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Understanding LSTM -- a tutorial into Long Short-Term Memory Recurrent Neural Networks

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Long Short-Term Memory Recurrent Neural Networks (LSTM-RNN) are one of the most powerful dynamic classifiers publicly known. The network itself and the related learning algorithms are reasonably well documented to get an idea how it works. This paper will shed more light into understanding how LSTM-RNNs evolved and why they work impressively well, focusing on the early, ground-breaking publications. We significantly improved documentation and fixed a number of errors and inconsistencies that accumulated in previous publications. To support understanding we as well revised and unified the notation used.
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– Understanding LSTM –
a tutorial into Long Short-Term Memory
Recurrent Neural Networks
Ralf C. Staudemeyer
Faculty of Computer Science
Schmalkalden University of Applied Sciences, Germany
Eric Rothstein Morris
(Singapore University of Technology and Design, Singapore
E-Mail: eric
September 23, 2019
Long Short-Term Memory Recurrent Neural Networks (LSTM-RNN)
are one of the most powerful dynamic classifiers publicly known. The net-
work itself and the related learning algorithms are reasonably well docu-
mented to get an idea how it works. This paper will shed more light into
understanding how LSTM-RNNs evolved and why they work impressively
well, focusing on the early, ground-breaking publications. We significantly
improved documentation and fixed a number of errors and inconsistencies
that accumulated in previous publications. To support understanding we
as well revised and unified the notation used.
1 Introduction
This article is an tutorial-like introduction initially developed as supplementary
material for lectures focused on Artificial Intelligence. The interested reader
can deepen his/her knowledge by understanding Long Short-Term Memory Re-
current Neural Networks (LSTM-RNN) considering its evolution since the early
nineties. Todays publications on LSTM-RNN use a slightly different notation
and a much more summarized representation of the derivations. Nevertheless
the authors found the presented approach very helpful and we are confident this
publication will find its audience.
Machine learning is concerned with the development of algorithms that au-
tomatically improve by practice. Ideally, the more the learning algorithm is run,
the better the algorithm becomes. It is the task of the learning algorithm to
create a classifier function from the training data presented. The performance
of this built classifier is then measured by applying it to previously unseen data.
Artificial Neural Networks (ANN) are inspired by biological learning sys-
tems and loosely model their basic functions. Biological learning systems are
arXiv:1909.09586v1 [cs.NE] 12 Sep 2019
complex webs of interconnected neurons. Neurons are simple units accepting
a vector of real-valued inputs and producing a single real-valued output. The
most common standard neural network type are feed-forward neural networks.
Here sets of neurons are organised in layers: one input layer, one output layer,
and at least one intermediate hidden layer. Feed-forward neural networks are
limited to static classification tasks. Therefore, they are limited to provide a
static mapping between input and output. To model time prediction tasks we
need a so-called dynamic classifier.
We can extend feed-forward neural networks towards dynamic classification.
To gain this property we need to feed signals from previous timesteps back into
the network. These networks with recurrent connections are called Recurrent
Neural Networks (RNN) [74], [75]. RNNs are limited to look back in time for
approximately ten timesteps [38], [56]. This is due to the fed back signal is
either vanishing or exploding. This issue was addressed with Long Short-Term
Memory Recurrent Neural Networks (LSTM-RNN) [22], [41], [23], [60]. LSTM
networks are to a certain extend biologically plausible [58] and capable to learn
more than 1,000 timesteps, depending on the complexity of the built network
In the early, ground-breaking papers by Hochreiter [41] and Graves [34], the
authors used different notations which made further development prone to errors
and inconvenient to follow. To address this we developed a unified notation and
did draw descriptive figures to support the interested reader in understanding
the related equations of the early publications.
In the following, we slowly dive into the world of neural networks and specifi-
cally LSTM-RNNs with a selection of its most promising extensions documented
so far. We successively explain how neural networks evolved from a single per-
ceptron to something as powerful as LSTM. This includes vanilla LSTM, al-
though not used in practice anymore, as the fundamental evolutionary step.
With this article, we support beginners in the machine learning community to
understand how LSTM works with the intention motivate its further develop-
This is the first document that covers LSTM and its extensions in such great
2 Notation
In this article we use the following notation:
The learning rate of the network is η.
A time unit is τ. Initial times of an epoch are denoted by t0and final
times by t.
The set of units of the network is N, with generic (unless stated otherwise)
units u, v, l, k N.
The set of input units is I, with input unit iI.
The set of output units is O, with output unit oO.
The set of non-input units is U.
The output of a unit u(also called the activation of u) is yu, and unlike
the input, it is a single value.
The set of units with connections to a unit u; i.e., its predecessors, is
Pre (u)
The set of units with connections from a unit u; i.e., its successors, is
Suc (u)
The weight that connects the unit vto the unit uis W[v,u].
The input of a unit ucoming from a unit vis denoted by X[v,u]
The weighted input of the unit uis zu.
The bias of the unit uis bu.
The state of the unit uis su.
The squashing function of the unit uis fu.
The error of the unit uis eu.
The error signal of the unit uis ϑu.
The output sensitivity of the unit kwith respect to the weight W[u,v]is
3 Perceptron and Delta Learning Rule
Artificial Neural Networks consist of a densely interconnected group of simple
neuron-like threshold switching units. Each unit takes a number of real-valued
inputs and produces a single real-valued output. Based on the connectivity
between the threshold units and element parameters, these networks can model
complex global behaviour.
3.1 The Perceptron
The most basic type of artificial neuron is called a perceptron. Perceptrons
consist of a number of external input links, a threshold, and a single external
output link. Additionally, perceptrons have an internal input, b, called bias. The
perceptron takes a vector of real-valued input values, all of which are weighted
by a multiplier. In a previous perceptron training phase, the perceptron learns
these weights on the basis of training data. It sums all weighted input values
and ‘fires’ if the resultant value is above a pre-defined threshold. The output of
the perceptron is always Boolean, and it is considered to have fired if the output
is ‘1’. The deactivated value of the perceptron is ‘1’, and the threshold value
is, in most cases, ‘0’.
As we only have one unit for the perceptron, we omit the subindexes that
refer to the unit. Given the input vector x=hx1, ..., xniand trained weights
W1, ..., Wn, the perceptron outputs y; which is computed by the formula
y=(1 if Pn
i=1 Wixi+b > 0;
1 otherwise.
We refer to z=Pn
i=1 Wixias the weighted input, and to s=z+bas the state
of the perceptron. For the perceptron to fire, its state smust exceed the value
of the threshold.
Single perceptron units can already represent a number of useful functions.
Examples are the Boolean functions AND, OR, NAND and NOR. Other func-
tions are only representable using networks of neurons. Single perceptrons are
limited to learning only functions that are linearly separable. In general, a prob-
lem is linear and the classes are linearly separable in an n-dimensional space if
the decision surface is an (n1)-dimensional hyperplane.
The general structure of a perceptron is shown in Figure 1.
Figure 1: The general structure of the most basic type of artificial neuron,
called a perceptron. Single perceptrons are limited to learning linearly separable
3.2 Linear Separability
To understand linear separability, it is helpful to visualise the possible inputs
of a perceptron on the axes of a two-dimensional graph. Figure 2 shows repre-
sentations of the Boolean functions OR and XOR. The OR function is linearly
separable, whereas the XOR function is not. In the figure, pluses are used for
an input where the perceptron fires and minuses, where it does not. If the
pluses and minuses can be completely separated by a single line, the problem
is linearly separable in two dimensions. The weights of the trained perceptron
should represent that line.
0 1
0 1
+ +
logical XOR
0 0
(linearly separable) (not linearly separable)
InputInput Output
1 1
1 2 1 2
logical OR
Figure 2: Representations of the Boolean functions OR and XOR. The figures
show that the OR function is linearly separable, whereas the XOR function is
3.3 The Delta Learning Rule
Perceptron training is learning by imitation, which is called ‘supervised learn-
ing’. During the training phase, the perceptron produces an output and com-
pares it with a derived output value provided by the training data. In cases
of misclassification, it then modifies the weights accordingly. [55] show that in
a finite time, the perceptron will converge to reproduce the correct behaviour,
provided that the training examples are linearly separable. Convergence is not
assured if the training data is not linearly separable.
A variety of training algorithms for perceptrons exist, of which the most
common are the perceptron learning rule and the delta learning rule. Both
start with random weights and both guarantee convergence to an acceptable
hypothesis. Using the perceptron learning rule algorithm, the perceptron can
learn from a set of samples A sample is a pair hx, diwhere xis the input and
dis its label. For the sample hx, di, given the input x=hx1, . . . , xni, the old
weight vector W=hW1, . . . , Wniis updated to the new vector W0using the
i=Wi+ ∆Wi,
where yis the output calculated using the input xand the weights Wand η
is the learning rate. The learning rate is a constant that controls the degree to
which the weights are changed. As stated before, the initial weight vector W0
has random values. The algorithm will only converge towards an optimum if
the training data is linearly separable, and the learning rate is sufficiently small.
The perceptron rule fails if the training examples are not linearly separable.
The delta learning rule was specifically designed to handle linearly separable
and linearly non-separable training examples. It also calculates the errors be-
tween calculated output and output data from training samples, and modifies
the weights accordingly. The modification of weights is achieved by using the
gradient optimisation descent algorithm, which alters them in the direction that
produces the steepest descent along the error surface towards the global min-
imum error. The delta learning rule is the basis of the error backpropagation
algorithm, which we will discuss later in this section.
3.4 The Sigmoid Threshold Unit
The sigmoid threshold unit is a different kind of artificial neuron, very similar
to the perceptron, but uses a sigmoid function to calculate the output. The
output yis computed by the formula
(1 el×s),
where bis the bias and lis a positive constant that determines the steepness
of the sigmoid function. The major effect on the perceptron is that the output
of the sigmoid threshold unit now has more than two possible values; now, the
output is “squashed” by a continuous function that ranges between 0 and 1.
Accordingly, the function 1
(1el×s)is called the ‘squashing’ function, because it
maps a very large input domain onto a small range of outputs. For a low total
input value, the output of the sigmoid function is close to zero, whereas it is close
to one for a high total input value. The slope of the sigmoid function is adjusted
by the threshold value. The advantage of neural networks using sigmoid units
is that they are capable of representing non-linear functions. Cascaded linear
units, like the perceptron, are limited to representing linear functions. A sigmoid
threshold unit is sketched in Figure 3.
Figure 3: The sigmoid threshold unit is capable of representing non-linear func-
tions. Its output is a continuous function of its input, which ranges between 0
and 1.
4 Feed-Forward Neural Networks and Backprop-
In feed-forward neural networks (FFNNs), sets of neurons are organised in lay-
ers, where each neuron computes a weighted sum of its inputs. Input neurons
take signals from the environment, and output neurons present signals to the
environment. Neurons that are not directly connected to the environment, but
which are connected to other neurons, are called hidden neurons.
Feed-forward neural networks are loop-free and fully connected. This means
that each neuron provides an input to each neuron in the following layer, and
that none of the weights give an input to a neuron in a previous layer.
The simplest type of neural feed-forward networks are single-layer perceptron
networks. Single-layer neural networks consist of a set of input neurons, defined
as the input layer, and a set of output neurons, defined as the output layer. The
outputs of the input-layer neurons are directly connected to the neurons of the
output layer. The weights are applied to the connections between the input and
output layer.
In the single-layer perceptron network, every single perceptron calculates
the sum of the products of the weights and the inputs. The perceptron fires ‘1’
if the value is above the threshold value; otherwise, the perceptron takes the
deactivated value, which is usually ‘-1’. The threshold value is typically zero.
Sets of neurons organised in several layers can form multilayer, forward-
connected networks. The input and output layers are connected via at least one
hidden layer, built from set(s) of hidden neurons. The multilayer feed-forward
neural network sketched in Figure 4, with one input layer and three output
layers (two hidden and one output), is classified as a 3-layer feed-forward neural
network. For most problems, feed-forward neural networks with more than two
layers offer no advantage.
Multilayer feed-forward networks using sigmoid threshold functions are able
to express non-linear decision surfaces. Any function can be closely approxi-
mated by these networks, given enough hidden units.
Figure 4: A multilayer feed-forward neural network with one input layer, two
hidden layers, and an output layer. Using neurons with sigmoid threshold func-
tions, these neural networks are able to express non-linear decision surfaces.
The most common neural network learning technique is the error backprop-
agation algorithm. It uses gradient descent to learn the weights in multilayer
networks. It works in small iterative steps, starting backwards from the output
layer towards the input layer. A requirement is that the activation function of
the neuron is differentiable.
Usually, the weights of a feed-forward neural network are initialised to small,
normalised random numbers using bias values. Then, error backpropagation
applies all training samples to the neural network and computes the input and
output of each unit for all (hidden and) output layers.
The set of units of the network is N,ItHtO, where tis disjoint
union, and I, H, O are the sets of input, hidden and output units, respectively.
We denote input units by i, hidden units by hand output units by o. For
convenience, we define the set of non-input units U,HtO. For a non-input
unit uU, the input to uis denoted by xu, its state by su, its bias by buand
its output by yu. Given units u, v U, the weight that connects uwith vis
denoted by Wuv.
To model the external input that the neural network receives, we use the
external input vector x=hx1, . . . , xni. For each component of the external
input vector we find a corresponding input unit that models it, so the output
of the ith input unit should be equal ith component of the input to the network
(i.e., xi), and consequently |I|=n.
For the non-input unit uU, the output of u, written yu, is defined using
the sigmoid activation function by
1 + esu(1)
where suis the state of u, and it is defined by
su=zu+bu; (2)
where buis the bias of u, and zuis the weighted input of u, defined in turn by
W[v,u]X[v,u],with vPre (u)
where X[v,u]is the information that vpasses as input to u, and Pre (u) is the set
of units vthat preceed u; that is, input units, and hidden units that feed their
outputs yv(see Equation (1)) multiplied by the corresponding weight W[v,u]to
the unit u.
Starting from the input layer, the inputs are propagated forwards through
the network until the output units are reached at the output layer. Then,
the output units produce an observable output (the network output) y. More
precisely, for oO, its output yocorresponds to the oth component of y.
Next, the backpropagation learning algorithm propagates the error back-
wards, and the weights and biases are updated such that we reduce the error
with respect to the present training sample. Starting from the output layer,
the algorithm compares the network output yowith the corresponding desired
target output do. It calculates the error eofor each output neuron using some
error function to be minimised. The error eois computed as
eo= (doyo)
and we have the following notion of overall error of the network
To update the weight W[u,v], we will use the formula
where ηis the learning rate. We now make use of the factors ∂yu
∂yuand su
calculate the weight update by deriving the error with respect to the activation,
and the activation in terms of the state, and in turn the derivative of the state
with respect to the weight:
The derivative of the error with respect to the activation for output units is
now, the derivative of the activation with respect to the state for output units
is ∂yo
=yo(1 yo),
and the derivative of the state with respect to a weight that connects the hidden
unit hto the output unit ois
Let us define, for the output unit o, the error signal by
for output units we have that
ϑo= (doyo)yo(1 yo),(5)
and we see that we can update the weight between the hidden unit hand the
output unit oby
Now, for a hidden unit h, if we consider that its notion of error is related
to how much it contributed to the production of a faulty output, then we can
backpropagate the error from the output units that hsends signals to; more pre-
cisely, for an input unit i, we need to expand the equation ∆W[i,h]=η∂E
with oSuc (h).
where Suc (h) is the set of units that succeed h; that is, the units that are fed
with the output of has part of their input. By solving the partial derivatives,
we obtain
oϑoW[h,o]yh(1 yh)yi.
Figure 5: This figure shows a feed-forward neural network.
If we define the error signal of the hidden unit hby
oϑoW[h,o]yh(1 yh); with oSuc (h),
then we have a uniform expression for weight change; that is,
We calculate ∆W[v,u]again and again until all network outputs are within
an acceptable range, or some other terminating condition is reached.
5 Recurrent Neural Networks
Recurrent neural networks (RNNs) [74, 75] are dynamic systems; they have an
internal state at each time step of the classification. This is due to circular
connections between higher- and lower-layer neurons and optional self-feedback
connections. These feedback connections enable RNNs to propagate data from
earlier events to current processing steps. Thus, RNNs build a memory of time
series events.
5.1 Basic Architecture
RNNs range from partly to fully connected, and two simple RNNs are suggested
by [46] and [16]. The Elman network is similar to a three-layer neural network,
but additionally, the outputs of the hidden layer are saved in so-called ‘context
cells’. The output of a context cell is circularly fed back to the hidden neuron
along with the originating signal. Every hidden neuron has its own context cell
and receives input both from the input layer and the context cells. Elman net-
works can be trained with standard error backpropagation, the output from the
context cells being simply regarded as an additional input. Figures 5 and 6 show
a standard feed-forward network in comparison with such an Elman network.
Figure 6: This figure shows an Elman neural network.
Figure 7: This figure shows a partially recurrent neural network with self-
feedback in the hidden layer.
Jordan networks have a similar structure to Elman networks, but the context
cells are instead fed by the output layer. A partial recurrent neural network with
a fully connected recurrent hidden layer is shown in Figure 7. Figure 8 shows a
fully connected RNN.
RNNs need to be trained differently to the feed-forward neural networks
(FFNNs) described in Section 4. This is because, for RNNs, we need to propa-
gate information through the recurrent connections in-between steps. The most
common and well-documented learning algorithms for training RNNs in tempo-
ral, supervised learning tasks are backpropagation through time (BPTT) and
real-time recurrent learning (RTRL). In BPTT, the network is unfolded in time
to construct an FFNN. Then, the generalised delta rule is applied to update the
weights. This is an offline learning algorithm in the sense that we first collect
the data and then build the model from the system. In RTRL, the gradient
information is forward propagated. Here, the data is collected online from the
Figure 8: This figure shows a fully recurrent neural network (RNN) with self-
feedback connections.
system and the model is learned during collection. Therefore, RTRL is an online
learning algorithm.
6 Training Recurrent Neural Networks
The most common methods to train recurrent neural networks are Backpropa-
gation Through Time (BPTT) [62, 74, 75] and Real-Time Recurrent Learning
(RTRL) [75, 76], whereas BPTT is the most common method. The main differ-
ence between BPTT and RTRL is the way the weight changes are calculated.
The original formulation of LSTM-RNNs used a combination of BPTT and
RTRL. Therefore we cover both learning algorithms in short.
6.1 Backpropagation Through Time
The BPTT algorithm makes use of the fact that, for a finite period of time, there
is an FFNN with identical behaviour for every RNN. To obtain this FFNN, we
need to unfold the RNN in time. Figure 9a shows a simple, fully recurrent
neural network with a single two-neuron layer. The corresponding feed-forward
neural network, shown in Figure 9b, requires a separate layer for each time step
with the same weights for all layers. If weights are identical to the RNN, both
networks show the same behaviour.
The unfolded network can be trained using the backpropagation algorithm
described in Section 4. At the end of a training sequence, the network is unfolded
in time. The error is calculated for the output units with existing target values
using some chosen error measure. Then, the error is injected backwards into
the network and the weight updates for all time steps calculated. The weights
in the recurrent version of the network are updated with the sum of its deltas
over all time steps.
We calculate the error signal for a unit for all time steps in a single pass,
using the following iterative backpropagation algorithm. We consider discrete
time steps 1,2,3..., indexed by the variable τ. The network starts at a point in
time t0and runs until a final time t. This time frame between t0and tis called
an epoch. Let Ube the set of non input units, and let fube the differentiable,
non-linear squashing function of the unit uU; the output yu(τ) of uat time
Figure 9: Figure a shows a simple fully recurrent neural network with a two-
neuron layer. The same network unfolded over time with a separate layer for
each time step is shown in Figure b. The latter representation is a feed-forward
neural network.
τis given by
yu(τ) = fu(zu(τ)) (6)
with the weighted input
zu(τ+ 1) = X
W[u,l]X[l,u](τ+ 1),with lPre (u)
W[u,v]yv(τ) + X
W[u,i]yi(τ+ 1)
where vUPre (u) and iI, the set of input units. Note that the inputs to u
at time τ+1 are of two types: the environmental input that arrives at time τ+1
via the input units, and the recurrent output from all non-input units in the
network produced at time τ. If the network is fully connected, then UPre (u)
is equal to the set Uof non-input units. Let T(τ) be the set of non-input units
for which, at time τ, the output value yu(τ) of the unit uT(τ) should match
some target value du(τ). The cost function is the summed error Etotal (t0, t) for
the epoch t0, t0+ 1, . . . , t, which we want to minimise using a learning algorithm.
Such total error is defined by
Etotal(t0, t) =
E(τ), (8)
with the error E(τ) at time τdefined using the squared error as an objective
function by
E(τ) = 1
(eu(τ))2, (9)
and with the error eu(τ) of the non-input unit uat time τdefined by
eu(τ) = (du(τ)yu(τ) if uT(τ),
0 otherwise. (10)
To adjust the weights, we use the error signal ϑu(τ) of a non-input unit uat a
time τ, which is defined by
ϑu(τ) = E(τ)
When we unroll ϑuover time, we obtain the equality
ϑu(τ) = (f0
u(zu(τ))eu(τ) if τ=t,
u(zu(τ)) PkUW[k,u]ϑk(τ+ 1)if t0τ < t .(12)
After the backpropagation computation is performed down to time t0, we cal-
culate the weight update ∆W[u,v]in the recurrent version of the network. This
is done by summing the corresponding weight updates for all time steps:
W[u,v]=ηEtotal(t0, t)
∂Etotal (t0, t)
BPTT is described in more detail in [74], [62] and [76].
6.2 Real-Time Recurrent Learning
The RTRL algorithm does not require error propagation. All the information
necessary to compute the gradient is collected as the input stream is presented
to the network. This makes a dedicated training interval obsolete. The algo-
rithm comes at significant computational cost per update cycle, and the stored
information is non-local; i.e., we need an additional notion called sensitivity
of the output, which we’ll explain later. Nevertheless, the memory required
depends only on the size of the network and not on the size of the input.
Following the notation from the previous section, we will now define for the
network units vIUand u, k U, and the time steps t0τt. Unlike
BPTT, in RTRL we assume the existence of a label dk(τ) at every time τ
(given that it is an online algorithm) for every non-input unit k, so the training
objective is to minimise the overall network error, which is given at time step τ
E(τ) = 1
We conclude from Equation 8 that the gradient of the total error is also the
sum of the gradient for all previous time steps and the current time step:
WEtotal(t0, t + 1) = WEtotal (t0, t) + WE(t+ 1).
During presentation of the time series to the network, we need to accumulate
the values of the gradient at each time step. Thus, we can also keep track of
the weight changes ∆W[u,v](τ). After presentation, the overall weight change
for W[u,v]is then given by
To get the weight changes we need to calculate
W[u,v](τ) = ηE(τ)
for each time step t. After expanding this equation via gradient descent and by
applying Equation 9, we find that
W[u,v](τ) = ηX
(dk(τ)yk(τ)) ∂yk(τ)
Since the error ek(τ) = dk(τ)yk(τ) is always known, we need to find a way
to calculate the second factor only. We define the quantity
uv(τ) = ∂yk(τ)
which measures the sensitivity of the output of unit kat time τto a small change
in the weight W[u,v], in due consideration of the effect of such a change in the
weight over the entire network trajectory from time t0to t. The weight W[u,v]
does not have to be connected to unit k, which makes the algorithm non-local.
Local changes in the network can have an effect anywhere in the network.
In RTRL, the gradient information is forward-propagated. Using Equa-
tions 6 and 7, the output yk(t+ 1) at time step t+ 1 is given by
yk(t+ 1) = fk(zk(t+ 1)) (16)
with the weighted input
zk(t+ 1) = X
W[k,l]X[k,l](t+ 1),with lPre (k)
W[k,v]yv(t) + X
W[k,i]yi(t+ 1).
By differentiating Equations 15, 16 and 17, we can calculate results for all
time steps t+ 1 with
uv(t+ 1) = yk(t+ 1)
W[k,l]X[k,l](t+ 1)
k(zk(t+ 1))
W[k,l]X[k,l](t+ 1)
k(zk(t+ 1))
X[k,l](t+ 1)
∂X[k,l](t+ 1)
k(zk(t+ 1))
δukX[u,v ](t+ 1) +
∂yi(t+ 1)
| {z }
= 0 because yi(t+ 1)
is independent of W[u,v]
k(zk(t+ 1)) "δuk X[u,v](t+ 1) + X
where δuk is the Kronecker delta; that is,
δuk =(1 if u=k
0 if otherwise,
Assuming that the initial state of the network has no functional dependency on
the weights, the derivative for the first time step is
uv(t0) = yk(t0)
= 0 . (19)
Equation 18 shows how pk
uv(t+ 1) can be calculated in terms of pk
uv(t). In
this sense, the learning algorithm becomes incremental, so that we can learn
as we receive new inputs (in real time), and we no longer need to perform
back-propagation through time.
Knowing the initial value for pk
uv at time t0from Equation 19, we can re-
cursively calculate the quantities pk
uv for the first and all subsequent time steps
using Equation 18. Note that pk
uv(τ) uses the values of W[u,v]at t0, and not
values in-between t0and τ. Combining these values with the error vector e(τ)
for that time step, using Equation 14, we can finally calculate the negative error
gradient 5W E(τ). The final weight change for W[u,v]can be calculated using
Equations 14 and 13.
A more detailed description of the RTRL algorithm is given in [75] and [76].
7 Solving the Vanishing Error Problem
Standard RNN cannot bridge more than 5–10 time steps ([22]). This is due to
that back-propagated error signals tend to either grow or shrink with every time
step. Over many time steps the error therefore typically blows-up or vanishes
([5, 42]). Blown-up error signals lead straight to oscillating weights, whereas
with a vanishing error, learning takes an unacceptable amount of time, or does
not work at all.
The explanation of how gradients are computed by the standard backpropa-
gation algorithm and the basic vanishing error analysis is as follows: we update
weights after the network has trained from time t0to time tusing the formula
W[u,v]=ηEtotal(t0, t)
∂Etotal (t0, t)
where the backpropagated error signal at time τ(with t0τ < t) of the unit u
ϑu(τ) = f0
u(zu(τ)) X
Wvuϑv(τ+ 1)!.(20)
Consequently, given a fully recurrent neural network with a set of non-input
units U, the error signal that occurs at any chosen output-layer neuron oO,
at time-step τ, is propagated back through time for tt0time-steps, with t0< t
to an arbitrary neuron v. This causes the error to be scaled by the following
v(zv(t0))W[o,v]if tt0= 1,
v(zv(t0)) PuU
∂ϑo(t)W[u,v]if tt0>1
To solve the above equation, we unroll it over time. For t0τt, let uτbe
a non-input-layer neuron in one of the replicas in the unrolled network at time
τ. Now, by setting ut=vand ut0=o, we obtain the equation
... X
ut1U t
uτ(zuτ(tτ+t0))W[uτ,uτ1]!. (21)
Observing Equation 21, it follows that if
uτ(zuτ(tτ+t0))W[uτ,uτ1]|>1 (22)
for all τ, then the product will grow exponentially, causing the error to blow-up;
moreover, conflicting error signals arriving at neuron vcan lead to oscillating
weights and unstable learning. If now
uτ(zuτ(tτ+t0))W[uτ,uτ1]|<1 (23)
for all τ, then the product decreases exponentially, causing the error to vanish,
preventing the network from learning within an acceptable time period. Finally,
the equation X
shows that if the local error vanishes, then the global error also vanishes.
A more detailed theoretical analysis of the problem with long-term depen-
dencies is presented in [39]. The paper also briefly outlines several proposals on
how to address this problem.
8 Long Short-Term Neural Networks
One solution that addresses the vanishing error problem is a gradient-based
method called long short-term memory (LSTM) published by [41], [42], [22]
and [23]. LSTM can learn how to bridge minimal time lags of more than 1,000
discrete time steps. The solution uses constant error carousels (CECs), which
enforce a constant error flow within special cells. Access to the cells is handled
by multiplicative gate units, which learn when to grant access.
8.1 Constant Error Carousel
Suppose that we have only one unit uwith a single connection to itself. The
local error back flow of uat a single time-step τfollows from Equation 20 and
is given by
ϑu(τ) = f0
u(zu(τ))W[u,u]ϑu(τ+ 1).
From Equations 22 and 23 we see that, in order to ensure a constant error flow
through u, we need to have
u(zu(τ))W[u,u]= 1.0
and by integration we have
fu(zu(τ)) = zu(τ)
From this, we learn that fumust be linear, and that u’s activation must remain
constant over time; i.e.,
yu(τ+ 1) = fu(zu(τ+ 1)) = fu(yu(τ)W[u,u]) = yu(τ).
This is ensured by using the identity function fu=id, and by setting W[u,u]=
1.0. This preservation of error is called the constant error carousel (CEC), and
it is the central feature of LSTM, where short-term memory storage is achieved
for extended periods of time. Clearly, we still need to handle the connections
from other units to the unit u, and this is where the different components of
LSTM networks come into the picture.
8.2 Memory Blocks
In the absence of new inputs to the cell, we now know that the CEC’s backflow
remains constant. However, as part of a neural network, the CEC is not only
connected to itself, but also to other units in the neural network. We need
to take these additional weighted inputs and outputs into account. Incoming
connections to neuron ucan have conflicting weight update signals, because
the same weight is used for storing and ignoring inputs. For weighted output
connections from neuron u, the same weights can be used to both retrieve u’s
contents and prevent u’s output flow to other neurons in the network.
To address the problem of conflicting weight updates, LSTM extends the
CEC with input and output gates connected to the network input layer and
to other memory cells. This results in a more complex LSTM unit, called a
memory block; its standard architecture is shown in Figure 11.
The input gates, which are simple sigmoid threshold units with an activation
function range of [0,1], control the signals from the network to the memory cell
by scaling them appropriately; when the gate is closed, activation is close to
zero. Additionally, these can learn to protect the contents stored in ufrom
disturbance by irrelevant signals. The activation of a CEC by the input gate is
defined as the cell state. The output gates can learn how to control access to
the memory cell contents, which protects other memory cells from disturbances
originating from u. So we can see that the basic function of multiplicative gate
units is to either allow or deny access to constant error flow through the CEC.
9 Training LSTM-RNNs - the Hybrid Learning
In order to preserve the CEC in LSTM memory block cells, the original formu-
lation of LSTM used a combination of two learning algorithms: BPTT to train
network components located after cells, and RTRL to train network components
located before and including cells. The latter units work with RTRL because
there are some partial derivatives (related to the state of the cell) that need to
be computed during every step, no matter if a target value is given or not at that
step. For now, we only allow the gradient of the cell to be propagated through
time, truncating the rest of the gradients for the other recurrent connections.
We define discrete time steps in the form τ= 1,2,3, .... Each step has a
forward pass and a backward pass; in the forward pass the output/activation
of all units are calculated, whereas in the backward pass, the calculation of the
error signals for all weights is performed.
9.1 The Forward Pass
Let Mbe the set of memory blocks. Let mcbe the c-th memory cell in the
memory block m, and W[u,v]be a weight connecting unit uto unit v.
In the original formulation of LSTM, each memory block mis associated with
one input gate inmand one output gate outm. The internal state of a memory
cell mcat time τ+ 1 is updated according to its state smc(τ) and according
to the weighted input zmc(τ+ 1) multiplied by the activation of the input gate
yinm(τ+ 1). Then, we use the activation of the output gate zoutm(τ+ 1) to
calculate the activation of the cell ymc(τ+ 1).
The activation yinmof the input gate inmis computed as
yinm(τ+ 1) = finm(zinm(τ+ 1)) (24)
Figure 10: A standard LSTM memory block. The block contains (at least) one
cell with a recurrent self-connection (CEC) and weight of ‘1’. The state of the
cell is denoted as sc. Read and write access is regulated by the input gate, yin,
and the output gate, yout. The internal cell state is calculated by multiplying
the result of the squashed input, g, by the result of the input gate, yin , and
then adding the state of the last time step, sc(t1). Finally, the cell output
is calculated by multiplying the cell state, sc, by the activation of the output
gate, yout.
outputs to
next layer
ymc(τ+ 1)
youtm(τ+ 1) output
smc(τ+ 1) CEC 1.0
yinm(τ+ 1) input
yi(τ+ 1)
u: non-input unit
i: input unit
Figure 11: A standard LSTM memory block. The block contains (at least)
one cell with a recurrent self-connection (CEC) and weight of ‘1’. The state
of the cell is denoted as sc. Read and write access is regulated by the input
gate, yin , and the output gate, yout . The internal cell state is calculated by
multiplying the result of the squashed input, g(x), by the result of the input
gate and then adding the state of the current time step, smc(τ), to the next,
smc(τ+ 1). Finally, the cell output is calculated by multiplying the cell state
by the activation of the output gate.
Figure 12: A three cell LSTM memory block with recurrent self-connections
with the input gate input
zinm(τ+ 1) = X
W[inm,u]X[u,inm](τ+ 1),with uPre (inm),
W[inm,v]yv(τ) + X
W[inm,i]yi(τ+ 1).(25)
The activation of the output gate outmis
youtm(τ+ 1) = foutm(zoutm(τ+ 1)) (26)
with the output gate input
zoutm(τ+ 1) = X
W[outm,u]X[u,outm](τ+ 1),with uPre (outm).
W[outm,v]yv(τ) + X
W[outm,i]yi(τ+ 1).(27)
The results of the gates are scaled using the non-linear squashing function
finm=foutm=f, defined by
f(s) = 1
1 + es(28)
so that they are within the range [0,1]. Thus, the input for the memory cell
will only be able to pass if the signal at the input gate is sufficiently close to ‘1’.
For a memory cell mcin the memory block m, the weighted input zmc(τ+1)
is defined by
zmc(τ+ 1) = X
W[mc,u]X[u,mc](τ+ 1),with uPre (mc).
W[mc,v]yv(τ) + X
W[mc,i]yi(τ+ 1).(29)
As we mentioned before, the internal state smc(τ+1) of the unit in the memory
cell at time τ+ 1 is computed differently; the weighted input is squashed and
then multiplied by the activation of the input gate, and then the state of the
last time step smc(τ) is added. The corresponding equation is
smc(τ+ 1) = smc(τ) + yinm(τ+ 1)g(zmc(τ+ 1)) (30)
with smc(0) = 0 and the non-linear squashing function for the cell input
g(z) = 4
1 + ez2 (31)
which, in this case, scales the result to the range [2,2].
The output ymcis now calculated by squashing and multiplying the cell state
smcby the activation of the output gate youtm:
ymc(τ+ 1) = youtm(τ+ 1)h(smc(τ+ 1)).(32)
with the non-linear squashing function
h(z) = 2
1 + ez1 (33)
with range [1,1].
Assuming a layered, recurrent neural network with standard input, standard
output and hidden layer consisting of memory blocks, the activation of the
output unit ois computed as
yo(τ+ 1) = fo(zo(τ+ 1)) (34)
zo(τ+ 1) = X
W[o,u]yu(τ+ 1).(35)
where Gis the set of gate units, and we can again use the logistic sigmoid in
Equation 28 as a squashing function fo.
9.2 Forget Gates
The self-connection in a standard LSTM network has a fixed weight set to ‘1’ in
order to preserve the cell state over time. Unfortunately, the cell states smtend
to grow linearly during the progression of a time series presented in a continuous
input stream. The main negative effect is that the entire memory cell loses its
memorising capability, and begins to function like an ordinary RNN network
By manually resetting the state of the cell at the beginning of each sequence,
the cell state growth can be limited, but this is not practical for continuous input
where there is no distinguishable end, or subdivision is very complex and error
To address this problem, [22] suggested that an adaptive forget gate could
be attached to the self-connection. Forget gates can learn to reset the internal
state of the memory cell when the stored information is no longer needed. To
this end, we replace the weight ‘1.0’ of the self-connection from the CEC with
a multiplicative, forget gate activation yϕ, which is computed using a similar
method as for the other gates:
yϕm(τ+ 1) = fϕm(zϕm(τ+ 1) + bϕm) , (36)
where fis the squashing function from Equation 28 with a range [0,1], bϕmis
the bias of the forget gate, and
zϕm(τ+ 1) = X
W[ϕm,u]X[u,ϕm](τ+ 1),with uPre (ϕm).
W[ϕm,v]yv(τ) + X
W[ϕm,i]yi(τ+ 1).(37)
Originally, bϕmis set to 0, however, following the recommendation by [47], we
fix bϕmto 1, in order to improve the performance of LSTM (see Section 10.3).
The updated equation for calculating the internal cell state smcis
smc(τ+ 1) = smc(τ)yϕm(τ+ 1)
| {z }
=1 without
forget gate
+yinm(τ+ 1)g(zmc(τ+ 1)) (38)
with smc(0) = 0 and using the squashing function in Equation 31, with a range
[2,2]. The extended forward pass is given simply by exchanging Equation 30
for Equation 38.
The bias weights of input and output gates are initialised with negative
values, and the weights of the forget gate are initialised with positive values.
From this, it follows that at the beginning of training, the forget gate activation
will be close to ‘1.0’. The memory cell will behave like a standard LSTM memory
cell without a forget gate. This prevents the LSTM memory cell from forgetting,
before it has actually learned anything.
9.3 Backward Pass
LSTM incorporates elements from both BPTT and RTRL. Thus, we separate
units into two types: those units whose weight changes are computed using a
variation of BPTT (i.e, output units, hidden units, and the output gates), and
those whose weight changes are computed using a variation of RTRL (i.e., the
input gates, the forget gates and the cells).
Following the notation used in previous sections, and using Equations 8
and 10, the overall network error at time step τis
E(τ) = 1
| {z }
)2. (39)
Let us first consider units that work with BPTT. We define the notion of
individual error of a unit uat time τby
ϑu(τ) = ∂E(τ)
where zuis the weighted input of the unit. We can expand the notion of weight
contribution as follows
W[u,v](τ) = ηE(τ)
The factor zu(τ)
∂W[u,v]corresponds to the input signal that comes from the unit vto
the unit u. However, depending on the nature of u, the individual error varies.
If uis equal to an output unit o, then
ϑo(τ) = f0
thus, the weight contribution of output units is
W[o,v](τ) = ηϑo(τ)X[v ,o](τ).
Now, if uis equal to a hidden unit hlocated between cells and output units,
ϑh(τ) = f0
h(zh(τ)) X
where Ois the set of output units, and the weight contribution of hidden units
W[h,v](τ) = ηϑh(τ)X[v ,h](τ).
Finally, if uis equal to the output gate outmof the memory block m, then
outm(zoutm(τ)) X
h(smc(τ)) X
where tr
= means the equality only holds if the error is truncated so that it does
not propagate “too much”; that is, it prevents the error from propagating back
to the unit via its own feedback connection. Finally, the weight contribution for
output gates is
W[outm,v](τ) = ηϑoutm(τ)X[v ,outm](τ).
Let us now consider units that work with RTRL. In this case, the individual
errors of the input gate and the forget gate revolve around the individual error
of the cells in the memory block. We define the individual error of the cell mc
of the memory block mby
∂smc(τ)+ϑmc(τ+ 1)yϕm(τ+ 1)
| {z }
recurrent connection
∂smc(τ) X
∂zo(τ)!+ϑmc(τ+ 1)yϕm(τ+ 1)
=youtm(τ)h0(smc(τ)) X
W[o,mc]ϑo(τ)!+ϑmc(τ+ 1)yϕm(τ+ 1).
Note that this equation does not consider the recurrent connection between
the cell and other units, propagating back in time only the error through its
recurrent connection (accounting for the influence of the forget gate). We use
the following partial derivatives to expand the weight contribution for the cell
as follows
W[mc,v](τ) = ηE(τ)
and the weight contribution for forget and input gates as follows
W[u,v](τ) = ηE(τ)
Now, we need to define what is the value of ∂smc(τ+1)
∂W[u,v]. As expected, these also
depend on the nature of the unit u. If uis equal to the cell mc, then
∂smc(τ+ 1)
yϕm(τ+1)+g0(zmc(τ+1))finm(zinm(τ+1))yv(τ). (44)
Now, if uis equal to the input gate inm, then
∂smc(τ+ 1)
inm(zinm(τ+1))yv(τ). (45)
Finally, if uis equal to a forget gate ϕm, then
∂smc(τ+ 1)
yϕm(τ+ 1) + smc(τ)f0
ϕm(zϕm(τ+ 1))yv(τ). (46)
with smc(0) = 0. A more detailed version of the LSTM backward pass with
forget gates is described in [22].
9.4 Complexity
In this section, we present a complexity measure following the same principles
that Gers used in [22]; namely, we assume that every memory block contains the
same number of cells (usually one), and that output units only receive signals
from cell units and not from other units in the network. Let B, C, I n, Out be
the number of of memory blocks, memory cells in each block, input units and
output units, respectively. Now, for each memory block we need to resolve the
(recurrent) connections for each cell, input gate, forget gate and output gate.
Solving these connections yields a complexity measure of
(B. C )
| {z }
+ (B. C )
| {z }
input gates
+ (B. C )
| {z }
forget gates
+B. C
output gates
O B2. C 2.(47)
We also need to solve the connections from input units and to output units;
these are, respectively
In. B. S O (I n. B. S),(48)
Out. B. S ∼ O (Out. B. S).(49)
The numbers B, C, I n and Out do not change as the network executes, and, at
each step, the number of weight updates is bounded by the number of connec-
tions; thus, we can say that LSTM’s computational complexity per step and
weight is O(1).
9.5 Strengths and limitations of LSTM-RNNs
According to [23], LSTM excels on tasks in which a limited amount of data
must be remembered for a long time. This property is attributed to the use
of memory blocks. Memory blocks are interesting constructions: they have
access control in the form of input and output gates; which prevent irrelevant
information from entering or leaving the memory block. Memory blocks also
have a forget gate which weights the information inside the cells, so whenever
previous information becomes irrelevant for some cells, the forget gate can reset
the state of the different cell inside the block. Forget gates also enable continuous
prediction [54], because they can make cells completely forget their previous
state; preventing biases in prediction.
Like other algorithms, LSTM requires the topology of the network to be
fixed a priori. The number of memory blocks in networks does not change dy-
namically, so the memory of the network is ultimately limited. Moreover, [23]
point out that it is unlikely to overcome this limitation by increasing the net-
work size homogeneously, and suggest that modularisation promotes effective
learning. The process of modularisation is, however, “not generally clear”.
10 Problem specific topologies
LSTM-RNN permits many different variants and topologies. These partially
problem specific and can be derived [3] from the basic method [41], [21] cov-
ered in Section 8 and 9. More recently the basic method is referenced to as
‘vanilla’ LSTM, which used in practise these days only with various extensions
and modifications. In the following sections we cover the most common in use,
namely bidirectional LSTM (BLSTM-CTC) ([34], [27], [31]), Grid LSTM (or
N-LSTM) [49] and Gated Recurrent Unit (GRU) ([10], [13]). There are vari-
ous variants of Grid LSTM. The most important to note are Multidimensional
LSTM ( [29], [35]), Stacked LSTM ([18], [33], [68]). Specifically we would like to
also point out the more recent variant Sequence-to-Sequence ([68], [36], [8], [80],
[69]) and attention-based learning [12], which are both important to mention in
the context of cognitive learning tasks.
10.1 Bidirectional LSTM
Conventional RNNs analyse, for any given point in a sequence, just one di-
rection during processing: the past. The work published in [34] explores the
possibility of analysing both the future as well as the past of a given point in
the context of LSTM. At a very high level, bidirectional means that the input
is presented forwards and backwards to two separate LSTM networks, both of
which are connected to the same output layer. According to [34], bidirectional
training possesses an architectural advantage over unidirectional training if used
to classify phonemes.
Bidirectional LSTM removes the one-step truncation originally present in
LSTM, and implements a full error gradient calculation. This full error gradient
approach eased the implementation of bidirectional LSTM, and allowed it to be
trained using standard BPTT.
In 2006 [28] introduced an RNN objective function named Connectionist
Temporal Classification (CTC). The advantage of CTC is that it enables the
LSTM-RNN to handle input data not segmented into sequences. This is impor-
tant if the correct segmentation of data is difficult to achieve (e.g. separation
of letters in handwriting). Later this lead to the now common variant BLSTM-
CTC as documented by [52, 19, 27].
10.2 Grid LSTM
Grid LSTM presented by [49] is an attempt to generalise the advantages of
LSTM – including its ability to select or ignore inputs– into deep networks of
a unified architecture. An N-dimensional grid LSTM or N-LSTM is a network
arranged in a grid of Ndimensions, with LSTM cells along and in-between some
(or all) of the dimensions, enabling communication among consecutive layers.
Grid LSTM is analogous to the stacked LSTM [33], but it adds cells along the
depth dimension too, i.e., in-between layers. Additionally, N-LSTM networks
with N > 2 are analogous to multidimensional LSTM [29], but they differ
again by the cells along the depth dimension, and by the ability of grid LSTM
networks to modulate the interaction among layers such that it is not prone to
the instability present in Multidimensional LSTM.
Consider a trained LSTM network with weights W, whose hidden cells emit
a collection of signals represented by the vector ~yhand whose memory units
emit a collection of signals represented by the vector ~yh. Whenever this LSTM
network is provided an input vector ~x, there is a change in the signals emitted
by both hidden units and memory cells; let ~yh
0and ~sm
0represent the new values
of signals. Let Pbe a projection matrix, the concatenation of the new input
signals and the recurrent signals is given by
X=P ~x
An LSTM transform, which changes the values of hidden and memory signals
as previously mentioned, can be formulated as follows:
(X, ~sm)W
0, ~sm
0) (51)
Before we explain in detail the architecture of Grid LSTM blocks, we quickly
review Stacked LSTM and Multidimensional LSTM architectures.
10.2.1 Stacked LSTM
A stacked LSTM [33], as its name suggests, stacks LSTM layers on top of each
other in order to increase capacity. At a high level, to stack NLSTM networks,
we make the first network have X1as defined in Equation (52), but we make
the i-th network have Xidefined by
instead, replacing the input signals ~x with the hidden signals from the previous
LSTM transform, effectively “stacking” them.
10.2.2 Multidimensional LSTM
In Multidimensional LSTM networks [29], inputs are structured in an N-dimensional
grid instead of being sequences of values; for example, a solid expressed as a
three-dimensional array of voxels. To use this structure of inputs, Multidimen-
sional LSTM networks increase the number of recurrent connections from 1 to
N; thus, an N-dimensional LSTM receives Nhidden vectors ~yh1, . . . , ~yhNand
Nmemory vectors ~sm1, . . . , ~smNas input, then the network outputs a single
hidden vector ~yhand a single memory vector ~sm. For multidimensional LSTM
networks, we define Xby
P ~x
and the memory signal vector ~smis calculated using
where is the Hadamard product, ~ϕ is a vector consisting of Nforget signals
(one for each ~yhi), and ~
inmand ~zmrespectively correspond to the signals of
the input gate and the weighted input of the memory cell (see Equation (38) to
compare Equation (54) with the standard calculation of ~sm).
10.2.3 Grid LSTM Blocks
Due to the high number of connections, large multidimensional LSTM net-
works are usually unstable [49]. Grid LSTM offers an alternate way of comput-
ing the new memory vector. However, unlike multidimensional LSTM, a Grid
LSTM block outputs Nhidden vectors ~yh
1, . . . , ~yh
Nand Nmemory vectors
1, . . . , ~sm
Nthat are all distinct. To do so, the model concatenates the hidden
vectors from the Ndimensions as follows
The grid LSTM block computes NLSTM transforms, one for each dimension,
as follows
(X, ~sm1)W1
1, ~sm
(X, ~smN)WN
N, ~sm
Each transform applies standard LSTM across its respective dimension. Having
Xas input to all transforms represents the sharing of hidden signals across the
different dimension of the grid; note that each transform independently manages
its memory signals.
10.3 Gated Recurrent Unit (GRU)
[10] propose the Gated Recurrent Unit (GRU) architecture for RNN as an al-
ternative to LSTM. GRU has empirically been found to outperform LSTM on
nearly all tasks, except language modelling with naive initialization [47]. GRU
units, unlike LSTM memory blocks, do not have a memory cell; although they
do have gating units: a reset gate and an update gate. More precisely, let H
be the set of GRU units; if uH, then we define the activation yresu(τ+ 1) of
the reset gate resuat time τ+ 1 by
yresu(τ+ 1) = fresu(sresu(τ+ 1)) ,(57)
where fresuis the squashing function of the reset gate (usually a sigmoid func-
tion), and sresu(τ+ 1) is the state of the reset gate resuat time τ+ 1, which
is defined by
sresu(τ+ 1) = zresu(τ+ 1) + bresu,(58)
where bresuis the bias of the reset gate, and zresu(τ+ 1) is the weighted input
of the reset gate at time τ+ 1, which is in turn defined by
zresu(τ+ 1) = X
W[resu,u]X[u,resu](τ+ 1),with uPre (resu) ; (59)
W[resu,h]yh(τ) + X
W[resu,i]yi(τ+ 1),(60)
where Iis the set of input units.
Similarly, we define define the activation yupdu(τ+ 1) of the update gate
upduat time τ+ 1 by
yupdu(τ+ 1) = fupdusupdu(τ+ 1)(61)
where fupduis the squashing function of the update gate (again, usually a sig-
moid function), and supdu(τ+ 1) is the state of the update gate upduat time
τ+ 1, defined by
supdu(τ+ 1) = zupdu(τ+ 1) + bupdu,(62)
where bupduis the bias of the update gate, and zupdu(τ+ 1) is the weighted input
of the update gate at time τ+ 1, which in turn is defined by
zupdu(τ+ 1) = X
W[updu,u]X[u,updu](τ+ 1),with uPre (updu) ; (63)
W[updu,h]yh(τ) + X
W[updu,i]yi(τ+ 1),(64)
GRU reset and input gates behave like normal units in a recurrent network.
The main characteristic of GRU is the way the activation of the GRU units is
defined. A GRU unit uHhas an associated candidate activation eyu(τ+ 1)
at time τ+ 1, formally defined by
eyu(τ+ 1) = fu
W[u,i]yi(τ+ 1)
| {z }
External input at time τ+ 1
+yresu(τ+ 1) X
| {z }
Gated recurrent connection
where fuis usually tanh, and the activation yu(τ+ 1) of the GRU unit uat
time τ+ 1 is defined by
yu(τ+ 1) = yupdu(τ+ 1)yu(τ) + (1 yupdu(τ+ 1))eyu(τ+ 1) (66)
Note the similarities between Equations (38) and (66). The factor yupdu(τ+ 1)
appears to emulate the function of the forget gate of LSTM, while the factor
(1yupdu(τ+ 1)) appears to emulate the function of the the input gate of LSTM.
11 Applications of LSTM-RNN
In this final section we cover a selection of well-known publications which proved
relevant over time.
11.1 Early learning tasks
In early experiments LSTM proved applicable to various learning tasks, pre-
viously considered impossible to learn. This included recalling high precision
real numbers over extended noisy sequences [41], learning context free lan-
guages [21], and various tasks that require precise timing and counting [23].
In [43] LSTM was successfully introduced to meta-learning with a program
search tasks to approximate a learning algorithm for quadratic functions. The
successful application of reinforcement learning to solve non-Markovian learning
tasks with long-term dependencies was shown by [2].
11.2 Cognitive learning tasks
LSTM-RNNs proved great strengths in solving a large variety of cognitive learn-
ing tasks. Speech and handwriting recognition, and more recently machine
translation are the most predominant in literature. Other cognitive learn-
ing tasks include emotion recognition from speech [78], text generation [67],
handwriting generation [24], constituency parsing [71], and conversational mod-
elling [72].
11.2.1 Speech recognition
A first indication of the capabilities of neural networks in tasks related to nat-
ural language was given by [4] with a neural language modelling task. In 2003
good results applying standard LSTM-RNN networks with a mix of LSTM and
sigmoidal units to speech recognition tasks were obtained by [25, 26]. Better
results comparable to Hidden-Markov-Model (HMM)-based systems [7] were
achieved using bidirectional training with BLSTM [6, 34]. A variant named
BLSTM-CTC [28, 19, 17] finally outperformed HMMs, with recent improve-
ments documented in [44, 77]. A deep variant of stacked BLSTM-CTC was
used in 2013 by [33] and later extended with a modified CTC objective func-
tion by [30], both achieving outstanding results. The performance of different
LSTM-RNN architectures on large vocabulary speech recognition tasks was in-
vestigated by [63], with best results using an LSTM/HMM hybrid architecture.
Comparable results were achieved by [20].
More recently LSTM was improving results using the sequence-to-sequence
framework ([68]) and attention-based learning ([11] [12]). In 2015 [8] introduced
an specialised architecture for speech recognition with two functions, the first
called ‘listener’ and the latter called ‘attend and spell’. The ‘listener’ function
uses BLSTM with a pyramid structure (pBLSTM), similar to clockwork RNNs
introduced by [50]. The other function, ‘attend and spell’, uses an attention-
based LSTM transducer developed by [1] and [12]. Both functions are trained
with methods introduced in the sequence-to-sequence framework [68] and in
attention-based learning [1].
11.2.2 Handwriting recognition
In 2007 [52] introduced BLSTM-CTC and applied it to online handwriting recog-
nition, with results later outperforming Hidden-Markov-based recognition sys-
tems presented by [32]. [27] combined BLSTM-CTC with a probabilistic lan-
guage model and by this developed a system capable of directly transcribing raw
online handwriting data. In a real-world use case this system showed a very high
automation rate with an error rate comparable to a human on this kind of task
( [57]). In another approach [35] combined BLSTM-CTC with multidimensional
LSTM and applied it to an offline handwriting recognition task, as well outper-
forming classifiers based on Hidden-Markov models. In 2013 [81, 61] applied the
very successful regularisation method dropout as proposed by [37, 64]).
11.2.3 Machine translation
In 2014 [10] the authors applied the RNN encoder-decoder neural network ar-
chitecture to machine translation and improved the performance of a statistical
machine translation system. The RNN Encoder-Decoder architecture is based
on an approach communicated by [48]. A very similar deep LSTM architecture,
referred to as sequence-to-sequence learning, was investigated by [68] confirming
these results. [53] addressed the rare word problem using sequence-to-sequence,
which improves the ability to translate words not in the vocabulary. The archi-
tecture was further improved by [1] addressing issues related to the translation
of long sentences by implementing an attention mechanism into the decoder.
11.2.4 Image processing
In 2012 BSLTM was applied to keyword spotting and mode detection distin-
guishing different types of content in handwritten documents, such as text,
formulas, diagrams and figures, outperforming HMMs and SVMs [44, 45, 59].
At approximately the same period of time [51] investigated the classification
of high-resolution images from the ImageNet database with considerable better
results then previous approaches. In 2015 the more recent LSTM variant using
the Sequence-to-Sequence framework was successfully trained by [73, 79] to gen-
erate natural sentences in plain English describing images. Also in 2015 [14] the
authors combined LSTMs with a deep hierarchical visual feature extractor and
applied the model to image interpretation and classification tasks, like activity
recognition and image/video description.
11.3 Other learning tasks
Early papers applied LSTM-RNN to a number of real world problems pushing its
evolution further. Covered problems include protein secondary structure predic-
tion [40, 9] and music generation [15]. Network security was covered in [65, 66]
were the authors apply LSTM-RNN to the DARPA intrusion detection dataset.
In [80, 70] the authors apply computational tasks to LSTM-RNN. In 2014
the authors of [80] evaluate short computer programs using the Sequence-to-
Sequence framework. One year later the authors of [70] use a modified version
of the framework to learn solutions of combinatorial optimisation problems.
12 Conclusions
In this article, we covered the derivation of LSTM in detail, summarising the
most relevant literature. Specifically, we highlighted the vanishing error prob-
lem, which is a serious shortcoming of RNNs. LSTM provides a possible solution
to this problem by introducing a constant error flow through the internal states
of special memory cells. In this way, LSTM is able to tackle long time-lag prob-
lems, bridging time intervals in excess of 1,000 time steps. Finally, we introduced
two LSTM extensions that enable LSTM to learn self-resets and precise timing.
With self-resets, LSTM is able to free memory of irrelevant information.
This work was mainly pushed as a private pro ject from Ralf C. Staudemeyer
spanning a period of ten years from 2007–17. During the time 2013–15 it was
partially supported by post-doctoral fellowship research funds provided by the
South African National Research Foundation, Rhodes University, the University
of South Africa, and the University of Passau. The co-author Eric Rothstein
Morris picked-up the loose ends, developed the unified notation for this article
in 2015–16.
We acknowledge support for this work from Ralf’s Ph.D. supervisor Chris-
tian W. Omlin for raising the authors interest to investigate the capabilities of
Long Short-Term Memory Recurrent Neural Networks. Very special thanks go
to Arne Janza for doing the internal review. Without his dedicated support to
eliminate a number of hard to find logical inconsistencies this publication would
not have found its way to the reader.
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ResearchGate has not been able to resolve any citations for this publication.
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In this paper we compare different types of recurrent units in recurrent neural networks (RNNs). Especially, we focus on more sophisticated units that implement a gating mechanism, such as a long short-term memory (LSTM) unit and a recently proposed gated recurrent unit (GRU). We evaluate these recurrent units on the tasks of polyphonic music modeling and speech signal modeling. Our experiments revealed that these advanced recurrent units are indeed better than more traditional recurrent units such as tanh units. Also, we found GRU to be comparable to LSTM.
Sequences have become first class citizens in supervised learning thanks to the resurgence of recurrent neural networks. Many complex tasks that require mapping from or to a sequence of observations can now be formulated with the sequence-to-sequence (seq2seq) framework which employs the chain rule to efficiently represent the joint probability of sequences. In many cases, however, variable sized inputs and/or outputs might not be naturally expressed as sequences. For instance, it is not clear how to input a set of numbers into a model where the task is to sort them; similarly, we do not know how to organize outputs when they correspond to random variables and the task is to model their unknown joint probability. In this paper, we first show using various examples that the order in which we organize input and/or output data matters significantly when learning an underlying model. We then discuss an extension of the seq2seq framework that goes beyond sequences and handles input sets in a principled way. In addition, we propose a loss which, by searching over possible orders during training, deals with the lack of structure of output sets. We show empirical evidence of our claims regarding ordering, and on the modifications to the seq2seq framework on benchmark language modeling and parsing tasks, as well as two artificial tasks -- sorting numbers and estimating the joint probability of unknown graphical models.
This paper introduces Grid Long Short-Term Memory, a network of LSTM cells arranged in a multidimensional grid that can be applied to vectors, sequences or higher dimensional data such as images. The network differs from existing deep LSTM architectures in that the cells are connected between network layers as well as along the spatiotemporal dimensions of the data. It therefore provides a unified way of using LSTM for both deep and sequential computation. We apply the model to algorithmic tasks such as integer addition and determining the parity of random binary vectors. It is able to solve these problems for 15-digit integers and 250-bit vectors respectively. We then give results for three empirical tasks. We find that 2D Grid LSTM achieves 1.47 bits per character on the Wikipedia character prediction benchmark, which is state-of-the-art among neural approaches. We also observe that a two-dimensional translation model based on Grid LSTM outperforms a phrase-based reference system on a Chinese-to-English translation task, and that 3D Grid LSTM yields a near state-of-the-art error rate of 0.32% on MNIST.
Long Short-Term Memory (LSTM) is a specific recurrent neural network (RNN) architecture that was designed to model temporal sequences and their long-range dependencies more accurately than conventional RNNs. In this paper, we explore LSTM RNN architectures for large scale acoustic modeling in speech recognition. We recently showed that LSTM RNNs are more effective than DNNs and conventional RNNs for acoustic modeling, considering moderately-sized models trained on a single machine. Here, we introduce the first distributed training of LSTM RNNs using asynchronous stochastic gradient descent optimization on a large cluster of machines. We show that a two-layer deep LSTM RNN where each LSTM layer has a linear recurrent projection layer can exceed state-of-the-art speech recognition performance. This architecture makes more effective use of model parameters than the others considered, converges quickly, and outperforms a deep feed forward neural network having an order of magnitude more parameters.
We introduce a new neural architecture to learn the conditional probability of an output sequence with elements that are discrete tokens corresponding to positions in an input sequence. Such problems cannot be trivially addressed by existent approaches such as sequence-to-sequence and Neural Turing Machines, because the number of target classes in each step of the output depends on the length of the input, which is variable. Problems such as sorting variable sized sequences, and various combinatorial optimization problems belong to this class. Our model solves the problem of variable size output dictionaries using a recently proposed mechanism of neural attention. It differs from the previous attention attempts in that, instead of using attention to blend hidden units of an encoder to a context vector at each decoder step, it uses attention as a pointer to select a member of the input sequence as the output. We call this architecture a Pointer Net (Ptr-Net). We show Ptr-Nets can be used to learn approximate solutions to three challenging geometric problems -- finding planar convex hulls, computing Delaunay triangulations, and the planar Travelling Salesman Problem -- using training examples alone. Ptr-Nets not only improve over sequence-to-sequence with input attention, but also allow us to generalize to variable size output dictionaries. We show that the learnt models generalize beyond the maximum lengths they were trained on. We hope our results on these tasks will encourage a broader exploration of neural learning for discrete problems.
Neural Machine Translation (NMT) has recently attracted a lot of attention due to the very high performance achieved by deep neural networks in other domains. An inherent weakness in existing NMT systems is their inability to correctly translate rare words: end-to-end NMTs tend to have relatively small vocabularies with a single "unknown-word" symbol representing every possible out-of-vocabulary (OOV) word. In this paper, we propose and implement a simple technique to address this problem. We train an NMT system on data that is augmented by the output of a word alignment algorithm, allowing the NMT system to output, for each OOV word in the target sentence, its corresponding word in the source sentence. This information is later utilized in a post-processing step that translates every OOV word using a dictionary. Our experiments on the WMT'14 English to French translation task show that this simple method provides a substantial improvement over an equivalent NMT system that does not use this technique. The performance of our system achieves a BLEU score of 36.9, which improves the previous best end-to-end NMT by 2.1 points. Our model matches the performance of the state-of-the-art system while using three times less data.