Dropout: A Simple Way to Prevent Neural Networks from Overfitting Nitish Srivastava

Dropout: A Simple Way to Prevent Neural Networks from Overfitting Nitish Srivastava
Journal of Machine Learning Research 15 (2014) 1929-1958
Submitted 11/13; Published 6/14
Dropout: A Simple Way to Prevent Neural Networks from
Overfitting
Nitish Srivastava
Geoffrey Hinton
Alex Krizhevsky
Ilya Sutskever
Ruslan Salakhutdinov
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
Department of Computer Science
University of Toronto
10 Kings College Road, Rm 3302
Toronto, Ontario, M5S 3G4, Canada.
Editor: Yoshua Bengio
Abstract
Deep neural nets with a large number of parameters are very powerful machine learning
systems. However, overfitting is a serious problem in such networks. Large networks are also
slow to use, making it difficult to deal with overfitting by combining the predictions of many
different large neural nets at test time. Dropout is a technique for addressing this problem.
The key idea is to randomly drop units (along with their connections) from the neural
network during training. This prevents units from co-adapting too much. During training,
dropout samples from an exponential number of different “thinned” networks. At test time,
it is easy to approximate the effect of averaging the predictions of all these thinned networks
by simply using a single unthinned network that has smaller weights. This significantly
reduces overfitting and gives major improvements over other regularization methods. We
show that dropout improves the performance of neural networks on supervised learning
tasks in vision, speech recognition, document classification and computational biology,
obtaining state-of-the-art results on many benchmark data sets.
Keywords: neural networks, regularization, model combination, deep learning
1. Introduction
Deep neural networks contain multiple non-linear hidden layers and this makes them very
expressive models that can learn very complicated relationships between their inputs and
outputs. With limited training data, however, many of these complicated relationships
will be the result of sampling noise, so they will exist in the training set but not in real
test data even if it is drawn from the same distribution. This leads to overfitting and many
methods have been developed for reducing it. These include stopping the training as soon as
performance on a validation set starts to get worse, introducing weight penalties of various
kinds such as L1 and L2 regularization and soft weight sharing (Nowlan and Hinton, 1992).
With unlimited computation, the best way to “regularize” a fixed-sized model is to
average the predictions of all possible settings of the parameters, weighting each setting by
c
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Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever and Ruslan Salakhutdinov.
Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
(a) Standard Neural Net
(b) After applying dropout.
Figure 1: Dropout Neural Net Model. Left: A standard neural net with 2 hidden layers. Right:
An example of a thinned net produced by applying dropout to the network on the left.
Crossed units have been dropped.
its posterior probability given the training data. This can sometimes be approximated quite
well for simple or small models (Xiong et al., 2011; Salakhutdinov and Mnih, 2008), but we
would like to approach the performance of the Bayesian gold standard using considerably
less computation. We propose to do this by approximating an equally weighted geometric
mean of the predictions of an exponential number of learned models that share parameters.
Model combination nearly always improves the performance of machine learning methods. With large neural networks, however, the obvious idea of averaging the outputs of
many separately trained nets is prohibitively expensive. Combining several models is most
helpful when the individual models are different from each other and in order to make
neural net models different, they should either have different architectures or be trained
on different data. Training many different architectures is hard because finding optimal
hyperparameters for each architecture is a daunting task and training each large network
requires a lot of computation. Moreover, large networks normally require large amounts of
training data and there may not be enough data available to train different networks on
different subsets of the data. Even if one was able to train many different large networks,
using them all at test time is infeasible in applications where it is important to respond
quickly.
Dropout is a technique that addresses both these issues. It prevents overfitting and
provides a way of approximately combining exponentially many different neural network
architectures efficiently. The term “dropout” refers to dropping out units (hidden and
visible) in a neural network. By dropping a unit out, we mean temporarily removing it from
the network, along with all its incoming and outgoing connections, as shown in Figure 1.
The choice of which units to drop is random. In the simplest case, each unit is retained with
a fixed probability p independent of other units, where p can be chosen using a validation
set or can simply be set at 0.5, which seems to be close to optimal for a wide range of
networks and tasks. For the input units, however, the optimal probability of retention is
usually closer to 1 than to 0.5.
1930
Dropout
w
Present with
probability p
pw
Always
present
-
(a) At training time
(b) At test time
Figure 2: Left: A unit at training time that is present with probability p and is connected to units
in the next layer with weights w. Right: At test time, the unit is always present and
the weights are multiplied by p. The output at test time is same as the expected output
at training time.
Applying dropout to a neural network amounts to sampling a “thinned” network from
it. The thinned network consists of all the units that survived dropout (Figure 1b). A
neural net with n units, can be seen as a collection of 2n possible thinned neural networks.
These networks all share weights so that the total number of parameters is still O(n2 ), or
less. For each presentation of each training case, a new thinned network is sampled and
trained. So training a neural network with dropout can be seen as training a collection of 2n
thinned networks with extensive weight sharing, where each thinned network gets trained
very rarely, if at all.
At test time, it is not feasible to explicitly average the predictions from exponentially
many thinned models. However, a very simple approximate averaging method works well in
practice. The idea is to use a single neural net at test time without dropout. The weights
of this network are scaled-down versions of the trained weights. If a unit is retained with
probability p during training, the outgoing weights of that unit are multiplied by p at test
time as shown in Figure 2. This ensures that for any hidden unit the expected output (under
the distribution used to drop units at training time) is the same as the actual output at
test time. By doing this scaling, 2n networks with shared weights can be combined into
a single neural network to be used at test time. We found that training a network with
dropout and using this approximate averaging method at test time leads to significantly
lower generalization error on a wide variety of classification problems compared to training
with other regularization methods.
The idea of dropout is not limited to feed-forward neural nets. It can be more generally
applied to graphical models such as Boltzmann Machines. In this paper, we introduce
the dropout Restricted Boltzmann Machine model and compare it to standard Restricted
Boltzmann Machines (RBM). Our experiments show that dropout RBMs are better than
standard RBMs in certain respects.
This paper is structured as follows. Section 2 describes the motivation for this idea.
Section 3 describes relevant previous work. Section 4 formally describes the dropout model.
Section 5 gives an algorithm for training dropout networks. In Section 6, we present our
experimental results where we apply dropout to problems in different domains and compare
it with other forms of regularization and model combination. Section 7 analyzes the effect of
dropout on different properties of a neural network and describes how dropout interacts with
the network’s hyperparameters. Section 8 describes the Dropout RBM model. In Section 9
we explore the idea of marginalizing dropout. In Appendix A we present a practical guide
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Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
for training dropout nets. This includes a detailed analysis of the practical considerations
involved in choosing hyperparameters when training dropout networks.
2. Motivation
A motivation for dropout comes from a theory of the role of sex in evolution (Livnat et al.,
2010). Sexual reproduction involves taking half the genes of one parent and half of the
other, adding a very small amount of random mutation, and combining them to produce an
offspring. The asexual alternative is to create an offspring with a slightly mutated copy of
the parent’s genes. It seems plausible that asexual reproduction should be a better way to
optimize individual fitness because a good set of genes that have come to work well together
can be passed on directly to the offspring. On the other hand, sexual reproduction is likely
to break up these co-adapted sets of genes, especially if these sets are large and, intuitively,
this should decrease the fitness of organisms that have already evolved complicated coadaptations. However, sexual reproduction is the way most advanced organisms evolved.
One possible explanation for the superiority of sexual reproduction is that, over the long
term, the criterion for natural selection may not be individual fitness but rather mix-ability
of genes. The ability of a set of genes to be able to work well with another random set of
genes makes them more robust. Since a gene cannot rely on a large set of partners to be
present at all times, it must learn to do something useful on its own or in collaboration with
a small number of other genes. According to this theory, the role of sexual reproduction
is not just to allow useful new genes to spread throughout the population, but also to
facilitate this process by reducing complex co-adaptations that would reduce the chance of
a new gene improving the fitness of an individual. Similarly, each hidden unit in a neural
network trained with dropout must learn to work with a randomly chosen sample of other
units. This should make each hidden unit more robust and drive it towards creating useful
features on its own without relying on other hidden units to correct its mistakes. However,
the hidden units within a layer will still learn to do different things from each other. One
might imagine that the net would become robust against dropout by making many copies
of each hidden unit, but this is a poor solution for exactly the same reason as replica codes
are a poor way to deal with a noisy channel.
A closely related, but slightly different motivation for dropout comes from thinking
about successful conspiracies. Ten conspiracies each involving five people is probably a
better way to create havoc than one big conspiracy that requires fifty people to all play
their parts correctly. If conditions do not change and there is plenty of time for rehearsal, a
big conspiracy can work well, but with non-stationary conditions, the smaller the conspiracy
the greater its chance of still working. Complex co-adaptations can be trained to work well
on a training set, but on novel test data they are far more likely to fail than multiple simpler
co-adaptations that achieve the same thing.
3. Related Work
Dropout can be interpreted as a way of regularizing a neural network by adding noise to
its hidden units. The idea of adding noise to the states of units has previously been used in
the context of Denoising Autoencoders (DAEs) by Vincent et al. (2008, 2010) where noise
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Dropout
is added to the input units of an autoencoder and the network is trained to reconstruct the
noise-free input. Our work extends this idea by showing that dropout can be effectively
applied in the hidden layers as well and that it can be interpreted as a form of model
averaging. We also show that adding noise is not only useful for unsupervised feature
learning but can also be extended to supervised learning problems. In fact, our method can
be applied to other neuron-based architectures, for example, Boltzmann Machines. While
5% noise typically works best for DAEs, we found that our weight scaling procedure applied
at test time enables us to use much higher noise levels. Dropping out 20% of the input units
and 50% of the hidden units was often found to be optimal.
Since dropout can be seen as a stochastic regularization technique, it is natural to
consider its deterministic counterpart which is obtained by marginalizing out the noise. In
this paper, we show that, in simple cases, dropout can be analytically marginalized out
to obtain deterministic regularization methods. Recently, van der Maaten et al. (2013)
also explored deterministic regularizers corresponding to different exponential-family noise
distributions, including dropout (which they refer to as “blankout noise”). However, they
apply noise to the inputs and only explore models with no hidden layers. Wang and Manning
(2013) proposed a method for speeding up dropout by marginalizing dropout noise. Chen
et al. (2012) explored marginalization in the context of denoising autoencoders.
In dropout, we minimize the loss function stochastically under a noise distribution.
This can be seen as minimizing an expected loss function. Previous work of Globerson and
Roweis (2006); Dekel et al. (2010) explored an alternate setting where the loss is minimized
when an adversary gets to pick which units to drop. Here, instead of a noise distribution,
the maximum number of units that can be dropped is fixed. However, this work also does
not explore models with hidden units.
4. Model Description
This section describes the dropout neural network model. Consider a neural network with
L hidden layers. Let l ∈ {1, . . . , L} index the hidden layers of the network. Let z(l) denote
the vector of inputs into layer l, y(l) denote the vector of outputs from layer l (y(0) = x is
the input). W (l) and b(l) are the weights and biases at layer l. The feed-forward operation
of a standard neural network (Figure 3a) can be described as (for l ∈ {0, . . . , L − 1} and
any hidden unit i)
(l+1)
zi
(l+1)
yi
(l+1) l
= wi
(l+1)
y + bi
(l+1)
= f (zi
,
),
where f is any activation function, for example, f (x) = 1/ (1 + exp(−x)).
With dropout, the feed-forward operation becomes (Figure 3b)
(l)
rj
e (l) = r(l) ∗ y(l) ,
y
(l+1)
zi
(l+1)
yi
∼ Bernoulli(p),
(l+1) l
= wi
(l+1)
= f (zi
1933
(l+1)
e + bi
y
).
,
Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
+1
+1
(l)
r3
(l+1)
(l+1)
bi
bi
(l)
(l)
(l)
y3
y3
(l+1)
wi
(l+1)
zi
f
(l)
(l+1)
r2
yi
(l)
ye3
(l)
y2
(l+1)
wi
(l+1)
zi
f
(l+1)
yi
(l)
y2
ye2
(l)
r1
(l)
y1
(l)
y1
(a) Standard network
(l)
ye1
(b) Dropout network
Figure 3: Comparison of the basic operations of a standard and dropout network.
Here ∗ denotes an element-wise product. For any layer l, r(l) is a vector of independent
Bernoulli random variables each of which has probability p of being 1. This vector is
sampled and multiplied element-wise with the outputs of that layer, y(l) , to create the
e (l) . The thinned outputs are then used as input to the next layer. This
thinned outputs y
process is applied at each layer. This amounts to sampling a sub-network from a larger
network. For learning, the derivatives of the loss function are backpropagated through the
(l)
sub-network. At test time, the weights are scaled as Wtest = pW (l) as shown in Figure 2.
The resulting neural network is used without dropout.
5. Learning Dropout Nets
This section describes a procedure for training dropout neural nets.
5.1 Backpropagation
Dropout neural networks can be trained using stochastic gradient descent in a manner similar to standard neural nets. The only difference is that for each training case in a mini-batch,
we sample a thinned network by dropping out units. Forward and backpropagation for that
training case are done only on this thinned network. The gradients for each parameter are
averaged over the training cases in each mini-batch. Any training case which does not use a
parameter contributes a gradient of zero for that parameter. Many methods have been used
to improve stochastic gradient descent such as momentum, annealed learning rates and L2
weight decay. Those were found to be useful for dropout neural networks as well.
One particular form of regularization was found to be especially useful for dropout—
constraining the norm of the incoming weight vector at each hidden unit to be upper
bounded by a fixed constant c. In other words, if w represents the vector of weights incident
on any hidden unit, the neural network was optimized under the constraint ||w||2 ≤ c. This
constraint was imposed during optimization by projecting w onto the surface of a ball of
radius c, whenever w went out of it. This is also called max-norm regularization since it
implies that the maximum value that the norm of any weight can take is c. The constant
1934
Dropout
c is a tunable hyperparameter, which is determined using a validation set. Max-norm
regularization has been previously used in the context of collaborative filtering (Srebro and
Shraibman, 2005). It typically improves the performance of stochastic gradient descent
training of deep neural nets, even when no dropout is used.
Although dropout alone gives significant improvements, using dropout along with maxnorm regularization, large decaying learning rates and high momentum provides a significant
boost over just using dropout. A possible justification is that constraining weight vectors
to lie inside a ball of fixed radius makes it possible to use a huge learning rate without the
possibility of weights blowing up. The noise provided by dropout then allows the optimization process to explore different regions of the weight space that would have otherwise been
difficult to reach. As the learning rate decays, the optimization takes shorter steps, thereby
doing less exploration and eventually settles into a minimum.
5.2 Unsupervised Pretraining
Neural networks can be pretrained using stacks of RBMs (Hinton and Salakhutdinov, 2006),
autoencoders (Vincent et al., 2010) or Deep Boltzmann Machines (Salakhutdinov and Hinton, 2009). Pretraining is an effective way of making use of unlabeled data. Pretraining
followed by finetuning with backpropagation has been shown to give significant performance
boosts over finetuning from random initializations in certain cases.
Dropout can be applied to finetune nets that have been pretrained using these techniques. The pretraining procedure stays the same. The weights obtained from pretraining
should be scaled up by a factor of 1/p. This makes sure that for each unit, the expected
output from it under random dropout will be the same as the output during pretraining.
We were initially concerned that the stochastic nature of dropout might wipe out the information in the pretrained weights. This did happen when the learning rates used during
finetuning were comparable to the best learning rates for randomly initialized nets. However, when the learning rates were chosen to be smaller, the information in the pretrained
weights seemed to be retained and we were able to get improvements in terms of the final
generalization error compared to not using dropout when finetuning.
6. Experimental Results
We trained dropout neural networks for classification problems on data sets in different
domains. We found that dropout improved generalization performance on all data sets
compared to neural networks that did not use dropout. Table 1 gives a brief description of
the data sets. The data sets are
• MNIST : A standard toy data set of handwritten digits.
• TIMIT : A standard speech benchmark for clean speech recognition.
• CIFAR-10 and CIFAR-100 : Tiny natural images (Krizhevsky, 2009).
• Street View House Numbers data set (SVHN) : Images of house numbers collected by
Google Street View (Netzer et al., 2011).
• ImageNet : A large collection of natural images.
• Reuters-RCV1 : A collection of Reuters newswire articles.
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Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
• Alternative Splicing data set: RNA features for predicting alternative gene splicing
(Xiong et al., 2011).
We chose a diverse set of data sets to demonstrate that dropout is a general technique
for improving neural nets and is not specific to any particular application domain. In this
section, we present some key results that show the effectiveness of dropout. A more detailed
description of all the experiments and data sets is provided in Appendix B.
Data Set
Domain
Dimensionality
Training Set
Test Set
MNIST
SVHN
CIFAR-10/100
ImageNet (ILSVRC-2012)
TIMIT
Reuters-RCV1
Alternative Splicing
Vision
Vision
Vision
Vision
Speech
Text
Genetics
784 (28 × 28 grayscale)
3072 (32 × 32 color)
3072 (32 × 32 color)
65536 (256 × 256 color)
2520 (120-dim, 21 frames)
2000
1014
60K
600K
60K
1.2M
1.1M frames
200K
2932
10K
26K
10K
150K
58K frames
200K
733
Table 1: Overview of the data sets used in this paper.
6.1 Results on Image Data Sets
We used five image data sets to evaluate dropout—MNIST, SVHN, CIFAR-10, CIFAR-100
and ImageNet. These data sets include different image types and training set sizes. Models
which achieve state-of-the-art results on all of these data sets use dropout.
6.1.1 MNIST
Unit
Type
Method
Standard Neural Net (Simard et al., 2003)
SVM Gaussian kernel
Dropout NN
Dropout NN
Dropout NN + max-norm constraint
Dropout NN + max-norm constraint
Dropout NN + max-norm constraint
Dropout NN + max-norm constraint
Dropout NN + max-norm constraint (Goodfellow
et al., 2013)
Logistic
NA
Logistic
ReLU
ReLU
ReLU
ReLU
ReLU
DBN + finetuning (Hinton and Salakhutdinov, 2006)
DBM + finetuning (Salakhutdinov and Hinton, 2009)
DBN + dropout finetuning
DBM + dropout finetuning
Logistic
Logistic
Logistic
Logistic
Maxout
Architecture
Error
%
2 layers, 800 units
NA
3 layers, 1024 units
3 layers, 1024 units
3 layers, 1024 units
3 layers, 2048 units
2 layers, 4096 units
2 layers, 8192 units
2 layers, (5 × 240)
units
1.60
1.40
1.35
1.25
1.06
1.04
1.01
0.95
500-500-2000
500-500-2000
500-500-2000
500-500-2000
1.18
0.96
0.92
0.79
0.94
Table 2: Comparison of different models on MNIST.
The MNIST data set consists of 28 × 28 pixel handwritten digit images. The task is
to classify the images into 10 digit classes. Table 2 compares the performance of dropout
with other techniques. The best performing neural networks for the permutation invariant
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Dropout
Classification Error %
setting that do not use dropout or unsupervised pretraining achieve an error of about
1.60% (Simard et al., 2003). With dropout the error reduces to 1.35%. Replacing logistic
units with rectified linear units (ReLUs) (Jarrett et al., 2009) further reduces the error to
1.25%. Adding max-norm regularization again reduces it to 1.06%. Increasing the size of
the network leads to better results. A neural net with 2 layers and 8192 units per layer
gets down to 0.95% error. Note that this network has more than 65 million parameters and
is being trained on a data set of size 60,000. Training a network of this size to give good
generalization error is very hard with standard regularization methods and early stopping.
Dropout, on the other hand, prevents overfitting, even in this case. It does not even need
early stopping. Goodfellow et al. (2013) showed that results can be further improved to
0.94% by replacing ReLU units with maxout units. All dropout nets use p = 0.5 for hidden
units and p = 0.8 for input units. More experimental details can be found in Appendix B.1.
Dropout nets pretrained with stacks of RBMs and Deep Boltzmann Machines also give
improvements as shown in Table 2. DBM—pretrained dropout nets achieve a test error of
0.79% which is the best performance ever reported for the permutation invariant setting.
We note that it possible to obtain better results by using 2-D spatial information and
augmenting the training set with distorted versions of images from the standard training
set. We demonstrate the effectiveness of dropout in that setting on more interesting data
sets.
In order to test the robustness of
dropout, classification experiments were
2.5
done with networks of many different architectures keeping all hyperparameters, inWithout dropout
cluding p, fixed. Figure 4 shows the test
2.0
@
error rates obtained for these different arR
@
chitectures as training progresses. The
1.5
same architectures trained with and withWith dropout
out dropout have drastically different test
R
@
errors as seen as by the two separate clus1.0
ters of trajectories. Dropout gives a huge
improvement across all architectures, with0
200000
400000
600000
800000
1000000
out using hyperparameters that were tuned
Number of weight updates
specifically for each architecture.
Figure 4: Test error for different architectures
with and without dropout. The net6.1.2 Street View House Numbers
works have 2 to 4 hidden layers each
with 1024 to 2048 units.
The Street View House Numbers (SVHN)
Data Set (Netzer et al., 2011) consists of
color images of house numbers collected by
Google Street View. Figure 5a shows some examples of images from this data set. The
part of the data set that we use in our experiments consists of 32 × 32 color images roughly
centered on a digit in a house number. The task is to identify that digit.
For this data set, we applied dropout to convolutional neural networks (LeCun et al.,
1989). The best architecture that we found has three convolutional layers followed by 2
fully connected hidden layers. All hidden units were ReLUs. Each convolutional layer was
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Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
Method
Error %
Binary Features (WDCH) (Netzer et al., 2011)
HOG (Netzer et al., 2011)
Stacked Sparse Autoencoders (Netzer et al., 2011)
KMeans (Netzer et al., 2011)
Multi-stage Conv Net with average pooling (Sermanet et al., 2012)
Multi-stage Conv Net + L2 pooling (Sermanet et al., 2012)
Multi-stage Conv Net + L4 pooling + padding (Sermanet et al., 2012)
Conv Net + max-pooling
Conv Net + max pooling + dropout in fully connected layers
Conv Net + stochastic pooling (Zeiler and Fergus, 2013)
Conv Net + max pooling + dropout in all layers
Conv Net + maxout (Goodfellow et al., 2013)
Human Performance
36.7
15.0
10.3
9.4
9.06
5.36
4.90
3.95
3.02
2.80
2.55
2.47
2.0
Table 3: Results on the Street View House Numbers data set.
followed by a max-pooling layer. Appendix B.2 describes the architecture in more detail.
Dropout was applied to all the layers of the network with the probability of retaining a hidden unit being p = (0.9, 0.75, 0.75, 0.5, 0.5, 0.5) for the different layers of the network (going
from input to convolutional layers to fully connected layers). Max-norm regularization was
used for weights in both convolutional and fully connected layers. Table 3 compares the
results obtained by different methods. We find that convolutional nets outperform other
methods. The best performing convolutional nets that do not use dropout achieve an error
rate of 3.95%. Adding dropout only to the fully connected layers reduces the error to 3.02%.
Adding dropout to the convolutional layers as well further reduces the error to 2.55%. Even
more gains can be obtained by using maxout units.
The additional gain in performance obtained by adding dropout in the convolutional
layers (3.02% to 2.55%) is worth noting. One may have presumed that since the convolutional layers don’t have a lot of parameters, overfitting is not a problem and therefore
dropout would not have much effect. However, dropout in the lower layers still helps because it provides noisy inputs for the higher fully connected layers which prevents them
from overfitting.
6.1.3 CIFAR-10 and CIFAR-100
The CIFAR-10 and CIFAR-100 data sets consist of 32 × 32 color images drawn from 10
and 100 categories respectively. Figure 5b shows some examples of images from this data
set. A detailed description of the data sets, input preprocessing, network architectures and
other experimental details is given in Appendix B.3. Table 4 shows the error rate obtained
by different methods on these data sets. Without any data augmentation, Snoek et al.
(2012) used Bayesian hyperparameter optimization to obtained an error rate of 14.98% on
CIFAR-10. Using dropout in the fully connected layers reduces that to 14.32% and adding
dropout in every layer further reduces the error to 12.61%. Goodfellow et al. (2013) showed
that the error is further reduced to 11.68% by replacing ReLU units with maxout units. On
CIFAR-100, dropout reduces the error from 43.48% to 37.20% which is a huge improvement.
No data augmentation was used for either data set (apart from the input dropout).
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Dropout
(a) Street View House Numbers (SVHN)
(b) CIFAR-10
Figure 5: Samples from image data sets. Each row corresponds to a different category.
Method
Conv
Conv
Conv
Conv
Conv
Conv
Net
Net
Net
Net
Net
Net
+
+
+
+
+
+
max pooling (hand tuned)
stochastic pooling (Zeiler and Fergus, 2013)
max pooling (Snoek et al., 2012)
max pooling + dropout fully connected layers
max pooling + dropout in all layers
maxout (Goodfellow et al., 2013)
CIFAR-10
CIFAR-100
15.60
15.13
14.98
14.32
12.61
11.68
43.48
42.51
41.26
37.20
38.57
Table 4: Error rates on CIFAR-10 and CIFAR-100.
6.1.4 ImageNet
ImageNet is a data set of over 15 million labeled high-resolution images belonging to roughly
22,000 categories. Starting in 2010, as part of the Pascal Visual Object Challenge, an annual
competition called the ImageNet Large-Scale Visual Recognition Challenge (ILSVRC) has
been held. A subset of ImageNet with roughly 1000 images in each of 1000 categories is
used in this challenge. Since the number of categories is rather large, it is conventional to
report two error rates: top-1 and top-5, where the top-5 error rate is the fraction of test
images for which the correct label is not among the five labels considered most probable by
the model. Figure 6 shows some predictions made by our model on a few test images.
ILSVRC-2010 is the only version of ILSVRC for which the test set labels are available, so
most of our experiments were performed on this data set. Table 5 compares the performance
of different methods. Convolutional nets with dropout outperform other methods by a large
margin. The architecture and implementation details are described in detail in Krizhevsky
et al. (2012).
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Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
Figure 6: Some ImageNet test cases with the 4 most probable labels as predicted by our model.
The length of the horizontal bars is proportional to the probability assigned to the labels
by the model. Pink indicates ground truth.
Model
Top-1
Top-5
47.1
45.7
37.5
28.2
25.7
17.0
Sparse Coding (Lin et al., 2010)
SIFT + Fisher Vectors (Sanchez and Perronnin, 2011)
Conv Net + dropout (Krizhevsky et al., 2012)
Table 5: Results on the ILSVRC-2010 test set.
Model
SVM on Fisher Vectors of Dense SIFT and Color Statistics
Avg of classifiers over FVs of SIFT, LBP, GIST and CSIFT
Conv Net + dropout (Krizhevsky et al., 2012)
Avg of 5 Conv Nets + dropout (Krizhevsky et al., 2012)
Top-1
(val)
Top-5
(val)
Top-5
(test)
40.7
38.1
18.2
16.4
27.3
26.2
16.4
Table 6: Results on the ILSVRC-2012 validation/test set.
Our model based on convolutional nets and dropout won the ILSVRC-2012 competition.
Since the labels for the test set are not available, we report our results on the test set for
the final submission and include the validation set results for different variations of our
model. Table 6 shows the results from the competition. While the best methods based on
standard vision features achieve a top-5 error rate of about 26%, convolutional nets with
dropout achieve a test error of about 16% which is a staggering difference. Figure 6 shows
some examples of predictions made by our model. We can see that the model makes very
reasonable predictions, even when its best guess is not correct.
6.2 Results on TIMIT
Next, we applied dropout to a speech recognition task. We use the TIMIT data set which
consists of recordings from 680 speakers covering 8 major dialects of American English
reading ten phonetically-rich sentences in a controlled noise-free environment. Dropout
neural networks were trained on windows of 21 log-filter bank frames to predict the label
of the central frame. No speaker dependent operations were performed. Appendix B.4
describes the data preprocessing and training details. Table 7 compares dropout neural
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Dropout
nets with other models. A 6-layer net gives a phone error rate of 23.4%. Dropout further
improves it to 21.8%. We also trained dropout nets starting from pretrained weights. A
4-layer net pretrained with a stack of RBMs get a phone error rate of 22.7%. With dropout,
this reduces to 19.7%. Similarly, for an 8-layer net the error reduces from 20.5% to 19.7%.
Method
Phone Error Rate%
NN (6 layers) (Mohamed et al., 2010)
Dropout NN (6 layers)
23.4
21.8
DBN-pretrained NN (4 layers)
DBN-pretrained NN (6 layers) (Mohamed et al., 2010)
DBN-pretrained NN (8 layers) (Mohamed et al., 2010)
mcRBM-DBN-pretrained NN (5 layers) (Dahl et al., 2010)
DBN-pretrained NN (4 layers) + dropout
DBN-pretrained NN (8 layers) + dropout
22.7
22.4
20.7
20.5
19.7
19.7
Table 7: Phone error rate on the TIMIT core test set.
6.3 Results on a Text Data Set
To test the usefulness of dropout in the text domain, we used dropout networks to train a
document classifier. We used a subset of the Reuters-RCV1 data set which is a collection of
over 800,000 newswire articles from Reuters. These articles cover a variety of topics. The
task is to take a bag of words representation of a document and classify it into 50 disjoint
topics. Appendix B.5 describes the setup in more detail. Our best neural net which did
not use dropout obtained an error rate of 31.05%. Adding dropout reduced the error to
29.62%. We found that the improvement was much smaller compared to that for the vision
and speech data sets.
6.4 Comparison with Bayesian Neural Networks
Dropout can be seen as a way of doing an equally-weighted averaging of exponentially many
models with shared weights. On the other hand, Bayesian neural networks (Neal, 1996) are
the proper way of doing model averaging over the space of neural network structures and
parameters. In dropout, each model is weighted equally, whereas in a Bayesian neural
network each model is weighted taking into account the prior and how well the model fits
the data, which is the more correct approach. Bayesian neural nets are extremely useful for
solving problems in domains where data is scarce such as medical diagnosis, genetics, drug
discovery and other computational biology applications. However, Bayesian neural nets are
slow to train and difficult to scale to very large network sizes. Besides, it is expensive to
get predictions from many large nets at test time. On the other hand, dropout neural nets
are much faster to train and use at test time. In this section, we report experiments that
compare Bayesian neural nets with dropout neural nets on a small data set where Bayesian
neural networks are known to perform well and obtain state-of-the-art results. The aim is
to analyze how much does dropout lose compared to Bayesian neural nets.
The data set that we use (Xiong et al., 2011) comes from the domain of genetics. The
task is to predict the occurrence of alternative splicing based on RNA features. Alternative
splicing is a significant cause of cellular diversity in mammalian tissues. Predicting the
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Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
Method
Code Quality (bits)
Neural Network (early stopping) (Xiong et al., 2011)
Regression, PCA (Xiong et al., 2011)
SVM, PCA (Xiong et al., 2011)
Neural Network with dropout
Bayesian Neural Network (Xiong et al., 2011)
440
463
487
567
623
Table 8: Results on the Alternative Splicing Data Set.
occurrence of alternate splicing in certain tissues under different conditions is important for
understanding many human diseases. Given the RNA features, the task is to predict the
probability of three splicing related events that biologists care about. The evaluation metric
is Code Quality which is a measure of the negative KL divergence between the target and
the predicted probability distributions (higher is better). Appendix B.6 includes a detailed
description of the data set and this performance metric.
Table 8 summarizes the performance of different models on this data set. Xiong et al.
(2011) used Bayesian neural nets for this task. As expected, we found that Bayesian neural
nets perform better than dropout. However, we see that dropout improves significantly
upon the performance of standard neural nets and outperforms all other methods. The
challenge in this data set is to prevent overfitting since the size of the training set is small.
One way to prevent overfitting is to reduce the input dimensionality using PCA. Thereafter,
standard techniques such as SVMs or logistic regression can be used. However, with dropout
we were able to prevent overfitting without the need to do dimensionality reduction. The
dropout nets are very large (1000s of hidden units) compared to a few tens of units in the
Bayesian network. This shows that dropout has a strong regularizing effect.
6.5 Comparison with Standard Regularizers
Several regularization methods have been proposed for preventing overfitting in neural networks. These include L2 weight decay (more generally Tikhonov regularization (Tikhonov,
1943)), lasso (Tibshirani, 1996), KL-sparsity and max-norm regularization. Dropout can
be seen as another way of regularizing neural networks. In this section we compare dropout
with some of these regularization methods using the MNIST data set.
The same network architecture (784-1024-1024-2048-10) with ReLUs was trained using stochastic gradient descent with different regularizations. Table 9 shows the results.
The values of different hyperparameters associated with each kind of regularization (decay
constants, target sparsity, dropout rate, max-norm upper bound) were obtained using a
validation set. We found that dropout combined with max-norm regularization gives the
lowest generalization error.
7. Salient Features
The experiments described in the previous section provide strong evidence that dropout
is a useful technique for improving neural networks. In this section, we closely examine
how dropout affects a neural network. We analyze the effect of dropout on the quality of
features produced. We see how dropout affects the sparsity of hidden unit activations. We
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Dropout
Method
Test Classification error %
L2
L2 + L1 applied towards the end of training
L2 + KL-sparsity
Max-norm
Dropout + L2
Dropout + Max-norm
1.62
1.60
1.55
1.35
1.25
1.05
Table 9: Comparison of different regularization methods on MNIST.
also see how the advantages obtained from dropout vary with the probability of retaining
units, size of the network and the size of the training set. These observations give some
insight into why dropout works so well.
7.1 Effect on Features
(a) Without dropout
(b) Dropout with p = 0.5.
Figure 7: Features learned on MNIST with one hidden layer autoencoders having 256 rectified
linear units.
In a standard neural network, the derivative received by each parameter tells it how it
should change so the final loss function is reduced, given what all other units are doing.
Therefore, units may change in a way that they fix up the mistakes of the other units.
This may lead to complex co-adaptations. This in turn leads to overfitting because these
co-adaptations do not generalize to unseen data. We hypothesize that for each hidden unit,
dropout prevents co-adaptation by making the presence of other hidden units unreliable.
Therefore, a hidden unit cannot rely on other specific units to correct its mistakes. It must
perform well in a wide variety of different contexts provided by the other hidden units. To
observe this effect directly, we look at the first level features learned by neural networks
trained on visual tasks with and without dropout.
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Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
Figure 7a shows features learned by an autoencoder on MNIST with a single hidden
layer of 256 rectified linear units without dropout. Figure 7b shows the features learned by
an identical autoencoder which used dropout in the hidden layer with p = 0.5. Both autoencoders had similar test reconstruction errors. However, it is apparent that the features
shown in Figure 7a have co-adapted in order to produce good reconstructions. Each hidden
unit on its own does not seem to be detecting a meaningful feature. On the other hand, in
Figure 7b, the hidden units seem to detect edges, strokes and spots in different parts of the
image. This shows that dropout does break up co-adaptations, which is probably the main
reason why it leads to lower generalization errors.
7.2 Effect on Sparsity
(a) Without dropout
(b) Dropout with p = 0.5.
Figure 8: Effect of dropout on sparsity. ReLUs were used for both models. Left: The histogram
of mean activations shows that most units have a mean activation of about 2.0. The
histogram of activations shows a huge mode away from zero. Clearly, a large fraction of
units have high activation. Right: The histogram of mean activations shows that most
units have a smaller mean mean activation of about 0.7. The histogram of activations
shows a sharp peak at zero. Very few units have high activation.
We found that as a side-effect of doing dropout, the activations of the hidden units
become sparse, even when no sparsity inducing regularizers are present. Thus, dropout automatically leads to sparse representations. To observe this effect, we take the autoencoders
trained in the previous section and look at the sparsity of hidden unit activations on a random mini-batch taken from the test set. Figure 8a and Figure 8b compare the sparsity for
the two models. In a good sparse model, there should only be a few highly activated units
for any data case. Moreover, the average activation of any unit across data cases should
be low. To assess both of these qualities, we plot two histograms for each model. For each
model, the histogram on the left shows the distribution of mean activations of hidden units
across the minibatch. The histogram on the right shows the distribution of activations of
the hidden units.
Comparing the histograms of activations we can see that fewer hidden units have high
activations in Figure 8b compared to Figure 8a, as seen by the significant mass away from
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Dropout
zero for the net that does not use dropout. The mean activations are also smaller for the
dropout net. The overall mean activation of hidden units is close to 2.0 for the autoencoder
without dropout but drops to around 0.7 when dropout is used.
7.3 Effect of Dropout Rate
Dropout has a tunable hyperparameter p (the probability of retaining a unit in the network).
In this section, we explore the effect of varying this hyperparameter. The comparison is
done in two situations.
1. The number of hidden units is held constant.
2. The number of hidden units is changed so that the expected number of hidden units
that will be retained after dropout is held constant.
In the first case, we train the same network architecture with different amounts of
dropout. We use a 784-2048-2048-2048-10 architecture. No input dropout was used. Figure 9a shows the test error obtained as a function of p. If the architecture is held constant,
having a small p means very few units will turn on during training. It can be seen that this
has led to underfitting since the training error is also high. We see that as p increases, the
error goes down. It becomes flat when 0.4 ≤ p ≤ 0.8 and then increases as p becomes close
to 1.
3.0
3.5
Test Error
Training Error
2.5
2.0
1.5
1.0
2.0
1.5
1.0
0.5
0.5
0.00.0
Test Error
Training Error
2.5
Classification Error %
Classification Error %
3.0
0.2
0.4
0.6
0.8
Probability of retaining a unit (p)
1.0
(a) Keeping n fixed.
0.00.0
0.2
0.4
0.6
0.8
Probability of retaining a unit (p)
1.0
(b) Keeping pn fixed.
Figure 9: Effect of changing dropout rates on MNIST.
Another interesting setting is the second case in which the quantity pn is held constant
where n is the number of hidden units in any particular layer. This means that networks
that have small p will have a large number of hidden units. Therefore, after applying
dropout, the expected number of units that are present will be the same across different
architectures. However, the test networks will be of different sizes. In our experiments,
we set pn = 256 for the first two hidden layers and pn = 512 for the last hidden layer.
Figure 9b shows the test error obtained as a function of p. We notice that the magnitude
of errors for small values of p has reduced by a lot compared to Figure 9a (for p = 0.1 it fell
from 2.7% to 1.7%). Values of p that are close to 0.6 seem to perform best for this choice
of pn but our usual default value of 0.5 is close to optimal.
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Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
7.4 Effect of Data Set Size
Classification Error %
One test of a good regularizer is that it should make it possible to get good generalization
error from models with a large number of parameters trained on small data sets. This
section explores the effect of changing the data set size when dropout is used with feedforward networks. Huge neural networks trained in the standard way overfit massively on
small data sets. To see if dropout can help, we run classification experiments on MNIST
and vary the amount of data given to the network.
The results of these experiments are
30
With dropout
shown in Figure 10. The network was given
Without dropout
data sets of size 100, 500, 1K, 5K, 10K
25
and 50K chosen randomly from the MNIST
20
training set. The same network architecture (784-1024-1024-2048-10) was used for
15
all data sets. Dropout with p = 0.5 was performed at all the hidden layers and p = 0.8
10
at the input layer. It can be observed that
for extremely small data sets (100, 500)
5
dropout does not give any improvements.
0
The model has enough parameters that it
104
105
102
103
Dataset size
can overfit on the training data, even with
all the noise coming from dropout. As the
Figure 10: Effect of varying data set size.
size of the data set is increased, the gain
from doing dropout increases up to a point and then declines. This suggests that for any
given architecture and dropout rate, there is a “sweet spot” corresponding to some amount
of data that is large enough to not be memorized in spite of the noise but not so large that
overfitting is not a problem anyways.
7.5 Monte-Carlo Model Averaging vs. Weight Scaling
Test Classification error %
The efficient test time procedure that we
1.35
Monte-Carlo Model Averaging
propose is to do an approximate model comApproximate averaging by weight scaling
1.30
bination by scaling down the weights of the
trained neural network. An expensive but
1.25
more correct way of averaging the models
1.20
is to sample k neural nets using dropout for
each test case and average their predictions.
1.15
As k → ∞, this Monte-Carlo model average
1.10
gets close to the true model average. It is interesting to see empirically how many sam1.05
ples k are needed to match the performance
of the approximate averaging method. By
1.000
20
40
60
80
100
120
Number of samples used for Monte-Carlo averaging (k)
computing the error for different values of k
we can see how quickly the error rate of the
finite-sample average approaches the error Figure 11: Monte-Carlo model averaging vs.
weight scaling.
rate of the true model average.
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Dropout
We again use the MNIST data set and do classification by averaging the predictions
of k randomly sampled neural networks. Figure 11 shows the test error rate obtained for
different values of k. This is compared with the error obtained using the weight scaling
method (shown as a horizontal line). It can be seen that around k = 50, the Monte-Carlo
method becomes as good as the approximate method. Thereafter, the Monte-Carlo method
is slightly better than the approximate method but well within one standard deviation of
it. This suggests that the weight scaling method is a fairly good approximation of the true
model average.
8. Dropout Restricted Boltzmann Machines
Besides feed-forward neural networks, dropout can also be applied to Restricted Boltzmann
Machines (RBM). In this section, we formally describe this model and show some results
to illustrate its key properties.
8.1 Model Description
Consider an RBM with visible units v ∈ {0, 1}D and hidden units h ∈ {0, 1}F . It defines
the following probability distribution
P (h, v; θ) =
1
exp(v> W h + a> h + b> v).
Z(θ)
Where θ = {W, a, b} represents the model parameters and Z is the partition function.
Dropout RBMs are RBMs augmented with a vector of binary random variables r ∈
{0, 1}F . Each random variable rj takes the value 1 with probability p, independent of
others. If rj takes the value 1, the hidden unit hj is retained, otherwise it is dropped from
the model. The joint distribution defined by a Dropout RBM can be expressed as
P (r, h, v; p, θ) = P (r; p)P (h, v|r; θ),
P (r; p) =
F
Y
prj (1 − p)1−rj ,
j=1
P (h, v|r; θ) =
F
Y
1
>
>
>
exp(v
W
h
+
a
h
+
b
v)
g(hj , rj ),
Z 0 (θ, r)
j=1
g(hj , rj ) =
1(rj = 1) + 1(rj = 0)1(hj = 0).
Z 0 (θ, r) is the normalization constant. g(hj , rj ) imposes the constraint that if rj = 0,
hj must be 0. The distribution over h, conditioned on v and r is factorial
P (h|r, v) =
F
Y
P (hj |rj , v),
j=1
P (hj = 1|rj , v) =
1(rj = 1)σ bj +
X
i
1947
Wij vi
!
.
Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
(a) Without dropout
(b) Dropout with p = 0.5.
Figure 12: Features learned on MNIST by 256 hidden unit RBMs. The features are ordered by L2
norm.
The distribution over v conditioned on h is same as that of an RBM
P (v|h) =
D
Y
P (vi |h),
i=1

P (vi = 1|h) = σ ai +
X
j

Wij hj  .
Conditioned on r, the distribution over {v, h} is same as the distribution that an RBM
would impose, except that the units for which rj = 0 are dropped from h. Therefore, the
Dropout RBM model can be seen as a mixture of exponentially many RBMs with shared
weights each using a different subset of h.
8.2 Learning Dropout RBMs
Learning algorithms developed for RBMs such as Contrastive Divergence (Hinton et al.,
2006) can be directly applied for learning Dropout RBMs. The only difference is that r is
first sampled and only the hidden units that are retained are used for training. Similar to
dropout neural networks, a different r is sampled for each training case in every minibatch.
In our experiments, we use CD-1 for training dropout RBMs.
8.3 Effect on Features
Dropout in feed-forward networks improved the quality of features by reducing co-adaptations.
This section explores whether this effect transfers to Dropout RBMs as well.
Figure 12a shows features learned by a binary RBM with 256 hidden units. Figure 12b
shows features learned by a dropout RBM with the same number of hidden units. Features
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Dropout
(a) Without dropout
(b) Dropout with p = 0.5.
Figure 13: Effect of dropout on sparsity. Left: The activation histogram shows that a large number of units have activations away from zero. Right: A large number of units have
activations close to zero and very few units have high activation.
learned by the dropout RBM appear qualitatively different in the sense that they seem to
capture features that are coarser compared to the sharply defined stroke-like features in the
standard RBM. There seem to be very few dead units in the dropout RBM relative to the
standard RBM.
8.4 Effect on Sparsity
Next, we investigate the effect of dropout RBM training on sparsity of the hidden unit
activations. Figure 13a shows the histograms of hidden unit activations and their means on
a test mini-batch after training an RBM. Figure 13b shows the same for dropout RBMs.
The histograms clearly indicate that the dropout RBMs learn much sparser representations
than standard RBMs even when no additional sparsity inducing regularizer is present.
9. Marginalizing Dropout
Dropout can be seen as a way of adding noise to the states of hidden units in a neural
network. In this section, we explore the class of models that arise as a result of marginalizing
this noise. These models can be seen as deterministic versions of dropout. In contrast to
standard (“Monte-Carlo”) dropout, these models do not need random bits and it is possible
to get gradients for the marginalized loss functions. In this section, we briefly explore these
models.
Deterministic algorithms have been proposed that try to learn models that are robust to
feature deletion at test time (Globerson and Roweis, 2006). Marginalization in the context
of denoising autoencoders has been explored previously (Chen et al., 2012). The marginalization of dropout noise in the context of linear regression was discussed in Srivastava (2013).
Wang and Manning (2013) further explored the idea of marginalizing dropout to speed-up
training. van der Maaten et al. (2013) investigated different input noise distributions and
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Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
the regularizers obtained by marginalizing this noise. Wager et al. (2013) describes how
dropout can be seen as an adaptive regularizer.
9.1 Linear Regression
First we explore a very simple case of applying dropout to the classical problem of linear
regression. Let X ∈ RN ×D be a data matrix of N data points. y ∈ RN be a vector of
targets. Linear regression tries to find a w ∈ RD that minimizes
||y − Xw||2 .
When the input X is dropped out such that any input dimension is retained with
probability p, the input can be expressed as R ∗ X where R ∈ {0, 1}N ×D is a random matrix
with Rij ∼ Bernoulli(p) and ∗ denotes an element-wise product. Marginalizing the noise,
the objective function becomes
minimize ER∼Bernoulli(p) ||y − (R ∗ X)w||2 .
w
This reduces to
minimize
w
||y − pXw||2 + p(1 − p)||Γw||2 ,
where Γ = (diag(X > X))1/2 . Therefore, dropout with linear regression is equivalent, in
expectation, to ridge regression with a particular form for Γ. This form of Γ essentially
scales the weight cost for weight wi by the standard deviation of the ith dimension of the
data. If a particular data dimension varies a lot, the regularizer tries to squeeze its weight
more.
Another interesting way to look at this objective is to absorb the factor of p into w.
This leads to the following form
minimize
w
e 2+
||y − X w||
1−p
e 2,
||Γw||
p
e = pw. This makes the dependence of the regularization constant on p explicit.
where w
For p close to 1, all the inputs are retained and the regularization constant is small. As
more dropout is done (by decreasing p), the regularization constant grows larger.
9.2 Logistic Regression and Deep Networks
For logistic regression and deep neural nets, it is hard to obtain a closed form marginalized
model. However, Wang and Manning (2013) showed that in the context of dropout applied
to logistic regression, the corresponding marginalized model can be trained approximately.
Under reasonable assumptions, the distributions over the inputs to the logistic unit and over
the gradients of the marginalized model are Gaussian. Their means and variances can be
computed efficiently. This approximate marginalization outperforms Monte-Carlo dropout
in terms of training time and generalization performance.
However, the assumptions involved in this technique become successively weaker as more
layers are added. Therefore, the results are not directly applicable to deep networks.
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Dropout
Data Set
Architecture
Bernoulli dropout
Gaussian dropout
MNIST
CIFAR-10
2 layers, 1024 units each
3 conv + 2 fully connected layers
1.08 ± 0.04
12.6 ± 0.1
0.95 ± 0.04
12.5 ± 0.1
Table 10: Comparison of classification error % with Bernoulli and Gaussian dropout. For MNIST,
the Bernoulli model uses p = 0.5 for the hidden units and p = 0.8 for the input units.
For CIFAR-10, we use p = (0.9, 0.75, 0.75, 0.5, 0.5, 0.5) going from the
q input layer to the
top. The value of σ for the Gaussian dropout models was set to be
averaged over 10 different random seeds.
1−p
p .
Results were
10. Multiplicative Gaussian Noise
Dropout involves multiplying hidden activations by Bernoulli distributed random variables
which take the value 1 with probability p and 0 otherwise. This idea can be generalized
by multiplying the activations with random variables drawn from other distributions. We
recently discovered that multiplying by a random variable drawn from N (1, 1) works just
as well, or perhaps better than using Bernoulli noise. This new form of dropout amounts
to adding a Gaussian distributed random variable with zero mean and standard deviation
equal to the activation of the unit. That is, each hidden activation hi is perturbed to
hi + hi r where r ∼ N (0, 1), or equivalently hi r0 where r0 ∼ N (1, 1). We can generalize
this to r0 ∼ N (1, σ 2 ) where σ becomes an additional hyperparameter to tune, just like p
was in the standard (Bernoulli) dropout. The expected value of the activations remains
unchanged, therefore no weight scaling is required at test time.
In this paper, we described dropout as a method where we retain units with probability p
at training time and scale down the weights by multiplying them by a factor of p at test time.
Another way to achieve the same effect is to scale up the retained activations by multiplying
by 1/p at training time and not modifying the weights at test time. These methods are
equivalent with appropriate scaling of the learning rate and weight initializations at each
layer.
Therefore, dropout can be seen as multiplying hi by a Bernoulli random variable rb that
takes the value 1/p with probability p and 0 otherwise. E[rb ] = 1 and V ar[rb ] = (1 − p)/p.
For the Gaussian multiplicative noise, if we set σ 2 = (1 − p)/p, we end up multiplying
hi by a random variable rg , where E[rg ] = 1 and V ar[rg ] = (1 − p)/p. Therefore, both
forms of dropout can be set up so that the random variable being multiplied by has the
same mean and variance. However, given these first and second order moments, rg has the
highest entropy and rb has the lowest. Both these extremes work well, although preliminary
experimental results shown in Table 10 suggest that the high entropy case mightq
work
slightly better. For each layer, the value of σ in the Gaussian model was set to be 1−p
p
using the p from the corresponding layer in the Bernoulli model.
11. Conclusion
Dropout is a technique for improving neural networks by reducing overfitting. Standard
backpropagation learning builds up brittle co-adaptations that work for the training data
but do not generalize to unseen data. Random dropout breaks up these co-adaptations by
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Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
making the presence of any particular hidden unit unreliable. This technique was found
to improve the performance of neural nets in a wide variety of application domains including object classification, digit recognition, speech recognition, document classification and
analysis of computational biology data. This suggests that dropout is a general technique
and is not specific to any domain. Methods that use dropout achieve state-of-the-art results on SVHN, ImageNet, CIFAR-100 and MNIST. Dropout considerably improved the
performance of standard neural nets on other data sets as well.
This idea can be extended to Restricted Boltzmann Machines and other graphical models. The central idea of dropout is to take a large model that overfits easily and repeatedly
sample and train smaller sub-models from it. RBMs easily fit into this framework. We developed Dropout RBMs and empirically showed that they have certain desirable properties.
One of the drawbacks of dropout is that it increases training time. A dropout network
typically takes 2-3 times longer to train than a standard neural network of the same architecture. A major cause of this increase is that the parameter updates are very noisy.
Each training case effectively tries to train a different random architecture. Therefore, the
gradients that are being computed are not gradients of the final architecture that will be
used at test time. Therefore, it is not surprising that training takes a long time. However,
it is likely that this stochasticity prevents overfitting. This creates a trade-off between overfitting and training time. With more training time, one can use high dropout and suffer less
overfitting. However, one way to obtain some of the benefits of dropout without stochasticity is to marginalize the noise to obtain a regularizer that does the same thing as the
dropout procedure, in expectation. We showed that for linear regression this regularizer is
a modified form of L2 regularization. For more complicated models, it is not obvious how to
obtain an equivalent regularizer. Speeding up dropout is an interesting direction for future
work.
Acknowledgments
This research was supported by OGS, NSERC and an Early Researcher Award.
Appendix A. A Practical Guide for Training Dropout Networks
Neural networks are infamous for requiring extensive hyperparameter tuning. Dropout
networks are no exception. In this section, we describe heuristics that might be useful for
applying dropout.
A.1 Network Size
It is to be expected that dropping units will reduce the capacity of a neural network. If
n is the number of hidden units in any layer and p is the probability of retaining a unit,
then instead of n hidden units, only pn units will be present after dropout, in expectation.
Moreover, this set of pn units will be different each time and the units are not allowed to
build co-adaptations freely. Therefore, if an n-sized layer is optimal for a standard neural
net on any given task, a good dropout net should have at least n/p units. We found this to
be a useful heuristic for setting the number of hidden units in both convolutional and fully
connected networks.
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Dropout
A.2 Learning Rate and Momentum
Dropout introduces a significant amount of noise in the gradients compared to standard
stochastic gradient descent. Therefore, a lot of gradients tend to cancel each other. In
order to make up for this, a dropout net should typically use 10-100 times the learning rate
that was optimal for a standard neural net. Another way to reduce the effect the noise is
to use a high momentum. While momentum values of 0.9 are common for standard nets,
with dropout we found that values around 0.95 to 0.99 work quite a lot better. Using high
learning rate and/or momentum significantly speed up learning.
A.3 Max-norm Regularization
Though large momentum and learning rate speed up learning, they sometimes cause the
network weights to grow very large. To prevent this, we can use max-norm regularization.
This constrains the norm of the vector of incoming weights at each hidden unit to be bound
by a constant c. Typical values of c range from 3 to 4.
A.4 Dropout Rate
Dropout introduces an extra hyperparameter—the probability of retaining a unit p. This
hyperparameter controls the intensity of dropout. p = 1, implies no dropout and low values
of p mean more dropout. Typical values of p for hidden units are in the range 0.5 to 0.8.
For input layers, the choice depends on the kind of input. For real-valued inputs (image
patches or speech frames), a typical value is 0.8. For hidden layers, the choice of p is coupled
with the choice of number of hidden units n. Smaller p requires big n which slows down
the training and leads to underfitting. Large p may not produce enough dropout to prevent
overfitting.
Appendix B. Detailed Description of Experiments and Data Sets
.
This section describes the network architectures and training details for the experimental
results reported in this paper. The code for reproducing these results can be obtained from
http://www.cs.toronto.edu/~nitish/dropout. The implementation is GPU-based. We
used the excellent CUDA libraries—cudamat (Mnih, 2009) and cuda-convnet (Krizhevsky
et al., 2012) to implement our networks.
B.1 MNIST
The MNIST data set consists of 60,000 training and 10,000 test examples each representing
a 28×28 digit image. We held out 10,000 random training images for validation. Hyperparameters were tuned on the validation set such that the best validation error was produced
after 1 million weight updates. The validation set was then combined with the training set
and training was done for 1 million weight updates. This net was used to evaluate the performance on the test set. This way of using the validation set was chosen because we found
that it was easy to set up hyperparameters so that early stopping was not required at all.
Therefore, once the hyperparameters were fixed, it made sense to combine the validation
and training sets and train for a very long time.
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Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
The architectures shown in Figure 4 include all combinations of 2, 3, and 4 layer networks
with 1024 and 2048 units in each layer. Thus, there are six architectures in all. For all the
architectures (including the ones reported in Table 2), we used p = 0.5 in all hidden layers
and p = 0.8 in the input layer. A final momentum of 0.95 and weight constraints with c = 2
was used in all the layers.
To test the limits of dropout’s regularization power, we also experimented with 2 and 3
layer nets having 4096 and 8192 units. 2 layer nets gave improvements as shown in Table 2.
However, the three layer nets performed slightly worse than 2 layer ones with the same
level of dropout. When we increased dropout, performance improved but not enough to
outperform the 2 layer nets.
B.2 SVHN
The SVHN data set consists of approximately 600,000 training images and 26,000 test
images. The training set consists of two parts—A standard labeled training set and another
set of labeled examples that are easy. A validation set was constructed by taking examples
from both the parts. Two-thirds of it were taken from the standard set (400 per class) and
one-third from the extra set (200 per class), a total of 6000 samples. This same process
is used by Sermanet et al. (2012). The inputs were RGB pixels normalized to have zero
mean and unit variance. Other preprocessing techniques such as global or local contrast
normalization or ZCA whitening did not give any noticeable improvements.
The best architecture that we found uses three convolutional layers each followed by
a max-pooling layer. The convolutional layers have 96, 128 and 256 filters respectively.
Each convolutional layer has a 5 × 5 receptive field applied with a stride of 1 pixel. Each
max pooling layer pools 3 × 3 regions at strides of 2 pixels. The convolutional layers are
followed by two fully connected hidden layers having 2048 units each. All units use the
rectified linear activation function. Dropout was applied to all the layers of the network
with the probability of retaining the unit being p = (0.9, 0.75, 0.75, 0.5, 0.5, 0.5) for the
different layers of the network (going from input to convolutional layers to fully connected
layers). In addition, the max-norm constraint with c = 4 was used for all the weights. A
momentum of 0.95 was used in all the layers. These hyperparameters were tuned using a
validation set. Since the training set was quite large, we did not combine the validation
set with the training set for final training. We reported test error of the model that had
smallest validation error.
B.3 CIFAR-10 and CIFAR-100
The CIFAR-10 and CIFAR-100 data sets consists of 50,000 training and 10,000 test images
each. They have 10 and 100 image categories respectively. These are 32 × 32 color images.
We used 5,000 of the training images for validation. We followed the procedure similar
to MNIST, where we found the best hyperparameters using the validation set and then
combined it with the training set. The images were preprocessed by doing global contrast
normalization in each color channel followed by ZCA whitening. Global contrast normalization means that for image and each color channel in that image, we compute the mean
of the pixel intensities and subtract it from the channel. ZCA whitening means that we
mean center the data, rotate it onto its principle components, normalize each component
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Dropout
and then rotate it back. The network architecture and dropout rates are same as that for
SVHN, except the learning rates for the input layer which had to be set to smaller values.
B.4 TIMIT
The open source Kaldi toolkit (Povey et al., 2011) was used to preprocess the data into logfilter banks. A monophone system was trained to do a forced alignment and to get labels for
speech frames. Dropout neural networks were trained on windows of 21 consecutive frames
to predict the label of the central frame. No speaker dependent operations were performed.
The inputs were mean centered and normalized to have unit variance.
We used probability of retention p = 0.8 in the input layers and 0.5 in the hidden layers.
Max-norm constraint with c = 4 was used in all the layers. A momentum of 0.95 with a
high learning rate of 0.1 was used. The learning rate was decayed as 0 (1 + t/T )−1 . For
DBN pretraining, we trained RBMs using CD-1. The variance of each input unit for the
Gaussian RBM was fixed to 1. For finetuning the DBN with dropout, we found that in
order to get the best results it was important to use a smaller learning rate (about 0.01).
Adding max-norm constraints did not give any improvements.
B.5 Reuters
The Reuters RCV1 corpus contains more than 800,000 documents categorized into 103
classes. These classes are arranged in a tree hierarchy. We created a subset of this data set
consisting of 402,738 articles and a vocabulary of 2000 words comprising of 50 categories
in which each document belongs to exactly one class. The data was split into equal sized
training and test sets. We tried many network architectures and found that dropout gave
improvements in classification accuracy over all of them. However, the improvement was
not as significant as that for the image and speech data sets. This might be explained by
the fact that this data set is quite big (more than 200,000 training examples) and overfitting
is not a very serious problem.
B.6 Alternative Splicing
The alternative splicing data set consists of data for 3665 cassette exons, 1014 RNA features
and 4 tissue types derived from 27 mouse tissues. For each input, the target consists of 4
softmax units (one for tissue type). Each softmax unit has 3 states (inc, exc, nc) which are
of the biological importance. For each softmax unit, the aim is to predict a distribution over
these 3 states that matches the observed distribution from wet lab experiments as closely
as possible. The evaluation metric is Code Quality which is defined as
|data points|
X
i=1
X
X
t∈tissue types s∈{inc, exc, nc}
psi,t log(
qts (ri )
),
p̄s
where, psi,t is the target probability for state s and tissue type t in input i; qts (ri ) is the
predicted probability for state s in tissue type t for input ri and p̄s is the average of psi,t
over i and t.
A two layer dropout network with 1024 units in each layer was trained on this data set.
A value of p = 0.5 was used for the hidden layer and p = 0.7 for the input layer. Max-norm
regularization with high decaying learning rates was used. Results were averaged across the
same 5 folds used by Xiong et al. (2011).
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Srivastava, Hinton, Krizhevsky, Sutskever and Salakhutdinov
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