Layer Normalization
Jimmy Lei Ba Jamie Ryan Kiros Geoffrey E. Hinton
University of Toronto University of Toronto University of Toronto
jimmy@psi.toronto.edu rkiros@cs.toronto.edu and Google Inc.
hinton@cs.toronto.edu
arXiv:1607.06450v1 [stat.ML] 21 Jul 2016
Abstract
Training state-of-the-art, deep neural networks is computationally expensive. One
way to reduce the training time is to normalize the activities of the neurons. A
recently introduced technique called batch normalization uses the distribution of
the summed input to a neuron over a mini-batch of training cases to compute a
mean and variance which are then used to normalize the summed input to that
neuron on each training case. This significantly reduces the training time in feed-
forward neural networks. However, the effect of batch normalization is dependent
on the mini-batch size and it is not obvious how to apply it to recurrent neural net-
works. In this paper, we transpose batch normalization into layer normalization by
computing the mean and variance used for normalization from all of the summed
inputs to the neurons in a layer on a single training case. Like batch normalization,
we also give each neuron its own adaptive bias and gain which are applied after
the normalization but before the non-linearity. Unlike batch normalization, layer
normalization performs exactly the same computation at training and test times.
It is also straightforward to apply to recurrent neural networks by computing the
normalization statistics separately at each time step. Layer normalization is very
effective at stabilizing the hidden state dynamics in recurrent networks. Empiri-
cally, we show that layer normalization can substantially reduce the training time
compared with previously published techniques.
1 Introduction
Deep neural networks trained with some version of Stochastic Gradient Descent have been shown
to substantially outperform previous approaches on various supervised learning tasks in computer
vision [Krizhevsky et al., 2012] and speech processing [Hinton et al., 2012]. But state-of-the-art
deep neural networks often require many days of training. It is possible to speed-up the learning
by computing gradients for different subsets of the training cases on different machines or splitting
the neural network itself over many machines [Dean et al., 2012], but this can require a lot of com-
munication and complex software. It also tends to lead to rapidly diminishing returns as the degree
of parallelization increases. An orthogonal approach is to modify the computations performed in
the forward pass of the neural net to make learning easier. Recently, batch normalization [Ioffe and
Szegedy, 2015] has been proposed to reduce training time by including additional normalization
stages in deep neural networks. The normalization standardizes each summed input using its mean
and its standard deviation across the training data. Feedforward neural networks trained using batch
normalization converge faster even with simple SGD. In addition to training time improvement, the
stochasticity from the batch statistics serves as a regularizer during training.
Despite its simplicity, batch normalization requires running averages of the summed input statis-
tics. In feed-forward networks with fixed depth, it is straightforward to store the statistics separately
for each hidden layer. However, the summed inputs to the recurrent neurons in a recurrent neu-
ral network (RNN) often vary with the length of the sequence so applying batch normalization to
RNNs appears to require different statistics for different time-steps. Furthermore, batch normaliza-
tion cannot be applied to online learning tasks or to extremely large distributed models where the
minibatches have to be small.
This paper introduces layer normalization, a simple normalization method to improve the training
speed for various neural network models. Unlike batch normalization, the proposed method directly
estimates the normalization statistics from the summed inputs to the neurons within a hidden layer
so the normalization does not introduce any new dependencies between training cases. We show that
layer normalization works well for RNNs and improves both the training time and the generalization
performance of several existing RNN models.
2 Background
A feed-forward neural network is a non-linear mapping from a input pattern x to an output vector
y. Consider the lth hidden layer in a deep feed-forward, neural network, and let al be the vector
representation of the summed inputs to the neurons in that layer. The summed inputs are computed
through a linear projection with the weight matrix W l and the bottom-up inputs hl given as follows:
>
ali = wil hl hl+1
i = f (ali + bli ) (1)
where f (·) is an element-wise non-linear function and wil is the incoming weights to the ith hidden
units and bli is the scalar bias parameter. The parameters in the neural network are learnt using
gradient-based optimization algorithms with the gradients being computed by back-propagation.
One of the challenges of deep learning is that the gradients with respect to the weights in one layer
are highly dependent on the outputs of the neurons in the previous layer especially if these outputs
change in a highly correlated way. Batch normalization [Ioffe and Szegedy, 2015] was proposed
to reduce such undesirable “covariate shift”. The method normalizes the summed inputs to each
hidden unit over the training cases. Specifically, for the ith summed input in the lth layer, the batch
normalization method rescales the summed inputs according to their variances under the distribution
of the data
gl
r h 2 i
āli = il ali − µli µli = E
l
σil =
ai E ali − µli (2)
σi x∼P (x) x∼P (x)
where āli is normalized summed inputs to the ith hidden unit in the lth layer and gi is a gain parame-
ter scaling the normalized activation before the non-linear activation function. Note the expectation
is under the whole training data distribution. It is typically impractical to compute the expectations
in Eq. (2) exactly, since it would require forward passes through the whole training dataset with the
current set of weights. Instead, µ and σ are estimated using the empirical samples from the current
mini-batch. This puts constraints on the size of a mini-batch and it is hard to apply to recurrent
neural networks.
3 Layer normalization
We now consider the layer normalization method which is designed to overcome the drawbacks of
batch normalization.
Notice that changes in the output of one layer will tend to cause highly correlated changes in the
summed inputs to the next layer, especially with ReLU units whose outputs can change by a lot.
This suggests the “covariate shift” problem can be reduced by fixing the mean and the variance of
the summed inputs within each layer. We, thus, compute the layer normalization statistics over all
the hidden units in the same layer as follows:
v
H u
u1 X H
l 1 X
l l
2
µ = ai σ =t ali − µl (3)
H i=1 H i=1
where H denotes the number of hidden units in a layer. The difference between Eq. (2) and Eq. (3)
is that under layer normalization, all the hidden units in a layer share the same normalization terms
µ and σ, but different training cases have different normalization terms. Unlike batch normalization,
layer normaliztion does not impose any constraint on the size of a mini-batch and it can be used in
the pure online regime with batch size 1.
2
3.1 Layer normalized recurrent neural networks
The recent sequence to sequence models [Sutskever et al., 2014] utilize compact recurrent neural
networks to solve sequential prediction problems in natural language processing. It is common
among the NLP tasks to have different sentence lengths for different training cases. This is easy to
deal with in an RNN because the same weights are used at every time-step. But when we apply batch
normalization to an RNN in the obvious way, we need to to compute and store separate statistics for
each time step in a sequence. This is problematic if a test sequence is longer than any of the training
sequences. Layer normalization does not have such problem because its normalization terms depend
only on the summed inputs to a layer at the current time-step. It also has only one set of gain and
bias parameters shared over all time-steps.
In a standard RNN, the summed inputs in the recurrent layer are computed from the current input
xt and previous vector of hidden states ht−1 which are computed as at = Whh ht−1 + Wxh xt . The
layer normalized recurrent layer re-centers and re-scales its activations using the extra normalization
terms similar to Eq. (3):
v
hg H u
u1 X H
t t t
i
t 1 X
t t 2
h =f t a −µ +b µ = ai σ =t (at − µt ) (4)
σ H i=1 H i=1 i
where Whh is the recurrent hidden to hidden weights and Wxh are the bottom up input to hidden
weights. is the element-wise multiplication between two vectors. b and g are defined as the bias
and gain parameters of the same dimension as ht .
In a standard RNN, there is a tendency for the average magnitude of the summed inputs to the recur-
rent units to either grow or shrink at every time-step, leading to exploding or vanishing gradients. In
a layer normalized RNN, the normalization terms make it invariant to re-scaling all of the summed
inputs to a layer, which results in much more stable hidden-to-hidden dynamics.
4 Related work
Batch normalization has been previously extended to recurrent neural networks [Laurent et al., 2015,
Amodei et al., 2015, Cooijmans et al., 2016]. The previous work [Cooijmans et al., 2016] suggests
the best performance of recurrent batch normalization is obtained by keeping independent normal-
ization statistics for each time-step. The authors show that initializing the gain parameter in the
recurrent batch normalization layer to 0.1 makes significant difference in the final performance of
the model. Our work is also related to weight normalization [Salimans and Kingma, 2016]. In
weight normalization, instead of the variance, the L2 norm of the incoming weights is used to
normalize the summed inputs to a neuron. Applying either weight normalization or batch normal-
ization using expected statistics is equivalent to have a different parameterization of the original
feed-forward neural network. Re-parameterization in the ReLU network was studied in the Path-
normalized SGD [Neyshabur et al., 2015]. Our proposed layer normalization method, however, is
not a re-parameterization of the original neural network. The layer normalized model, thus, has
different invariance properties than the other methods, that we will study in the following section.
5 Analysis
In this section, we investigate the invariance properties of different normalization schemes.
5.1 Invariance under weights and data transformations
The proposed layer normalization is related to batch normalization and weight normalization. Al-
though, their normalization scalars are computed differently, these methods can be summarized as
normalizing the summed inputs ai to a neuron through the two scalars µ and σ. They also learn an
adaptive bias b and gain g for each neuron after the normalization.
gi
hi = f ( (ai − µi ) + bi ) (5)
σi
Note that for layer normalization and batch normalization, µ and σ is computed according to Eq. 2
and 3. In weight normalization, µ is 0, and σ = kwk2 .
3
Weight matrix Weight matrix Weight vector Dataset Dataset Single training case
re-scaling re-centering re-scaling re-scaling re-centering re-scaling
Batch norm Invariant No Invariant Invariant Invariant No
Weight norm Invariant No Invariant No No No
Layer norm Invariant Invariant No Invariant No Invariant
Table 1: Invariance properties under the normalization methods.
Table 1 highlights the following invariance results for three normalization methods.
Weight re-scaling and re-centering: First, observe that under batch normalization and weight
normalization, any re-scaling to the incoming weights wi of a single neuron has no effect on the
normalized summed inputs to a neuron. To be precise, under batch and weight normalization, if
the weight vector is scaled by δ, the two scalar µ and σ will also be scaled by δ. The normalized
summed inputs stays the same before and after scaling. So the batch and weight normalization are
invariant to the re-scaling of the weights. Layer normalization, on the other hand, is not invariant
to the individual scaling of the single weight vectors. Instead, layer normalization is invariant to
scaling of the entire weight matrix and invariant to a shift to all of the incoming weights in the
weight matrix. Let there be two sets of model parameters θ, θ0 whose weight matrices W and W 0
differ by a scaling factor δ and all of the incoming weights in W 0 are also shifted by a constant
vector γ, that is W 0 = δW + 1γ > . Under layer normalization, the two models effectively compute
the same output:
g g
h0 =f ( 0 (W 0 x − µ0 ) + b) = f ( 0 (δW + 1γ > )x − µ0 + b)
σ σ
g
=f ( (W x − µ) + b) = h. (6)
σ
Notice that if normalization is only applied to the input before the weights, the model will not be
invariant to re-scaling and re-centering of the weights.
Data re-scaling and re-centering: We can show that all the normalization methods are invariant
to re-scaling the dataset by verifying that the summed inputs of neurons stays constant under the
changes. Furthermore, layer normalization is invariant to re-scaling of individual training cases,
because the normalization scalars µ and σ in Eq. (3) only depend on the current input data. Let x0
be a new data point obtained by re-scaling x by δ. Then we have,
gi gi
h0i =f ( 0 wi> x0 − µ0 + bi ) = f ( δwi> x − δµ + bi ) = hi .
(7)
σ δσ
It is easy to see re-scaling individual data points does not change the model’s prediction under layer
normalization. Similar to the re-centering of the weight matrix in layer normalization, we can also
show that batch normalization is invariant to re-centering of the dataset.
5.2 Geometry of parameter space during learning
We have investigated the invariance of the model’s prediction under re-centering and re-scaling of
the parameters. Learning, however, can behave very differently under different parameterizations,
even though the models express the same underlying function. In this section, we analyze learning
behavior through the geometry and the manifold of the parameter space. We show that the normal-
ization scalar σ can implicitly reduce learning rate and makes learning more stable.
5.2.1 Riemannian metric
The learnable parameters in a statistical model form a smooth manifold that consists of all possible
input-output relations of the model. For models whose output is a probability distribution, a natural
way to measure the separation of two points on this manifold is the Kullback-Leibler divergence
between their model output distributions. Under the KL divergence metric, the parameter space is a
Riemannian manifold.
The curvature of a Riemannian manifold is entirely captured by its Riemannian metric, whose
quadratic form is denoted as ds2 . That is the infinitesimal distance in the tangent space at a point
in the parameter space. Intuitively, it measures the changes in the model output from the parameter
space along a tangent direction. The Riemannian metric under KL was previously studied [Amari,
1998] and was shown to be well approximated under second order Taylor expansion using the Fisher
4
information matrix:
1
ds2 = DKL P (y | x; θ)kP (y | x; θ + δ) ≈ δ > F (θ)δ,
(8)
" 2 #
>
∂ log P (y | x; θ) ∂ log P (y | x; θ)
F (θ) = E , (9)
x∼P (x),y∼P (y | x) ∂θ ∂θ
where, δ is a small change to the parameters. The Riemannian metric above presents a geometric
view of parameter spaces. The following analysis of the Riemannian metric provides some insight
into how normalization methods could help in training neural networks.
5.2.2 The geometry of normalized generalized linear models
We focus our geometric analysis on the generalized linear model. The results from the following
analysis can be easily applied to understand deep neural networks with block-diagonal approxima-
tion to the Fisher information matrix, where each block corresponds to the parameters for a single
neuron.
A generalized linear model (GLM) can be regarded as parameterizing an output distribution from
the exponential family using a weight vector w and bias scalar b. To be consistent with the previous
sections, the log likelihood of the GLM can be written using the summed inputs a as the following:
(a + b)y − η(a + b)
log P (y | x; w, b) = + c(y, φ), (10)
φ
> 0
E[y | x] = f (a + b) = f (w x + b), Var[y | x] = φf (a + b), (11)
where, f (·) is the transfer function that is the analog of the non-linearity in neural networks, f 0 (·)
is the derivative of the transfer function, η(·) is a real valued function and c(·) is the log parti-
tion function. φ is a constant that scales the output variance. Assume a H-dimensional output
vector y = [y1 , y2 , · · · , yH ] is modeled using H independent GLMs and log P (y | x; W, b) =
PH
i=1 log P (yi | x; wi , bi ). Let W be the weight matrix whose rows are the weight vectors of the
individual GLMs, b denote the bias vector of length H and vec(·) denote the Kronecker vector op-
erator. The Fisher information matrix for the multi-dimensional GLM with respect to its parameters
θ = [w1> , b1 , · · · , wH
>
, bH ]> = vec([W, b]> ) is simply the expected Kronecker product of the data
features and the output covariance matrix:
>
Cov[y | x] xx x
F (θ) = E ⊗ . (12)
x∼P (x) φ2 x> 1
We obtain normalized GLMs by applying the normalization methods to the summed inputs a in
the original model through µ and σ. Without loss of generality, we denote F̄ as the Fisher infor-
mation matrix under the normalized multi-dimensional GLM with the additional gain parameters
θ = vec([W, b, g]> ):
g (a −µ )
gg
F̄11 · · · F̄1H i j >
σi σj χi χj χi σgii χi i σji σj j
. .. .. Cov[yi , yj | x] aj −µj
χ> gj
F̄ (θ) = . . . , F̄ij = E 1
.
2 j σj σ
x∼P (x) φ j
g
> j i (a −µ i ) a −µ (ai −µ i )(a j −µj )
F̄H1 · · · F̄HH χj σi σj i
σi
i
σi σj
(13)
∂µi ai − µi ∂σi
χi = x − − . (14)
∂wi σi ∂wi
Implicit learning rate reduction through the growth of the weight vector: Notice that, com-
paring to standard GLM, the block F̄ij along the weight vector wi direction is scaled by the gain
parameters and the normalization scalar σi . If the norm of the weight vector wi grows twice as large,
even though the model’s output remains the same, the Fisher information matrix will be different.
The curvature along the wi direction will change by a factor of 21 because the σi will also be twice
as large. As a result, for the same parameter update in the normalized model, the norm of the weight
vector effectively controls the learning rate for the weight vector. During learning, it is harder to
change the orientation of the weight vector with large norm. The normalization methods, therefore,
5
43 Image Retrieval (Validation) 78 Image Retrieval (Validation) 90 Image Retrieval (Validation)
42 77 89
mean Recall@10
41
mean Recall@1 mean Recall@5
40 76 88
39 75
87
38 74
37 73 86
36 Order-Embedding + LN Order-Embedding + LN Order-Embedding + LN
72 85
35 Order-Embedding Order-Embedding Order-Embedding
340 50 100 150 200 250 300 710 50 100 150 200 250 300 840 50 100 150 200 250 300
iteration x 300 iteration x 300 iteration x 300
(a) Recall@1 (b) Recall@5 (c) Recall@10
Figure 1: Recall@K curves using order-embeddings with and without layer normalization.
MSCOCO
Caption Retrieval Image Retrieval
Model R@1 R@5 R@10 Mean r R@1 R@5 R@10 Mean r
Sym [Vendrov et al., 2016] 45.4 88.7 5.8 36.3 85.8 9.0
OE [Vendrov et al., 2016] 46.7 88.9 5.7 37.9 85.9 8.1
OE (ours) 46.6 79.3 89.1 5.2 37.8 73.6 85.7 7.9
OE + LN 48.5 80.6 89.8 5.1 38.9 74.3 86.3 7.6
Table 2: Average results across 5 test splits for caption and image retrieval. R@K is Recall@K
(high is good). Mean r is the mean rank (low is good). Sym corresponds to the symmetric baseline
while OE indicates order-embeddings.
have an implicit “early stopping” effect on the weight vectors and help to stabilize learning towards
convergence.
Learning the magnitude of incoming weights: In normalized models, the magnitude of the incom-
ing weights is explicitly parameterized by the gain parameters. We compare how the model output
changes between updating the gain parameters in the normalized GLM and updating the magnitude
of the equivalent weights under original parameterization during learning. The direction along the
gain parameters in F̄ captures the geometry for the magnitude of the incoming weights. We show
that Riemannian metric along the magnitude of the incoming weights for the standard GLM is scaled
by the norm of its input, whereas learning the gain parameters for the batch normalized and layer
normalized models depends only on the magnitude of the prediction error. Learning the magnitude
of incoming weights in the normalized model is therefore, more robust to the scaling of the input
and its parameters than in the standard model. See Appendix for detailed derivations.
6 Experimental results
We perform experiments with layer normalization on 6 tasks, with a focus on recurrent neural net-
works: image-sentence ranking, question-answering, contextual language modelling, generative
modelling, handwriting sequence generation and MNIST classification. Unless otherwise noted,
the default initialization of layer normalization is to set the adaptive gains to 1 and the biases to 0 in
the experiments.
6.1 Order embeddings of images and language
In this experiment, we apply layer normalization to the recently proposed order-embeddings model
of Vendrov et al. [2016] for learning a joint embedding space of images and sentences. We follow
the same experimental protocol as Vendrov et al. [2016] and modify their publicly available code
to incorporate layer normalization 1 which utilizes Theano [Team et al., 2016]. Images and sen-
tences from the Microsoft COCO dataset [Lin et al., 2014] are embedded into a common vector
space, where a GRU [Cho et al., 2014] is used to encode sentences and the outputs of a pre-trained
VGG ConvNet [Simonyan and Zisserman, 2015] (10-crop) are used to encode images. The order-
embedding model represents images and sentences as a 2-level partial ordering and replaces the
cosine similarity scoring function used in Kiros et al. [2014] with an asymmetric one.
1
https://github.com/ivendrov/order-embedding
6
1.0 Attentive reader
LSTM
0.9 BN-LSTM
BN-everywhere
validation error rate
LN-LSTM
0.8
0.7
0.6
0.5
0.40 100 200 300 400 500 600 700 800
training steps (thousands)
Figure 2: Validation curves for the attentive reader model. BN results are taken from [Cooijmans
et al., 2016].
We trained two models: the baseline order-embedding model as well as the same model with layer
normalization applied to the GRU. After every 300 iterations, we compute Recall@K (R@K) values
on a held out validation set and save the model whenever R@K improves. The best performing
models are then evaluated on 5 separate test sets, each containing 1000 images and 5000 captions,
for which the mean results are reported. Both models use Adam [Kingma and Ba, 2014] with the
same initial hyperparameters and both models are trained using the same architectural choices as
used in Vendrov et al. [2016]. We refer the reader to the appendix for a description of how layer
normalization is applied to GRU.
Figure 1 illustrates the validation curves of the models, with and without layer normalization. We
plot R@1, R@5 and R@10 for the image retrieval task. We observe that layer normalization offers
a per-iteration speedup across all metrics and converges to its best validation model in 60% of the
time it takes the baseline model to do so. In Table 2, the test set results are reported from which we
observe that layer normalization also results in improved generalization over the original model. The
results we report are state-of-the-art for RNN embedding models, with only the structure-preserving
model of Wang et al. [2016] reporting better results on this task. However, they evaluate under
different conditions (1 test set instead of the mean over 5) and are thus not directly comparable.
6.2 Teaching machines to read and comprehend
In order to compare layer normalization to the recently proposed recurrent batch normalization
[Cooijmans et al., 2016], we train an unidirectional attentive reader model on the CNN corpus both
introduced by Hermann et al. [2015]. This is a question-answering task where a query description
about a passage must be answered by filling in a blank. The data is anonymized such that entities
are given randomized tokens to prevent degenerate solutions, which are consistently permuted dur-
ing training and evaluation. We follow the same experimental protocol as Cooijmans et al. [2016]
and modify their public code to incorporate layer normalization 2 which uses Theano [Team et al.,
2016]. We obtained the pre-processed dataset used by Cooijmans et al. [2016] which differs from
the original experiments of Hermann et al. [2015] in that each passage is limited to 4 sentences.
In Cooijmans et al. [2016], two variants of recurrent batch normalization are used: one where BN
is only applied to the LSTM while the other applies BN everywhere throughout the model. In our
experiment, we only apply layer normalization within the LSTM.
The results of this experiment are shown in Figure 2. We observe that layer normalization not only
trains faster but converges to a better validation result over both the baseline and BN variants. In
Cooijmans et al. [2016], it is argued that the scale parameter in BN must be carefully chosen and is
set to 0.1 in their experiments. We experimented with layer normalization for both 1.0 and 0.1 scale
initialization and found that the former model performed significantly better. This demonstrates that
layer normalization is not sensitive to the initial scale in the same way that recurrent BN is. 3
6.3 Skip-thought vectors
Skip-thoughts [Kiros et al., 2015] is a generalization of the skip-gram model [Mikolov et al., 2013]
for learning unsupervised distributed sentence representations. Given contiguous text, a sentence is
2
https://github.com/cooijmanstim/Attentive_reader/tree/bn
3
We only produce results on the validation set, as in the case of Cooijmans et al. [2016]
7
86.0 34 82
85.5 Skip-Thoughts + LN
33 Skip-Thoughts 80
85.0 32
Pearson x 100
Original 78
84.5
MSE x 100 Accuracy
31
84.0 76
83.5 30
Skip-Thoughts + LN 29 74 Skip-Thoughts + LN
83.0 Skip-Thoughts Skip-Thoughts
82.5 Original 28 72
Original
82.0 5 10 15 20 27 5 10 15 20 70 5 10 15 20
iteration x 50000 iteration x 50000 iteration x 50000
(a) SICK(r) (b) SICK(MSE) (c) MR
86 94.5 91
84 94.0 90
93.5 89
82 93.0 88
Accuracy Accuracy Accuracy
80 92.5 87
92.0
78 91.5 86
Skip-Thoughts + LN Skip-Thoughts + LN 85 Skip-Thoughts + LN
76 Skip-Thoughts 91.0 Skip-Thoughts Skip-Thoughts
Original 90.5 Original 84 Original
74 5 10 15 20 90.0 5 10 15 20 83 5 10 15 20
iteration x 50000 iteration x 50000 iteration x 50000
(d) CR (e) SUBJ (f) MPQA
Figure 3: Performance of skip-thought vectors with and without layer normalization on downstream
tasks as a function of training iterations. The original lines are the reported results in [Kiros et al.,
2015]. Plots with error use 10-fold cross validation. Best seen in color.
Method SICK(r) SICK(ρ) SICK(MSE) MR CR SUBJ MPQA
Original [Kiros et al., 2015] 0.848 0.778 0.287 75.5 79.3 92.1 86.9
Ours 0.842 0.767 0.298 77.3 81.8 92.6 87.9
Ours + LN 0.854 0.785 0.277 79.5 82.6 93.4 89.0
Ours + LN † 0.858 0.788 0.270 79.4 83.1 93.7 89.3
Table 3: Skip-thoughts results. The first two evaluation columns indicate Pearson and Spearman cor-
relation, the third is mean squared error and the remaining indicate classification accuracy. Higher is
better for all evaluations except MSE. Our models were trained for 1M iterations with the exception
of (†) which was trained for 1 month (approximately 1.7M iterations)
encoded with a encoder RNN and decoder RNNs are used to predict the surrounding sentences.
Kiros et al. [2015] showed that this model could produce generic sentence representations that
perform well on several tasks without being fine-tuned. However, training this model is time-
consuming, requiring several days of training in order to produce meaningful results.
In this experiment we determine to what effect layer normalization can speed up training. Using the
publicly available code of Kiros et al. [2015] 4 , we train two models on the BookCorpus dataset [Zhu
et al., 2015]: one with and one without layer normalization. These experiments are performed with
Theano [Team et al., 2016]. We adhere to the experimental setup used in Kiros et al. [2015], training
a 2400-dimensional sentence encoder with the same hyperparameters. Given the size of the states
used, it is conceivable layer normalization would produce slower per-iteration updates than without.
However, we found that provided CNMeM 5 is used, there was no significant difference between the
two models. We checkpoint both models after every 50,000 iterations and evaluate their performance
on five tasks: semantic-relatedness (SICK) [Marelli et al., 2014], movie review sentiment (MR)
[Pang and Lee, 2005], customer product reviews (CR) [Hu and Liu, 2004], subjectivity/objectivity
classification (SUBJ) [Pang and Lee, 2004] and opinion polarity (MPQA) [Wiebe et al., 2005]. We
plot the performance of both models for each checkpoint on all tasks to determine whether the
performance rate can be improved with LN.
The experimental results are illustrated in Figure 3. We observe that applying layer normalization
results both in speedup over the baseline as well as better final results after 1M iterations are per-
formed as shown in Table 3. We also let the model with layer normalization train for a total of a
month, resulting in further performance gains across all but one task. We note that the performance
4
https://github.com/ryankiros/skip-thoughts
5
https://github.com/NVIDIA/cnmem
8
0
−100 Baseline test
Negative Log Likelihood
−200 Baseline train
−300 LN test
−400 LN train
−500
−600
−700
−800
−900 0
10 10 1 10 2 10 3
Updates x 200
Figure 5: Handwriting sequence generation model negative log likelihood with and without layer
normalization. The models are trained with mini-batch size of 8 and sequence length of 500,
differences between the original reported results and ours are likely due to the fact that the publicly
available code does not condition at each timestep of the decoder, where the original model does.
6.4 Modeling binarized MNIST using DRAW
100
We also experimented with the generative modeling on the Baseline
MNIST dataset. Deep Recurrent Attention Writer (DRAW) WN
Test Variational Bound
[Gregor et al., 2015] has previously achieved the state-of-the- 95 LN
art performance on modeling the distribution of MNIST dig-
its. The model uses a differential attention mechanism and
a recurrent neural network to sequentially generate pieces of 90
an image. We evaluate the effect of layer normalization on
a DRAW model using 64 glimpses and 256 LSTM hidden 85
units. The model is trained with the default setting of Adam
[Kingma and Ba, 2014] optimizer and the minibatch size of
128. Previous publications on binarized MNIST have used 80
0 20 40 60 80 100
various training protocols to generate their datasets. In this Epoch
experiment, we used the fixed binarization from Larochelle
and Murray [2011]. The dataset has been split into 50,000 Figure 4: DRAW model test nega-
training, 10,000 validation and 10,000 test images. tive log likelihood with and without
Figure 4 shows the test variational bound for the first 100 layer normalization.
epoch. It highlights the speedup benefit of applying layer nor-
malization that the layer normalized DRAW converges almost twice as fast than the baseline model.
After 200 epoches, the baseline model converges to a variational log likelihood of 82.36 nats on the
test data and the layer normalization model obtains 82.09 nats.
6.5 Handwriting sequence generation
The previous experiments mostly examine RNNs on NLP tasks whose lengths are in the range of 10
to 40. To show the effectiveness of layer normalization on longer sequences, we performed hand-
writing generation tasks using the IAM Online Handwriting Database [Liwicki and Bunke, 2005].
IAM-OnDB consists of handwritten lines collected from 221 different writers. When given the input
character string, the goal is to predict a sequence of x and y pen co-ordinates of the corresponding
handwriting line on the whiteboard. There are, in total, 12179 handwriting line sequences. The input
string is typically more than 25 characters and the average handwriting line has a length around 700.
We used the same model architecture as in Section (5.2) of Graves [2013]. The model architecture
consists of three hidden layers of 400 LSTM cells, which produce 20 bivariate Gaussian mixture
components at the output layer, and a size 3 input layer. The character sequence was encoded with
one-hot vectors, and hence the window vectors were size 57. A mixture of 10 Gaussian functions
was used for the window parameters, requiring a size 30 parameter vector. The total number of
weights was increased to approximately 3.7M. The model is trained using mini-batches of size 8
and the Adam [Kingma and Ba, 2014] optimizer.
The combination of small mini-batch size and very long sequences makes it important to have very
stable hidden dynamics. Figure 5 shows that layer normalization converges to a comparable log
likelihood as the baseline model but is much faster.
9
100 100
10-1 10-1
10-2 10-2
Train NLL Train NLL
10-3 10-3
10-4 10-4
10-5 BatchNorm bz128 10-5 LayerNorm bz4
Baseline bz128 Baseline bz4
10-6 LayerNorm bz128 10-6 BatchNorm bz4
10-7 0 10 20 30 40 50 60 10-7 0 10 20 30 40 50 60
Epoch Epoch
0.025 0.025
0.020 0.020
Test Err. 0.015 Test Err. 0.015
0.010 BatchNorm bz128 0.010 LayerNorm bz4
Baseline bz128 Baseline bz4
LayerNorm bz128 BatchNorm bz4
0.005 0.005
0 10 20 30 40 50 60 0 10 20 30 40 50 60
Epoch Epoch
Figure 6: Permutation invariant MNIST 784-1000-1000-10 model negative log likelihood and test
error with layer normalization and batch normalization. (Left) The models are trained with batch-
size of 128. (Right) The models are trained with batch-size of 4.
6.6 Permutation invariant MNIST
In addition to RNNs, we investigated layer normalization in feed-forward networks. We show how
layer normalization compares with batch normalization on the well-studied permutation invariant
MNIST classification problem. From the previous analysis, layer normalization is invariant to input
re-scaling which is desirable for the internal hidden layers. But this is unnecessary for the logit
outputs where the prediction confidence is determined by the scale of the logits. We only apply
layer normalization to the fully-connected hidden layers that excludes the last softmax layer.
All the models were trained using 55000 training data points and the Adam [Kingma and Ba, 2014]
optimizer. For the smaller batch-size, the variance term for batch normalization is computed using
the unbiased estimator. The experimental results from Figure 6 highlight that layer normalization is
robust to the batch-sizes and exhibits a faster training convergence comparing to batch normalization
that is applied to all layers.
6.7 Convolutional Networks
We have also experimented with convolutional neural networks. In our preliminary experiments, we
observed that layer normalization offers a speedup over the baseline model without normalization,
but batch normalization outperforms the other methods. With fully connected layers, all the hidden
units in a layer tend to make similar contributions to the final prediction and re-centering and re-
scaling the summed inputs to a layer works well. However, the assumption of similar contributions
is no longer true for convolutional neural networks. The large number of the hidden units whose
receptive fields lie near the boundary of the image are rarely turned on and thus have very different
statistics from the rest of the hidden units within the same layer. We think further research is needed
to make layer normalization work well in ConvNets.
7 Conclusion
In this paper, we introduced layer normalization to speed-up the training of neural networks. We
provided a theoretical analysis that compared the invariance properties of layer normalization with
batch normalization and weight normalization. We showed that layer normalization is invariant to
per training-case feature shifting and scaling.
Empirically, we showed that recurrent neural networks benefit the most from the proposed method
especially for long sequences and small mini-batches.
Acknowledgments
This research was funded by grants from NSERC, CFI, and Google.
10
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12
Supplementary Material
Application of layer normalization to each experiment
This section describes how layer normalization is applied to each of the papers’ experiments. For
notation convenience, we define layer normalization as a function mapping LN : RD → RD with
two set of adaptive parameters, gains α and biases β:
(z − µ)
LN (z; α, β) = α + β, (15)
v σ
D u1 D
u
1 X X
µ= zi , σ=t (zi − µ)2 , (16)
D i=1 D i=1
where, zi is the ith element of the vector z.
Teaching machines to read and comprehend and handwriting sequence generation
The basic LSTM equations used for these experiment are given by:
ft
it
o = Wh ht−1 + Wx xt + b (17)
t
gt
ct = σ(ft ) ct−1 + σ(it ) tanh(gt ) (18)
ht = σ(ot ) tanh(ct ) (19)
The version that incorporates layer normalization is modified as follows:
ft
it
o = LN (Wh ht−1 ; α1 , β1 ) + LN (Wx xt ; α2 , β2 ) + b (20)
t
gt
ct = σ(ft ) ct−1 + σ(it ) tanh(gt ) (21)
ht = σ(ot ) tanh(LN (ct ; α3 , β3 )) (22)
where αi , βi are the additive and multiplicative parameters, respectively. Each αi is initialized to a
vector of zeros and each βi is initialized to a vector of ones.
Order embeddings and skip-thoughts
These experiments utilize a variant of gated recurrent unit which is defined as follows:
zt
= Wh ht−1 + Wx xt (23)
rt
ĥt = tanh(Wxt + σ(rt ) (Uht−1 )) (24)
ht = (1 − σ(zt ))ht−1 + σ(zt )ĥt (25)
Layer normalization is applied as follows:
zt
= LN (Wh ht−1 ; α1 , β1 ) + LN (Wx xt ; α2 , β2 ) (26)
rt
ĥt = tanh(LN (Wxt ; α3 , β3 ) + σ(rt ) LN (Uht−1 ; α4 , β4 )) (27)
ht = (1 − σ(zt ))ht−1 + σ(zt )ĥt (28)
just as before, αi is initialized to a vector of zeros and each βi is initialized to a vector of ones.
13
Modeling binarized MNIST using DRAW
The layer norm is only applied to the output of the LSTM hidden states in this experiment:
The version that incorporates layer normalization is modified as follows:
ft
it
o = Wh ht−1 + Wx xt + b (29)
t
gt
ct = σ(ft ) ct−1 + σ(it ) tanh(gt ) (30)
ht = σ(ot ) tanh(LN (ct ; α, β)) (31)
where α, β are the additive and multiplicative parameters, respectively. α is initialized to a vector
of zeros and β is initialized to a vector of ones.
Learning the magnitude of incoming weights
We now compare how gradient descent updates changing magnitude of the equivalent weights be-
tween the normalized GLM and original parameterization. The magnitude of the weights are ex-
plicitly parameterized using the gain parameter in the normalized model. Assume there is a gradient
update that changes norm of the weight vectors by δg . We can project the gradient updates to the
weight vector for the normal GLM. The KL metric, ie how much the gradient update changes the
model prediction, for the normalized model depends only on the magnitude of the prediction error.
Specifically,
under batch normalization:
2 1 > > > > 1 > Cov[y | x]
ds = vec([0, 0, δg ] ) F̄ (vec([W, b, g] ) vec([0, 0, δg ] ) = δg E δg .
2 2 x∼P (x) φ2
(32)
Under layer normalization:
1
ds2 = vec([0, 0, δg ]> )> F̄ (vec([W, b, g]> ) vec([0, 0, δg ]> )
2
2
Cov(y1 , y1 | x) (a1σ−µ) · · · Cov(y1 , yH | x) (a1 −µ)(a H −µ)
2 σ2
1 1 .. .. ..
= δg> 2 E
. δg
2 φ x∼P (x) . .
2
(aH −µ)(a1 −µ) (aH −µ)
Cov(yH , y1 | x) σ2 ··· Cov(yH , yH | x) σ2
(33)
Under weight normalization:
1
ds2 = vec([0, 0, δg ]> )> F̄ (vec([W, b, g]> ) vec([0, 0, δg ]> )
2
a2
Cov(y1 , y1 | x) kw11k2 ··· Cov(y1 , yH | x) kw1ak21 kw
aH
H k2
2
1 1
.. .. ..
= δg> 2 E δg . (34)
. . .
2 φ x∼P (x)
a2H
Cov(yH , y1 | x) kwHakH2 akw
1
1 k2
··· Cov(yH , yH | x) kwH k2
2
Whereas, the KL metric in the standard GLM is related to its activities ai = wi> x, that is depended
on both its current weights and input data. We project the gradient updates to the gain parameter δg i
of the ith neuron to its weight vector as δgi kwwiik2 in the standard GLM model:
1 wi wj wi wj
vec([δgi , 0, δgj , 0]> )> F ([wi> , bi , wj> , bj ]> ) vec([δgi , 0, δgj , 0]> )
2 kwi k2 kwi k2 kwi k2 kwj k2
δgi δgj ai aj
= E Cov(y ,
i jy | x) (35)
2φ2 x∼P (x) kwi k2 kwj k2
The batch normalized and layer normalized models are therefore more robust to the scaling of the
input and its parameters than the standard model.
14
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