Big Neural Networks Waste Capacity
Authors: Yann Dauphin, Yoshua Bengio
ICLR 2013

Abstract: This article exposes the failure of some big neural networks to leverage added capacity to reduce underfitting. Past research suggest diminishing returns when increasing the size of neural networks. Our experiments on ImageNet LSVRC-2010 show that this may be due to the fact that bigger networks underfit the training objective, sometimes performing worse on the training set than smaller networks. This suggests that the optimization method - first order gradient descent - fails at this regime. Directly attacking this problem, either through the optimization method or the choices of parametrization, may allow to improve the generalization error on large datasets, for which a large capacity is required.

Big Neural Networks Waste Capacity
Yann N. Dauphin & Yoshua Bengio
D´epartement d’informatique et de recherche op´erationnelle
Universit´e de Montr´eal, Montr´eal, QC, Canada
Abstract
This article exposes the failure of some big neural networks to leverage added
capacity to reduce underﬁtting. Past research suggest diminishing returns when
increasing the size of neural networks. Our experiments on ImageNet LSVRC-
2010 show that this may be due to the fact there are highly diminishing returns for
capacity in terms of training error, leading to underﬁtting. This suggests that the
optimization method - ﬁrst order gradient descent - fails at this regime. Directly
attacking this problem, either through the optimization method or the choices of
parametrization, may allow to improve the generalization error on large datasets,
for which a large capacity is required.
1 Introduction
Deep learning and neural networks have achieved state-of-the-art results on vision 1, language 2 ,
and audio-processing tasks 3. All these cases involved fairly large datasets, but in all these cases,
even larger ones could be used. One of the major challenges remains to extend neural networks on
a much larger scale, and with this objective in mind, this paper asks a simple question: is there an
optimization issue that prevents efﬁciently training larger networks?
Prior evidence of the failure of big networks in the litterature can be found for example in Coates
et al. (2011), which shows that increasing the capacity of certain neural net methods quickly reaches
a point of diminishing returns on the test error. These results have since been extended to other
types of auto-encoders and RBMs (Rifai et al., 2011; Courville et al., 2011). Furthermore, Coates
et al. (2011) shows that while neural net methods fail to leverage added capacity K-Means can.
This has allowed K-Means to reach state-of-the-art performance on CIFAR-10 for methods that do
not use artiﬁcial transformations. This is an unexpected result because K-Means is a much dumber
unsupervised learning algorithm when compared with RBMs and regularized auto-encoders. Coates
et al. (2011) argues that this is mainly due to K-Means making better use of added capacity, but it
does not explain why the neural net methods failed to do this.
2 Experimental Setup
We will perform experiments with the well known ImageNet LSVRC-2010 object detection dataset4.
The subset used in the Large Scale Visual Recognition Challenge 2010 contains 1000 object cate-
gories and 1.2 million training images.
1Krizhevsky et al. (2012) reduced by almost one half the error rate on the 1000-class ImageNet object
recognition benchmark
2Mikolov et al. (2011) reduced perplexity on WSJ by 40% and speech recognition absolute word error rate
by > 1%.
3For speech recognition, Seide et al. (2011) report relative word error rates decreasing by about 30% on
datasets of 309 hours
4http://www.image-net.org/challenges/LSVRC/2010/
1
arXiv:1301.3583v4  [cs.LG]  14 Mar 2013
This dataset has many attractive features:
1. The task is difﬁcult enough for current algorithms that there is still room for much improve-
ment. For instance, Krizhevsky et al. (2012) was able to reduce the error by half recently.
What’s more the state-of-the-art is at 15.3% error. Assuming minimal error in the human
labelling of the dataset, it should be possible to reach errors close to 0%.
2. Improvements on ImageNet are thought to be a good proxy for progress in object recogni-
tion (Deng et al., 2009).
3. It has a large number of examples. This is the setting that is commonly found in industry
where datasets reach billions of examples. Interestingly, as you increase the amount of
data, the training error converges to the generalization error. In other words, reducing
training error is well correlated with reducing generalization error, when large datasets are
available. Therefore, it stands to reason that resolving underﬁtting problems may yield
signiﬁcant improvements.
We use the features provided by the Large Scale Visual Recognition Challenge 20105. The images
are convolved with SIFT features, then K-Means is used to form a visual vocabulary of 1000 visual
words. Following the litterature, we report the Top-5 error rate only.
The experiments focus on the behavior of Multi-Layer Perceptrons (MLP) as capacity is increased.
This is done by increasing the number of hidden units in the network. The ﬁnal classiﬁcation layer of
the network is a softmax over possible classes ( softmax(x) = e−x/∑
i e−xi ). The hidden layers
use the logistic sigmoid activation function ( σ(x) = 1/(1+e−x)). We initialize the weights of the
hidden layer according to the formula proposed by Glorot and Bengio (2010). The parameters of
the classiﬁcation layer are initialized to 0, along with all the bias (offset) parameters of the MLP.
The hyper-parameters to tune are the learning rate and the number of hidden units. We are in-
terested in optimization performance so we cross-validate them based on the training error. We
use a grid search with the learning rates taken from {0.1,0.01}and the number of hiddens from
{1000,2000,5000,7000,10000,15000}. When we report the performance of a network with a
given number of units we choose the best learning rate. The learning rate is decreased by 5% every-
time the training error goes up aftern an epoch. We do not use any regularization because it would
typically not help to decrease the training set error. The number of epochs is set to 300 s that it is
large enough for the networks to converge.
The experiments are run on a cluster of Nvidia Geforce GTX 580 GPUs with the help of the Theano
library (Bergstra et al., 2010). We make use of HDF5 (Folk et al., 2011) to load the dataset in a
lazy fashion because of its large size. The shortest training experiment took 10 hours to run and the
longest took 28 hours.
3 Experimental Results
Figure 1 shows the evolution of the training error as the capacity is increased. The common intuition
is that this increased capacity will help ﬁt the training set - possibly to the detriment of generalization
error. For this reason practitioners have focused mainly on the problem of overﬁtting the dataset
when dealing with large networks - not underﬁtting. In fact, much research is concerned with proper
regularization of such large networks (Hinton et al., 2012, 2006).
However, Figure 2 reveals a problem in the training of big networks. This ﬁgure is the derivative of
the curve in Figure 1 (using the number of errors instead of the percentage). It may be interpreted
as the return on investment (ROI) for the addition of capacity. The Figure shows that the return
on investment of additional hidden units decreases fast, where in fact we would like it to be close
to constant. Increasing the capacity from 1000 to 2000 units, the ROI decreases by an order of
magnitude. It is harder and harder for the model to make use of additional capacity. The red line is
a baseline where the additional unit is used as a template matcher for one of the training errors. In
this case, the number of errors reduced per unit is at least 1. We see that the MLP does not manage
to beat this baseline after 5000 units, for the given number of training iterations.
5http://www.image-net.org/challenges/LSVRC/2010/download-public
2
Figure 1: Training error with respect to the capacity of a 1-layer sigmoidal neural network. This
curve seems to suggest we are correctly leveraging added capacity.
Figure 2: Return on investment on the addition of hidden units for a 1-hidden layer sigmoidal neural
network. The vertical axis is the number of training errors removed per additional hidden unit, after
300 epochs. We see here that it is harder and harder to use added capacity.
For reference, we also include the learning curves of the networks used for Figure 1 and 2 in Figure
3. We see that the curves for capacities above 5000 all converge towards the same point.
4 Future directions
This rapidly decreasing return on investment for capacity in big networks seems to be a failure of
ﬁrst order gradient descent.
In fact, we know that the ﬁrst order approximation fails when there are a lot of interactions between
hidden units. It may be that adding units increases the interactions between units and causes the
Hessian to be ill-conditioned. This reasoning suggests two research directions:
• methods that break interactions between large numbers of units. This helps the Hessian
to be better conditioned and will lead to better effectiveness for ﬁrst-order descent. This
3
Figure 3: Training error with respect to the number of epochs of gradient descent. Each line is a
1-hidden layer sigmoidal neural network with a different number of hidden units.
type of method can be implemented efﬁciently. Examples of this approach are sparsity and
orthognality penalties.
•methods that model interactions between hidden units. For example, second order methods
(Martens, 2010) and natural gradient methods (Le Roux et al., 2008). Typically, these are
expensive approaches and the challenge is in scaling them to large datasets, where stochas-
tic gradient approaches may dominate. The ideal target is a stochastic natural gradient or
stochastic second-order method.
The optimization failure may also be due to other reasons. For example, networks with more ca-
pacity have more local minima. Future work should investigate tests that help discriminate between
ill-conditioning issues and local minima issues.
Fixing this optimization problem may be the key to unlocking better performance of deep networks.
Based on past observations (Bengio, 2009; Erhan et al., 2010), we expect this optimization problem
to worsen for deeper networks, and our experimental setup should be extended to measure the effect
of depth. As we have noted earlier, improvements on the training set error should be well correlated
with generalization for large datasets.
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Peer review:
[Review]:
The net gets bigger, yet keeps underfitting the training set. Authors suspect that gradient descent is the culprit. An interesting study!