Stochastic Pooling for Regularization of Deep Convolutional Neural
    Networks
Authors: Matthew Zeiler, Rob Fergus
ICLR 2013

Abstract: We introduce a simple and effective method for regularizing large convolutional neural networks. We replace the conventional deterministic pooling operations with a stochastic procedure, randomly picking the activation within each pooling region according to a multinomial distribution, given by the activities within the pooling region. The approach is hyper-parameter free and can be combined with other regularization approaches, such as dropout and data augmentation. We achieve state-of-the-art performance on four image datasets, relative to other approaches that do not utilize data augmentation.

Stochastic Pooling for Regularization of
Deep Convolutional Neural Networks
Matthew D. Zeiler
Department of Computer Science
Courant Institute, New York University
zeiler@cs.nyu.edu
Rob Fergus
Department of Computer Science
Courant Institute, New York University
fergus@cs.nyu.edu
Abstract
We introduce a simple and effective method for regularizing large convolutional
neural networks. We replace the conventional deterministic pooling operations
with a stochastic procedure, randomly picking the activation within each pool-
ing region according to a multinomial distribution, given by the activities within
the pooling region. The approach is hyper-parameter free and can be combined
with other regularization approaches, such as dropout and data augmentation. We
achieve state-of-the-art performance on four image datasets, relative to other ap-
proaches that do not utilize data augmentation.
1 Introduction
Neural network models are prone to over-ﬁtting due to their high capacity. A range of regularization
techniques are used to prevent this, such as weight decay, weight tying and the augmentation of
the training set with transformed copies [9]. These allow the training of larger capacity models
than would otherwise be possible, which yield superior test performance compared to smaller un-
regularized models.
Dropout, recently proposed by Hintonet al.[2], is another regularization approach that stochastically
sets half the activations within a layer to zero for each training sample during training. It has been
shown to deliver signiﬁcant gains in performance across a wide range of problems, although the
reasons for its efﬁcacy are not yet fully understood.
A drawback to dropout is that it does not seem to have the same beneﬁts for convolutional layers,
which are common in many networks designed for vision tasks. In this paper, we propose a novel
type of regularization for convolutional layers that enables the training of larger models without
over-ﬁtting, and produces superior performance on recognition tasks.
The key idea is to make the pooling that occurs in each convolutional layer a stochastic process.
Conventional forms of pooling such as average and max are deterministic, the latter selecting the
largest activation in each pooling region. In our stochastic pooling, the selected activation is drawn
from a multinomial distribution formed by the activations within the pooling region.
An alternate view of stochastic pooling is that it is equivalent to standard max pooling but with many
copies of an input image, each having small local deformations. This is similar to explicit elastic
deformations of the input images [13], which delivers excellent MNIST performance. Other types
of data augmentation, such as ﬂipping and cropping differ in that they are global image transforma-
tions. Furthermore, using stochastic pooling in a multi-layer model gives an exponential number of
deformations since the selections in higher layers are independent of those below.
1
arXiv:1301.3557v1  [cs.LG]  16 Jan 2013
2 Review of Convolutional Networks
Our stochastic pooling scheme is designed for use in a standard convolutional neural network archi-
tecture. We ﬁrst review this model, along with conventional pooling schemes, before introducing
our novel stochastic pooling approach.
A classical convolutional network is composed of alternating layers of convolution and pooling
(i.e. subsampling). The aim of the ﬁrst convolutional layer is to extract patterns found within local
regions of the input images that are common throughout the dataset. This is done by convolving a
template or ﬁlter over the input image pixels, computing the inner product of the template at every
location in the image and outputting this as a feature map c, for each ﬁlter in the layer. This output
is a measure of how well the template matches each portion of the image. A non-linear functionf()
is then applied element-wise to each feature map c: a = f(c). The resulting activations aare then
passed to the pooling layer. This aggregates the information within a set of small local regions, R,
producing a pooled feature map s(of smaller size) as output. Denoting the aggregation function as
pool(), for each feature map cwe have:
sj = pool(f(ci)) ∀i∈Rj (1)
where Rj is pooling region jin feature map cand iis the index of each element within it.
The motivation behind pooling is that the activations in the pooled map sare less sensitive to the
precise locations of structures within the image than the original feature map c. In a multi-layer
model, the convolutional layers, which take the pooled maps as input, can thus extract features
that are increasingly invariant to local transformations of the input image. This is important for
classiﬁcation tasks, since these transformations obfuscate the object identity.
A range of functions can be used for f(), with tanh() and logistic functions being popular choices.
In this is paper we use a linear rectiﬁcation function f(c) = max(0,c) as the non-linearity. In
general, this has been shown [10] to have signiﬁcant beneﬁts over tanh() or logistic functions.
However, it is especially suited to our pooling mechanism since: (i) our formulation involves the
non-negativity of elements in the pooling regions and (ii) the clipping of negative responses intro-
duces zeros into the pooling regions, ensuring that the stochastic sampling is selecting from a few
speciﬁc locations (those with strong responses), rather than all possible locations in the region.
There are two conventional choices for pool(): average and max. The former takes the arithmetic
mean of the elements in each pooling region:
sj = 1
|Rj|
∑
i∈Rj
ai (2)
while the max operation selects the largest element:
sj = max
i∈Rj
ai (3)
Both types of pooling have drawbacks when training deep convolutional networks. In average pool-
ing, all elements in a pooling region are considered, even if many have low magnitude. When com-
bined with linear rectiﬁcation non-linearities, this has the effect of down-weighting strong activa-
tions since many zero elements are included in the average. Even worse, withtanh() non-linearities,
strong positive and negative activations can cancel each other out, leading to small pooled responses.
While max pooling does not suffer from these drawbacks, we ﬁnd it easily overﬁts the training set
in practice, making it hard to generalize well to test examples. Our proposed pooling scheme has
the advantages of max pooling but its stochastic nature helps prevent over-ﬁtting.
3 Stochastic Pooling
In stochastic pooling, we select the pooled map response by sampling from a multinomial distri-
bution formed from the activations of each pooling region. More precisely, we ﬁrst compute the
probabilities pfor each region jby normalizing the activations within the region:
pi = ai∑
k∈Rj ak
(4)
2
We then sample from the multinomial distribution based on pto pick a location lwithin the region.
The pooled activation is then simply al:
sj = al where l∼P(p1,...,p |Rj |) (5)
The procedure is illustrated in Fig. 1. The samples for each pooling region in each layer for each
training example are drawn independently to one another. When back-propagating through the
network this same selected location lis used to direct the gradient back through the pooling region,
analogous to back-propagation with max pooling.
Max pooling only captures the strongest activation of the ﬁlter template with the input for each
region. However, there may be additional activations in the same pooling region that should be
taken into account when passing information up the network and stochastic pooling ensures that
these non-maximal activations will also be utilized.
★!
a)!Image!
b)!Filter!
c)!Rec0ﬁed!Linear! e)!Probabili0es,!pi 
0! 0! 0!
0!0!
0!0!
1.6!
2.4!
0! 0! 0!
0!0!
0!0!
0.4!
0.6!
d)!Ac0va0ons,!ai 
1.6!
f)!Sampled!!
!!!!Ac0va0on,!s!
Sample!a!loca0on!
from!P():!e.g.!!l = 1 
Figure 1: Toy example illustrating stochastic pooling. a) Input image. b) Convolutional ﬁlter. c)
Rectiﬁed linear function. d) Resulting activations within a given pooling region. e) Probabilities
based on the activations. f) Sampled activation. Note that the selected element for the pooling region
may not be the largest element. Stochastic pooling can thus represent multi-modal distributions of
activations within a region.
3.1 Probabilistic Weighting at Test Time
Using stochastic pooling at test time introduces noise into the network’s predictions which we found
to degrade performance (see Section 4.7). Instead, we use a probabilistic form of averaging. In this,
the activations in each region are weighted by the probability pi (see Eqn. 4) and summed:
sj =
∑
i∈Rj
piai (6)
This differs from standard average pooling because each element has a potentially different weight-
ing and the denominator is the sum of activations ∑
i∈Rj ai, rather than the pooling region size
|Rj|. In practice, using conventional average (or sum) pooling results in a huge performance drop
(see Section 4.7).
Our probabilistic weighting can be viewed as a form of model averaging in which each setting of
the locations l in the pooling regions deﬁnes a new model. At training time, sampling to get new
locations produces a new model since the connection structure throughout the network is modiﬁed.
At test time, using the probabilities instead of sampling, we effectively get an estimate of averaging
over all of these possible models without having to instantiate them. Given a network architecture
with ddifferent pooling regions, each of size n, the number of possible models is nd where dcan
be in the 104-106 range and nis typically 4,9, or 16 for example (corresponding to 2 ×2, 3 ×3 or
4 ×4 pooling regions). This is a signiﬁcantly larger number than the model averaging that occurs
in dropout [2], where n = 2 always (since an activation is either present or not). In Section 4.7
we conﬁrm that using this probability weighting achieves similar performance compared to using a
large number of model instantiations, while requiring only one pass through the network.
Using the probabilities for sampling at training time and for weighting the activations at test time
leads to state-of-the-art performance on many common benchmarks, as we now demonstrate.
3
CIFAR&100)
CIFAR&10)
SVHN)
 MNIST)
mean) Local)CN) mean) mean)
a)) d)) e)) g))
h))f))c))b)) !" !" !"
Figure 2: A selection of images from each of the datasets we evaluated. The top row shows the
raw images while the bottom row are the preprocessed versions of the images we used for training.
The CIFAR datasets (f,h) show slight changes by subtracting the per pixel mean, whereas SVHN
(b) is almost indistinguishable from the original images. This prompted the use of local contrast
normalization (c) to normalize the extreme brightness variations and color changes for SVHN.
4 Experiments
4.1 Overview
We compare our method to average and max pooling on a variety of image classiﬁcation tasks. In
all experiments we use mini-batch gradient descent with momentum to optimize the cross entropy
between our network’s prediction of the class and the ground truth labels. For a given parameter x
at time tthe weight updates added to the parameters, ∆xt are ∆xt = 0.9∆xt−1 −ϵgt where gt is
the gradient of the cost function with respect to that parameter at time taveraged over the batch and
ϵis a learning rate set by hand.
All experiments were conducted using an extremely efﬁcient C++ GPU convolution library [6]
wrapped in MATLAB using GPUmat [14], which allowed for rapid development and experimenta-
tion. We begin with the same network layout as in Hinton et al.’s dropout work [2], which has 3
convolutional layers with 5x5 ﬁlters and 64 feature maps per layer with rectiﬁed linear units as their
outputs. We use this same model and train for 280 epochs in all experiments aside from one addi-
tional model in Section 4.5 that has 128 feature maps in layer 3 and is trained for 500 epochs. Unless
otherwise speciﬁed we use 3 ×3 pooling with stride 2 (i.e. neighboring pooling regions overlap by
1 element along the borders) for each of the 3 pooling layers. Additionally, after each pooling layer
there is a response normalization layer (as in [2]), which normalizes the pooling outputs at each
location over a subset of neighboring feature maps. This typically helps training by suppressing
extremely large outputs allowed by the rectiﬁed linear units as well as helps neighboring features
communicate. Finally, we use a single fully-connected layer with soft-max outputs to produce the
network’s class predictions. We applied this model to four different datasets: MNIST, CIFAR-10,
CIFAR-100 and Street View House Numbers (SVHN), see Fig. 2 for examples images.
4.2 CIFAR-10
We begin our experiments with the CIFAR-10 dataset where convolutional networks and methods
such as dropout are known to work well [2, 5]. This dataset is composed of 10 classes of natural
images with 50,000 training examples in total, 5,000 per class. Each image is an RGB image of size
32x32 taken from the tiny images dataset and labeled by hand. For this dataset we scale to [0,1]
and follow Hinton et al.’s [2] approach of subtracting the per-pixel mean computed over the dataset
from each image as shown in Fig. 2(f).
4
50 100 150 200 2500
5
10
15
20
25
30
35
Epochs
% Error
 
 
Avg (train)
Avg (test)
Max (train)
Max (test)
Stochastic (train)
Stochastic (test)
Figure 3: CIFAR-10 train and test error rates throughout training for average, max, and stochastic
pooling. Max and average pooling test errors plateau as those methods overﬁt. With stochastic
pooling, training error remains higher while test errors continue to decrease.1
Cross-validating with a set of 5,000 CIFAR-10 training images, we found a good value for the
learning rate ϵto be 10−2 for convolutional layers and 1 for the ﬁnal softmax output layer. These
rates were annealed linearly throughout training to 1/100thof their original values. Additionally,
we found a small weight decay of 0.001 to be optimal and was applied to all layers. These hyper-
parameter settings found through cross-validation were used for all other datasets in our experiments.
Using the same network architecture described above, we trained three models using average, max
and stochastic pooling respectively and compare their performance. Fig. 3 shows the progression of
train and test errors over 280 training epochs. Stochastic pooling avoids over-ﬁtting, unlike average
and max pooling, and produces less test errors. Table 1 compares the test performance of the three
pooling approaches to the current state-of-the-art result on CIFAR-10 which uses no data augmenta-
tion but adds dropout on an additional locally connected layer [2]. Stochastic pooling surpasses this
result by 0.47% using the same architecture but without requiring the locally connected layer.
Train Error % Test Error %
3-layer Conv. Net [2] – 16.6
3-layer Conv. Net + 1 Locally Conn. layer with dropout [2] – 15.6
Avg Pooling 1.92 19.24
Max Pooling 0.0 19.40
Stochastic Pooling 3.40 15.13
Table 1: CIFAR-10 Classiﬁcation performance for various pooling methods in our model compared
to the state-of-the-art performance [2] with and without dropout.
To determine the effect of the pooling region size on the behavior of the system with stochastic
pooling, we compare the CIFAR-10 train and test set performance for 5x5, 4x4, 3x3, and 2x2 pooling
sizes throughout the network in Fig. 4. The optimal size appears to be 3x3, with smaller regions over-
ﬁtting and larger regions possibly being too noisy during training. At all sizes the stochastic pooling
outperforms both max and average pooling.
4.3 MNIST
The MNIST digit classiﬁcation task is composed of 28x28 images of the 10 handwritten digits [8].
There are 60,000 training images with 10,000 test images in this benchmark. The images are scaled
to [0,1] and we do not perform any other pre-processing.
During training, the error using both stochastic pooling and max pooling dropped quickly, but the
latter completely overﬁt the training data. Weight decay prevented average pooling from over-ﬁtting,
but had an inferior performance to the other two methods. Table 2 compares the three pooling ap-
proaches to state-of-the-art methods on MNIST, which also utilize convolutional networks. Stochas-
1Weight decay prevented training errors from reaching 0 with average and stochastic pooling methods and
required the high number of epochs for training. All methods performed slightly better with weight decay.
5
16.55	  
15.13	  
15.71	  
15.86	  
3.18	  
3.4	  
4.38	  
6.4	  
21.11	  
19.53	  
18.59	  
19.25	  
0	  
0	  
0	  
0	  
20.74	  
19.52	  
18.83	  
19.38	  
0.25	  
1.8	  
4.88	  
9.08	  
0	   5	   10	   15	   20	   25	  
2x2	  
3x3	  
4x4	  
5x5	  
%	  Error	  
Pooling	  Region	  Size	  
Avg	  Train	  
Avg	  Test	  
Max	  Train	  
Max	  Test	  
Stochas>c	  Train	  
Stochas>c	  Test	  
Figure 4: CIFAR-10 train and test error rates for various pooling region sizes with each method.
tic pooling outperforms all other methods that do not use data augmentation methods such as jittering
or elastic distortions [7]. The current state-of-the-art single model approach by Ciresanet al.[1] uses
elastic distortions to augment the original training set. As stochastic pooling is a different type of
regularization, it could be combined with data augmentation to further improve performance.
Train Error % Test Error %
2-layer Conv. Net + 2-layer Classiﬁer [3] – 0.53
6-layer Conv. Net + 2-layer Classiﬁer + elastic distortions [1] – 0.35
Avg Pooling 0.57 0.83
Max Pooling 0.04 0.55
Stochastic Pooling 0.33 0.47
Table 2: MNIST Classiﬁcation performance for various pooling methods. Rows 1 & 2 show the
current state-of-the-art approaches.
4.4 CIFAR-100
The CIFAR-100 dataset is another subset of the tiny images dataset, but with 100 classes [5]. There
are 50,000 training examples in total (500 per class) and 10,000 test examples. As with the CIFAR-
10, we scale to [0,1] and subtract the per-pixel mean from each image as shown in Fig. 2(h). Due
to the limited number of training examples per class, typical pooling methods used in convolutional
networks do not perform well, as shown in Table 3. Stochastic pooling outperforms these methods
by preventing over-ﬁtting and surpasses what we believe to be the state-of-the-art method by2.66%.
Train Error % Test Error %
Receptive Field Learning [4] – 45.17
Avg Pooling 11.20 47.77
Max Pooling 0.17 50.90
Stochastic Pooling 21.22 42.51
Table 3: CIFAR-100 Classiﬁcation performance for various pooling methods compared to the state-
of-the-art method based on receptive ﬁeld learning.
4.5 Street View House Numbers
The Street View House Numbers (SVHN) dataset is composed of 604,388 images (using both the
difﬁcult training set and simpler extra set) and 26,032 test images [11]. The goal of this task is to
classify the digit in the center of each cropped 32x32 color image. This is a difﬁcult real world
problem since multiple digits may be visible within each image. The practical application of this is
to classify house numbers throughout Google’s street view database of images.
We found that subtracting the per-pixel mean from each image did not really modify the statistics
of the images (see Fig. 2(b)) and left large variations of brightness and color that could make clas-
6
siﬁcation more difﬁcult. Instead, we utilized local contrast normalization (as in [12]) on each of the
three RGB channels to pre-process the images Fig. 2(c). This normalized the brightness and color
variations and helped training proceed quickly on this relatively large dataset.
Despite having signiﬁcant amounts of training data, a large convolutional network can still overﬁt.
For this dataset, we train an additional model for 500 epochs with 64, 64 and 128 feature maps in
layers 1, 2 and 3 respectively. Our stochastic pooling helps to prevent overﬁtting even in this large
model (denoted 64-64-128 in Table 4), despite training for a long time. The existing state-of-the-
art on this dataset is the multi-stage convolutional network of Sermanet et al.[12], but stochastic
pooling beats this by 2.10% (relative gain of 43%).
Train Error % Test Error %
Multi-Stage Conv. Net + 2-layer Classiﬁer [12] – 5.03
Multi-Stage Conv. Net + 2-layer Classifer + padding [12] – 4.90
64-64-64 Avg Pooling 1.83 3.98
64-64-64 Max Pooling 0.38 3.65
64-64-64 Stochastic Pooling 1.72 3.13
64-64-128 Avg Pooling 1.65 3.72
64-64-128 Max Pooling 0.13 3.81
64-64-128 Stochastic Pooling 1.41 2.80
Table 4: SVHN Classiﬁcation performance for various pooling methods in our model with 64 or
128 layer 3 feature maps compared to state-of-the-art results with and without data augmentation.
4.6 Reduced Training Set Size
To further illustrate the ability of stochastic pooling to prevent over-ﬁtting, we reduced the training
set size on MINST and CIFAR-10 datasets. Fig. 5 shows test performance when training on a
random selection of only 1000, 2000, 3000, 5000, 10000, half, or the full training set. In most cases,
stochastic pooling overﬁts less than the other pooling approaches.
1000 2000 3000 5000 10000 30000 600000
1
2
3
4
5
6
7
8
9
# of Training Cases
% Error
 
 
Avg
Max
Stochastic
1000 2000 3000 5000 10000 25000 5000015
20
25
30
35
40
45
50
55
60
65
# of Training Cases
% Error
 
 
Avg
Max
Stochastic
Figure 5: Test error when training with reduced dataset sizes on MNIST (left) and CIFAR-10 (right).
Stochastic pooling generally overﬁts the least.
4.7 Importance of Model Averaging
To analyze the importance of stochastic sampling at training time and probability weighting at test
time, we use different methods of pooling when training and testing on CIFAR-10 (see Table 5).
Choosing the locations stochastically at test time degrades performance slightly as could be ex-
pected, however it still outperforms models where max or average pooling are used at test time. To
conﬁrm that probability weighting is a valid approximation to averaging many models, we draw N
samples of the pooling locations throughout the network and average the output probabilities from
those N models (denoted Stochastic-N in Table 5). As N increases, the results approach the prob-
ability weighting method, but have the obvious downside of an N-fold increase in computations.
Using a model trained with max or average pooling and using stochastic pooling at test time per-
forms poorly. This suggests that training with stochastic pooling, which incorporates non-maximal
elements and sampling noise, makes the model more robust at test time. Furthermore, if these non-
maximal elements are not utilized correctly or the scale produced by the pooling function is not
correct, such as if average pooling is used at test time, a drastic performance hit is seen.
7
When using probability weighting during training, the network easily over-ﬁts and performs sub-
optimally at test time using any of the pooling methods. However, the beneﬁts of probability
weighting at test time are seen when the model has speciﬁcally been trained to utilize it through
either probability weighting or stochastic pooling at training time.
Train Method Test Method Train Error % Test Error %
Stochastic Pooling Probability Weighting 3.20 15.20
Stochastic Pooling Stochastic Pooling 3.20 17.49
Stochastic Pooling Stochastic-10 Pooling 3.20 15.51
Stochastic Pooling Stochastic-100 Pooling 3.20 15.12
Stochastic Pooling Max Pooling 3.20 17.66
Stochastic Pooling Avg Pooling 3.20 53.50
Probability Weighting Probability Weighting 0.0 19.40
Probability Weighting Stochastic Pooling 0.0 24.00
Probability Weighting Max Pooling 0.0 22.45
Probability Weighting Avg Pooling 0.0 58.97
Max Pooling Max Pooling 0.0 19.40
Max Pooling Stochastic Pooling 0.0 32.75
Max Pooling Probability Weighting 0.0 30.00
Avg Pooling Avg Pooling 1.92 19.24
Avg Pooling Stochastic Pooling 1.92 44.25
Avg Pooling Probability Weighting 1.92 40.09
Table 5: CIFAR-10 Classiﬁcation performance for various train and test combinations of pooling
methods. The best performance is obtained by using stochastic pooling when training (to prevent
over-ﬁtting), while using the probability weighting at test time.
4.8 Visualizations
Some insight into the mechanism of stochastic pooling can be gained by using a deconvolutional
network of Zeiler et al.[15] to provide a novel visualization of our trained convolutional network.
The deconvolutional network has the same components (pooling, ﬁltering) as a convolutional net-
work but are inverted to act as a top-down decoder that maps the top-layer feature maps back to the
input pixels. The unpooling operation uses the stochastically chosen locations selected during the
forward pass. The deconvolution network ﬁlters (now applied to the feature maps, rather than the
input) are the transpose of the feed-forward ﬁlters, as in an auto-encoder with tied encoder/decoder
weights. We repeat this top-down process until the input pixel level is reached, producing the vi-
sualizations in Fig. 6. With max pooling, many of the input image edges are present, but average
pooling produces a reconstruction with no discernible structure. Fig. 6(a) shows 16 examples of
pixel-space reconstructions for different location samples throughout the network. The reconstruc-
tions are similar to the max pooling case, but as the pooling locations change they result in small
local deformations of the visualized image.
Despite the stochastic nature of the model, the multinomial distributions effectively capture the reg-
ularities of the data. To demonstrate this, we compare the outputs produced by a deconvolutional
network when sampling using the feedforward (FF) proabilities versus sampling from uniform (UN)
distributions. In contrast to Fig. 6(a) which uses only feedforward proabilities, Fig. 6(b-h) replace
one or more of the pooling layers’ distributions with uniform distributions. The feed forward proba-
bilities encode signiﬁcant structural information, especially in the lower layers of the model. Addi-
tional visualizations and videos of the sampling process are provided as supplementary material at
www.matthewzeiler.com/pubs/iclr2013/.
5 Discussion
We propose a simple and effective stochastic pooling strategy that can be combined with any other
forms of regularization such as weight decay, dropout, data augmentation, etc. to prevent over-
ﬁtting when training deep convolutional networks. The method is also intuitive, selecting from
information the network is already providing, as opposed to methods such as dropout which throw
information away. We show state-of-the-art performance on numerous datasets, when comparing to
other approaches that do not employ data augmentation. Furthermore, our method has negligible
computational overhead and no hyper-parameters to tune, thus can be swapped into to any existing
convolutional network architecture.
8
a)	  FF(3)	  –	  FF(2)	  –	  FF(1)	  	   b)	  UN(3)	  –	  FF(2)	  –	  FF(1)	   c)	  FF(3)	  –	  UN(2)	  –	  FF(1)	   d)	  FF(3)	  –	  FF(2)	  –	  UN(1)	  
h)	  UN(3)	  –	  UN(2)	  –	  UN(1)	  
e)	  UN(3)	  –	  UN(2)	  –	  FF(1)	  
 g)	  FF(3)	  –	  UN(2)	  –	  UN(1)	  
f)	  FF(3)	  –	  UN(2)	  –	  UN(1)	  
Image	  
Avg	  
Max	  
Figure 6: Top down visualizations from the third layer feature map activations for the horse image
(far left). Max and average pooling visualizations are also shown on the left. (a)–(h): Each image
in a 4x4 block is one instantiation of the pooling locations using stochastic pooling. For sampling
the locations, each layer (indicated in parenthesis) can either use: (i) the multinomial distribution
over a pooling region derived from the feed-forward (FF) activations as in Eqn. 4, or (ii) a uniform
(UN) distribution. We can see that the feed-forward probabilities encode much of the structure in
the image, as almost all of it is lost when uniform sampling is used, especially in the lower layers.
References
[1] D. C. Ciresan, U. Meier, J. Masci, L. M. Gambardella, and J. Schmidhuber. Flexible, high
performance convolutional neural networks for image classiﬁcation. In IJCAI, 2011.
[2] G.E. Hinton, N. Srivastave, A. Krizhevsky, I. Sutskever, and R. R. Salakhutdinov. Improving
neural networks by preventing co-adaptation of feature detectors. arXiv:1207.0580, 2012.
[3] K. Jarrett, K. Kavukcuoglu, M. Ranzato, and Y . LeCun. What is the best multi-stage architec-
ture for object recognition? In ICCV, 2009.
[4] Y . Jia and C. Huang. Beyond spatial pyramids: Receptive ﬁeld learning for pooled image
features. In NIPS Workshops, 2011.
[5] A. Krizhevsky. Learning multiple layers of featurs from tiny images. Technical Report TR-
2009, University of Toronto, 2009.
[6] A. Krizhevsky. cuda-convnet. http://code.google.com/p/cuda-convnet/, 2012.
[7] Y . LeCun. The MNIST database. http://yann.lecun.com/exdb/mnist/, 2012.
[8] Y . LeCun, L. Bottou, Y . Bengio, and P. Haffner. Gradient-based learning applied to document
recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
[9] G. Montavon, G. Orr, and K.-R. Muller, editors. Neural Networks: Tricks of the Trade.
Springer, San Francisco, 2012.
[10] V . Nair and G.E. Hinton. Rectiﬁed linear units improve restricted boltzmann machines. In
ICML, 2010.
[11] Y . Netzer, T. Wang, A. Coates, A. Bissacco, B. Wu, and A. Y . Ng. Reading digits in natural
images with unsupervised feature learning. In NIPS Workshop, 2011.
[12] P. Sermanet, S. Chintala, and Y . LeCun. Convolutional neural networks applied to house
numbers digit classiﬁcation. In ICPR, 2012.
[13] P. Simard, D. Steinkraus, and J. Platt. Best practices for convolutional neural networks applied
to visual document analysis. In ICDAR, 2003.
[14] http://gp-you.org/. GPUmat. http://sourceforge.net/projects/
gpumat/, 2012.
[15] M. Zeiler, G. Taylor, and R. Fergus. Adaptive deconvolutional networks for mid and high level
feature learning. In ICCV, 2011.
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Peer review:
[Review]:
The authors introduce a stochastic pooling method in the context of
convolutional neural networks, which replaces the traditionally used
average or max pooling operators. In the stochastic pooling a
multinomial distribution is created from input activations and used to
select the index of the activation to pass to the next layer of the
network. On first read, this method resembled that of 'probabilistic max
pooling' by Lee et. al in 'Convolutional Deep Belief Networks for
Scalable Unsupervised Learning of Hierarchical Representations',
however the context and execution are different.

During testing, the authors employ a separate pooling function that is a
weighted sum of the input activations and their corresponding
probabilities that would be used for index selection during training.
This pooling operator is speculated to work as a form of
regularization through model averaging. The authors substantiate this claim with results averaging multiple samples at each pool of the stochastic architectures and visualizations of images obtained from
reconstructions using deconvolutional networks.

Moreover, test set accuracies for this method are given for four
relevant datasets where it appears stochastic pooling CNNs are able to
achieve the best known performance on three. A good amount of detail
has been provided allowing the reader to reproduce the results.

As the sampling scheme proposed may be combined with other regularization techniques, it will be exciting to see how multiple forms of regularization can contribute or degrade test accuracies.

Some minor comments follow:

- Mini-batch size for training is not mentioned.

- Fig. 2 could be clearer on first read, e.g. if boxes were drawn
around (a,b,c), (e,f), and (g,h) to indicate they are operations on
the same dataset.

- In Section 4.2 it is noted that stochastic pooling avoids
over-fitting unlike averaging and max pooling, however in Fig. 3 it
certainly appears that the average and max techniques are not severely
over-fitting as in the typical network training case (with noticeable
degradation in test set performance). However, the network does train
to near zero error on the training set. It may be more accurate to state
that stochastic pooling promotes better generalization yet additional training epochs may make the over-fitting argument clearer.

- Fig. 3 also suggests that additional training may improve the final
reported test set error in the case of stochastic pooling. The
reference to state-of-the-art performance on CIFAR-10 is no longer
current.

- Section 4.8, sp 'proabilities'