_p = 1: [[0. 0. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0.]]

X = tf.reshape(tf.range(16, dtype=tf.float32), (2, 8))
print('dropout_p = 0:', dropout_layer(X, 0))
print('dropout_p = 0.5:', dropout_layer(X, 0.5))
print('dropout_p = 1:', dropout_layer(X, 1))

dropout_p = 0: tf.Tensor(
[[ 0.  1.  2.  3.  4.  5.  6.  7.]
[ 8.  9. 10. 11. 12. 13. 14. 15.]], shape=(2, 8), dtype=float32)
dropout_p = 0.5: tf.Tensor(
[[ 0.  0.  0.  6.  0.  0.  0. 14.]
[ 0. 18. 20. 22.  0.  0. 28. 30.]], shape=(2, 8), dtype=float32)
dropout_p = 1: tf.Tensor(
[[0. 0. 0. 0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0. 0. 0. 0.]], shape=(2, 8), dtype=float32)


### 5.6.2.1. 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.

class DropoutMLPScratch(d2l.Classifier):
def __init__(self, num_outputs, num_hiddens_1, num_hiddens_2,
dropout_1, dropout_2, lr):
super().__init__()
self.save_hyperparameters()
self.lin1 = nn.LazyLinear(num_hiddens_1)
self.lin2 = nn.LazyLinear(num_hiddens_2)
self.lin3 = nn.LazyLinear(num_outputs)
self.relu = nn.ReLU()

def forward(self, X):
H1 = self.relu(self.lin1(X.reshape((X.shape, -1))))
if self.training:
H1 = dropout_layer(H1, self.dropout_1)
H2 = self.relu(self.lin2(H1))
if self.training:
H2 = dropout_layer(H2, self.dropout_2)
return self.lin3(H2)

class DropoutMLPScratch(d2l.Classifier):
def __init__(self, num_outputs, num_hiddens_1, num_hiddens_2,
dropout_1, dropout_2, lr):
super().__init__()
self.save_hyperparameters()
self.lin1 = nn.Dense(num_hiddens_1, activation='relu')
self.lin2 = nn.Dense(num_hiddens_2, activation='relu')
self.lin3 = nn.Dense(num_outputs)
self.initialize()

def forward(self, X):
H1 = self.lin1(X)
H1 = dropout_layer(H1, self.dropout_1)
H2 = self.lin2(H1)
H2 = dropout_layer(H2, self.dropout_2)
return self.lin3(H2)

class DropoutMLPScratch(d2l.Classifier):
num_hiddens_1: int
num_hiddens_2: int
num_outputs: int
dropout_1: float
dropout_2: float
lr: float
training: bool = True

def setup(self):
self.lin1 = nn.Dense(self.num_hiddens_1)
self.lin2 = nn.Dense(self.num_hiddens_2)
self.lin3 = nn.Dense(self.num_outputs)
self.relu = nn.relu

def forward(self, X):
H1 = self.relu(self.lin1(X.reshape(X.shape, -1)))
if self.training:
H1 = dropout_layer(H1, self.dropout_1)
H2 = self.relu(self.lin2(H1))
if self.training:
H2 = dropout_layer(H2, self.dropout_2)
return self.lin3(H2)

class DropoutMLPScratch(d2l.Classifier):
def __init__(self, num_outputs, num_hiddens_1, num_hiddens_2,
dropout_1, dropout_2, lr):
super().__init__()
self.save_hyperparameters()
self.lin1 = tf.keras.layers.Dense(num_hiddens_1, activation='relu')
self.lin2 = tf.keras.layers.Dense(num_hiddens_2, activation='relu')
self.lin3 = tf.keras.layers.Dense(num_outputs)

def forward(self, X):
H1 = self.lin1(tf.reshape(X, (X.shape, -1)))
if self.training:
H1 = dropout_layer(H1, self.dropout_1)
H2 = self.lin2(H1)
if self.training:
H2 = dropout_layer(H2, self.dropout_2)
return self.lin3(H2)


### 5.6.2.2. Training¶

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

hparams = {'num_outputs':10, 'num_hiddens_1':256, 'num_hiddens_2':256,
'dropout_1':0.5, 'dropout_2':0.5, 'lr':0.1}
model = DropoutMLPScratch(**hparams)
data = d2l.FashionMNIST(batch_size=256)
trainer = d2l.Trainer(max_epochs=10)
trainer.fit(model, data) hparams = {'num_outputs':10, 'num_hiddens_1':256, 'num_hiddens_2':256,
'dropout_1':0.5, 'dropout_2':0.5, 'lr':0.1}
model = DropoutMLPScratch(**hparams)
data = d2l.FashionMNIST(batch_size=256)
trainer = d2l.Trainer(max_epochs=10)
trainer.fit(model, data) hparams = {'num_outputs':10, 'num_hiddens_1':256, 'num_hiddens_2':256,
'dropout_1':0.5, 'dropout_2':0.5, 'lr':0.1}
model = DropoutMLPScratch(**hparams)
data = d2l.FashionMNIST(batch_size=256)
trainer = d2l.Trainer(max_epochs=10)
trainer.fit(model, data) hparams = {'num_outputs':10, 'num_hiddens_1':256, 'num_hiddens_2':256,
'dropout_1':0.5, 'dropout_2':0.5, 'lr':0.1}
model = DropoutMLPScratch(**hparams)
data = d2l.FashionMNIST(batch_size=256)
trainer = d2l.Trainer(max_epochs=10)
trainer.fit(model, data) ## 5.6.3. Concise Implementation¶

With high-level APIs, all we need to do is add a Dropout layer after each fully connected layer, passing in the dropout probability as the only argument to its constructor. During training, the Dropout layer will randomly drop out outputs of the previous layer (or equivalently, the inputs to the subsequent layer) according to the specified dropout probability. When not in training mode, the Dropout layer simply passes the data through during testing.

class DropoutMLP(d2l.Classifier):
def __init__(self, num_outputs, num_hiddens_1, num_hiddens_2,
dropout_1, dropout_2, lr):
super().__init__()
self.save_hyperparameters()
self.net = nn.Sequential(
nn.Flatten(), nn.LazyLinear(num_hiddens_1), nn.ReLU(),
nn.Dropout(dropout_1), nn.LazyLinear(num_hiddens_2), nn.ReLU(),
nn.Dropout(dropout_2), nn.LazyLinear(num_outputs))

class DropoutMLP(d2l.Classifier):
def __init__(self, num_outputs, num_hiddens_1, num_hiddens_2,
dropout_1, dropout_2, lr):
super().__init__()
self.save_hyperparameters()
self.net = nn.Sequential()
nn.Dropout(dropout_1),
nn.Dense(num_hiddens_2, activation="relu"),
nn.Dropout(dropout_2),
nn.Dense(num_outputs))
self.net.initialize()

class DropoutMLP(d2l.Classifier):
num_hiddens_1: int
num_hiddens_2: int
num_outputs: int
dropout_1: float
dropout_2: float
lr: float
training: bool = True

@nn.compact
def __call__(self, X):
x = nn.relu(nn.Dense(self.num_hiddens_1)(X.reshape((X.shape, -1))))
x = nn.Dropout(self.dropout_1, deterministic=not self.training)(x)
x = nn.relu(nn.Dense(self.num_hiddens_2)(x))
x = nn.Dropout(self.dropout_2, deterministic=not self.training)(x)
return nn.Dense(self.num_outputs)(x)


Note that we need to redefine the loss function since a network with a dropout layer needs a PRNGKey when using Module.apply(), and this RNG seed should be explicitly named dropout. This key is used by the dropout layer in Flax to generate the random dropout mask internally. It is important to use a unique dropout_rng key with every epoch in the training loop, otherwise the generated dropout mask will not be stochastic and different between the epoch runs. This dropout_rng can be stored in the TrainState object (in the d2l.Trainer class defined in Section 3.2.4) as an attribute and with every epoch it is replaced with a new dropout_rng. We already handled this with the fit_epoch method defined in Section 3.4.

@d2l.add_to_class(d2l.Classifier)  #@save
@partial(jax.jit, static_argnums=(0, 5))
def loss(self, params, X, Y, state, averaged=True):
Y_hat = state.apply_fn({'params': params}, *X,
mutable=False,  # To be used later (e.g., batch norm)
rngs={'dropout': state.dropout_rng})
Y_hat = Y_hat.reshape((-1, Y_hat.shape[-1]))
Y = Y.reshape((-1,))
fn = optax.softmax_cross_entropy_with_integer_labels
# The returned empty dictionary is a placeholder for auxiliary data,
# which will be used later (e.g., for batch norm)
return (fn(Y_hat, Y).mean(), {}) if averaged else (fn(Y_hat, Y), {})

class DropoutMLP(d2l.Classifier):
def __init__(self, num_outputs, num_hiddens_1, num_hiddens_2,
dropout_1, dropout_2, lr):
super().__init__()
self.save_hyperparameters()
self.net = tf.keras.models.Sequential([
tf.keras.layers.Flatten(),
tf.keras.layers.Dense(num_hiddens_1, activation=tf.nn.relu),
tf.keras.layers.Dropout(dropout_1),
tf.keras.layers.Dense(num_hiddens_2, activation=tf.nn.relu),
tf.keras.layers.Dropout(dropout_2),
tf.keras.layers.Dense(num_outputs)])


Next, we train the model.

model = DropoutMLP(**hparams)
trainer.fit(model, data) model = DropoutMLP(**hparams)
trainer.fit(model, data) model = DropoutMLP(**hparams)
trainer.fit(model, data) model = DropoutMLP(**hparams)
trainer.fit(model, data) ## 5.6.4. Summary¶

Beyond controlling the number of dimensions and the size of the weight vector, dropout is yet another tool for avoiding overfitting. Often tools are used jointly. Note that dropout is used only during training: it replaces an activation $$h$$ with a random variable with expected value $$h$$.

## 5.6.5. Exercises¶

1. What happens if you change the dropout probabilities for the first and second layers? In particular, what happens if you switch the ones for both layers? Design an experiment to answer these questions, describe your results quantitatively, and summarize the qualitative takeaways.

2. Increase the number of epochs and compare the results obtained when using dropout with those when not using it.

3. What is the variance of the activations in each hidden layer when dropout is and is not applied? Draw a plot to show how this quantity evolves over time for both models.

4. Why is dropout not typically used at test time?

5. Using the model in this section as an example, compare the effects of using dropout and weight decay. What happens when dropout and weight decay are used at the same time? Are the results additive? Are there diminished returns (or worse)? Do they cancel each other out?

6. What happens if we apply dropout to the individual weights of the weight matrix rather than the activations?

7. Invent another technique for injecting random noise at each layer that is different from the standard dropout technique. Can you develop a method that outperforms dropout on the Fashion-MNIST dataset (for a fixed architecture)?