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# Introduction
In this notebook we will reproduce the results of
[Deep Speech: Scaling up end-to-end speech recognition](http://arxiv.org/abs/1412.5567).
The core of the system is a bidirectional recurrent neural network (BRNN)
trained to ingest speech spectrograms and generate English text transcriptions.
Let a single utterance <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//332cc365a4987aacce0ead01b8bdcc0b.svg?invert_in_darkmode" align=middle width=9.359955pt height=14.10255pt/> and label <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//deceeaf6940a8c7a5a02373728002b0f.svg?invert_in_darkmode" align=middle width=8.61696pt height=14.10255pt/> be sampled from a training set
<img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//7b79365abe0c9abfa5d858b170803f9c.svg?invert_in_darkmode" align=middle width=224.067195pt height=29.12679pt/>.
Each utterance, <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//ad769e751231d17313953f80471b27a4.svg?invert_in_darkmode" align=middle width=24.229095pt height=29.12679pt/> is a time-series of length <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//3f623f5f39889e228ea730e85d8200bd.svg?invert_in_darkmode" align=middle width=26.722575pt height=29.12679pt/>
where every time-slice is a vector of audio features,
<img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//537fe6f40a03a282859c43fb7aa2874f.svg?invert_in_darkmode" align=middle width=24.229095pt height=34.27314pt/> where <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//e5f174df537eb831a8326efc975aa4b4.svg?invert_in_darkmode" align=middle width=99.190245pt height=29.12679pt/>.
We use MFCC as our features; so <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//a16a13264cbafca76e1829d3e0765a69.svg?invert_in_darkmode" align=middle width=24.947835pt height=34.27314pt/> denotes the <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//2ec6e630f199f589a2402fdf3e0289d5.svg?invert_in_darkmode" align=middle width=8.2397205pt height=14.10255pt/>-th MFCC feature
in the audio frame at time <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//4f4f4e395762a3af4575de74c019ebb5.svg?invert_in_darkmode" align=middle width=5.913963pt height=20.1465pt/>. The goal of our BRNN is to convert an input
sequence <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//332cc365a4987aacce0ead01b8bdcc0b.svg?invert_in_darkmode" align=middle width=9.359955pt height=14.10255pt/> into a sequence of character probabilities for the transcription
<img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//deceeaf6940a8c7a5a02373728002b0f.svg?invert_in_darkmode" align=middle width=8.61696pt height=14.10255pt/>, with <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//f559f3f08ccdab2e684d621045c067e7.svg?invert_in_darkmode" align=middle width=94.37901pt height=24.56553pt/>,
where <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//e3955e81b908e83f304c8cb51fa01cdf.svg?invert_in_darkmode" align=middle width=303.646695pt height=24.56553pt/>.
(The significance of <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//f99b9a2c90cd48431a7372de2cf47523.svg?invert_in_darkmode" align=middle width=39.76665pt height=22.74591pt/> will be explained below.)
Our BRNN model is composed of <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//9612eecfec9dadf1a81d296bd2473777.svg?invert_in_darkmode" align=middle width=8.188554pt height=21.10812pt/> layers of hidden units.
For an input <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//332cc365a4987aacce0ead01b8bdcc0b.svg?invert_in_darkmode" align=middle width=9.359955pt height=14.10255pt/>, the hidden units at layer <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//2f2322dff5bde89c37bcae4116fe20a8.svg?invert_in_darkmode" align=middle width=5.2088685pt height=22.74591pt/> are denoted <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//977354073f08eeac0f95545d71c2b77a.svg?invert_in_darkmode" align=middle width=23.879625pt height=29.12679pt/> with the
convention that <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//4187f3f834c6a9c9221fd863bd0da600.svg?invert_in_darkmode" align=middle width=26.199525pt height=29.12679pt/> is the input. The first three layers are not recurrent.
For the first layer, at each time <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//4f4f4e395762a3af4575de74c019ebb5.svg?invert_in_darkmode" align=middle width=5.913963pt height=20.1465pt/>, the output depends on the MFCC frame
<img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//23776aad854f2d33e83e4f4cad44e1b9.svg?invert_in_darkmode" align=middle width=14.30715pt height=14.10255pt/> along with a context of <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//9b325b9e31e85137d1de765f43c0f8bc.svg?invert_in_darkmode" align=middle width=12.876435pt height=22.38192pt/> frames on each side.
(We typically use <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//3b3d517eaf7a36186d0f2c9bf921e14f.svg?invert_in_darkmode" align=middle width=88.45122pt height=24.56553pt/> for our experiments.)
The remaining non-recurrent layers operate on independent data for each time step.
Thus, for each time <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//4f4f4e395762a3af4575de74c019ebb5.svg?invert_in_darkmode" align=middle width=5.913963pt height=20.1465pt/>, the first <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//5dc642f297e291cfdde8982599601d7e.svg?invert_in_darkmode" align=middle width=8.188554pt height=21.10812pt/> layers are computed by:
<p align="center"><img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//f1441afd023f38a19efe100013abeb0e.svg?invert_in_darkmode" align=middle width=184.80825pt height=21.23088pt/></p>
where <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//621ab064e859e3477fc9c023ef25de33.svg?invert_in_darkmode" align=middle width=189.575595pt height=24.56553pt/> is a clipped rectified-linear (ReLu)
activation function and <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//e7e30f89a8c77e163e112c244ba28bf2.svg?invert_in_darkmode" align=middle width=32.19414pt height=29.12679pt/>, <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//0b8618b400785e741ea027ec23718169.svg?invert_in_darkmode" align=middle width=21.47244pt height=29.12679pt/> are the weight matrix and bias
parameters for layer <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//2f2322dff5bde89c37bcae4116fe20a8.svg?invert_in_darkmode" align=middle width=5.2088685pt height=22.74591pt/>. The fourth layer is a bidirectional recurrent
layer[[1](http://www.di.ufpe.br/~fnj/RNA/bibliografia/BRNN.pdf)].
This layer includes two sets of hidden units: a set with forward recurrence,
<img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//c51574e4f666a0701fd9e09d47458ba5.svg?invert_in_darkmode" align=middle width=27.347265pt height=29.12679pt/>, and a set with backward recurrence <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//b05e38531483454a1d5ee778ad02b2f5.svg?invert_in_darkmode" align=middle width=25.430625pt height=29.12679pt/>:
<p align="center"><img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//a36ac2d62f39ae1a8b5ef15a33913cbe.svg?invert_in_darkmode" align=middle width=267.0822pt height=22.88484pt/></p>
<p align="center"><img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//9e58ffe596dc1893dfbfad2efcf4d2a4.svg?invert_in_darkmode" align=middle width=263.0628pt height=22.88484pt/></p>
Note that <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//c51574e4f666a0701fd9e09d47458ba5.svg?invert_in_darkmode" align=middle width=27.347265pt height=29.12679pt/> must be computed sequentially from <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//a762a11a67cb9ba897faa6c3cacedbbc.svg?invert_in_darkmode" align=middle width=35.97231pt height=21.10812pt/> to <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//301d7bc34a1d89cb52b64ce4cf16322e.svg?invert_in_darkmode" align=middle width=54.50643pt height=29.12679pt/>
for the <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//77a3b857d53fb44e33b53e4c8b68351a.svg?invert_in_darkmode" align=middle width=5.642109pt height=21.60213pt/>-th utterance, while the units <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//b05e38531483454a1d5ee778ad02b2f5.svg?invert_in_darkmode" align=middle width=25.430625pt height=29.12679pt/> must be computed
sequentially in reverse from <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//301d7bc34a1d89cb52b64ce4cf16322e.svg?invert_in_darkmode" align=middle width=54.50643pt height=29.12679pt/> to <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//a762a11a67cb9ba897faa6c3cacedbbc.svg?invert_in_darkmode" align=middle width=35.97231pt height=21.10812pt/>.
The fifth (non-recurrent) layer takes both the forward and backward units as inputs
<p align="center"><img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//5b71e9714fd57cbed1518f54d9674924.svg?invert_in_darkmode" align=middle width=177.2958pt height=19.479405pt/></p>
where <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//5544c9e9f0c04f486d43b9a513f821f4.svg?invert_in_darkmode" align=middle width=122.65968pt height=29.12679pt/>. The output layer are standard logits that
correspond to the predicted character probabilities for each time slice <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//4f4f4e395762a3af4575de74c019ebb5.svg?invert_in_darkmode" align=middle width=5.913963pt height=20.1465pt/> and
character <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//63bb9849783d01d91403bc9a5fea12a2.svg?invert_in_darkmode" align=middle width=9.041505pt height=22.74591pt/> in the alphabet:
<p align="center"><img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//7ba8b2b4fec039363453efb55129fc6b.svg?invert_in_darkmode" align=middle width=222.98595pt height=24.324465pt/></p>
Here <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//99568e8e69332dfe7e459b1be0064905.svg?invert_in_darkmode" align=middle width=23.792175pt height=34.27314pt/> denotes the <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//63bb9849783d01d91403bc9a5fea12a2.svg?invert_in_darkmode" align=middle width=9.041505pt height=22.74591pt/>-th bias and <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//616b07a8d9f2b9a6cb5ebab869587fab.svg?invert_in_darkmode" align=middle width=82.459575pt height=34.27314pt/> the <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//63bb9849783d01d91403bc9a5fea12a2.svg?invert_in_darkmode" align=middle width=9.041505pt height=22.74591pt/>-th
element of the matrix product.
Once we have computed a prediction for <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//e04f274d8b6ee9ec252857546b8b73b5.svg?invert_in_darkmode" align=middle width=24.105015pt height=22.74591pt/>, we compute the CTC loss
[[2]](http://www.cs.toronto.edu/~graves/preprint.pdf) <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//b97650f5d17fe16c62e9b16a11767663.svg?invert_in_darkmode" align=middle width=45.906795pt height=31.42161pt/>
to measure the error in prediction. During training, we can evaluate the gradient
<img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//1e46e5cdcab1a717717603eb80c5061b.svg?invert_in_darkmode" align=middle width=59.554275pt height=31.42161pt/> with respect to the network outputs given the
ground-truth character sequence <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//deceeaf6940a8c7a5a02373728002b0f.svg?invert_in_darkmode" align=middle width=8.61696pt height=14.10255pt/>. From this point, computing the gradient
with respect to all of the model parameters may be done via back-propagation
through the rest of the network. We use the Adam method for training
[[3](http://arxiv.org/abs/1412.6980)].
The complete BRNN model is illustrated in the figure below.
![DeepSpeech BRNN](../images/rnn_fig-624x548.png)

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# Geometric Constants
This is about several constants related to the geometry of the network.
## n_steps
The network views each speech sample as a sequence of time-slices <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//537fe6f40a03a282859c43fb7aa2874f.svg?invert_in_darkmode" align=middle width=24.229095pt height=34.27314pt/> of
length <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//3f623f5f39889e228ea730e85d8200bd.svg?invert_in_darkmode" align=middle width=26.722575pt height=29.12679pt/>. As the speech samples vary in length, we know that <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//3f623f5f39889e228ea730e85d8200bd.svg?invert_in_darkmode" align=middle width=26.722575pt height=29.12679pt/>
need not equal <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//1741c589c4fdb782aa008664d3cf9bf7.svg?invert_in_darkmode" align=middle width=28.173255pt height=29.12679pt/> for <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//4d56de33ada8539cfdcff54baea8310e.svg?invert_in_darkmode" align=middle width=35.19351pt height=22.74591pt/>. For each batch, BRNN in TensorFlow needs
to know `n_steps` which is the maximum <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//3f623f5f39889e228ea730e85d8200bd.svg?invert_in_darkmode" align=middle width=26.722575pt height=29.12679pt/> for the batch.
## n_input
Each of the at maximum `n_steps` vectors is a vector of MFCC features of a
time-slice of the speech sample. We will make the number of MFCC features
dependent upon the sample rate of the data set. Generically, if the sample rate
is 8kHz we use 13 features. If the sample rate is 16kHz we use 26 features...
We capture the dimension of these vectors, equivalently the number of MFCC
features, in the variable `n_input`
## n_context
As previously mentioned, the BRNN is not simply fed the MFCC features of a given
time-slice. It is fed, in addition, a context of <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//3b3d517eaf7a36186d0f2c9bf921e14f.svg?invert_in_darkmode" align=middle width=88.45122pt height=24.56553pt/> frames on
either side of the frame in question. The number of frames in this context is
captured in the variable `n_context`
Next we will introduce constants that specify the geometry of some of the
non-recurrent layers of the network. We do this by simply specifying the number
of units in each of the layers
## n_hidden_1, n_hidden_2, n_hidden_5
`n_hidden_1` is the number of units in the first layer, `n_hidden_2` the number
of units in the second, and `n_hidden_5` the number in the fifth. We haven't
forgotten about the third or sixth layer. We will define their unit count below.
A LSTM BRNN consists of a pair of LSTM RNN's.
One LSTM RNN that works "forward in time"
<img src="../images/LSTM3-chain.png" alt="LSTM" width="800">
and a second LSTM RNN that works "backwards in time"
<img src="../images/LSTM3-chain.png" alt="LSTM" width="800">
The dimension of the cell state, the upper line connecting subsequent LSTM units,
is independent of the input dimension and the same for both the forward and
backward LSTM RNN.
## n_cell_dim
Hence, we are free to choose the dimension of this cell state independent of the
input dimension. We capture the cell state dimension in the variable `n_cell_dim`.
## n_hidden_3
The number of units in the third layer, which feeds in to the LSTM, is
determined by `n_cell_dim` as follows
```python
n_hidden_3 = 2 * n_cell_dim
```
## n_character
The variable `n_character` will hold the number of characters in the target
language plus one, for the <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//2566b4f99fd69f63a00f0abbc8415767.svg?invert_in_darkmode" align=middle width=44.315865pt height=22.74591pt/>.
For English it is the cardinality of the set
<img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//620afbde0635e244e162a8b0a2bcb9cb.svg?invert_in_darkmode" align=middle width=270.722595pt height=24.56553pt/>
we referred to earlier.
## n_hidden_6
The number of units in the sixth layer is determined by `n_character` as follows
```python
n_hidden_6 = n_character
```

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# Parallel Optimization
This is how we implement optimization of the DeepSpeech model across GPU's on a
single host. Parallel optimization can take on various forms. For example
one can use asynchronous updates of the model, synchronous updates of the model,
or some combination of the two.
## Asynchronous Parallel Optimization
In asynchronous parallel optimization, for example, one places the model
initially in CPU memory. Then each of the <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//5201385589993766eea584cd3aa6fa13.svg?invert_in_darkmode" align=middle width=12.876435pt height=22.38192pt/> GPU's obtains a mini-batch of data
along with the current model parameters. Using this mini-batch each GPU then
computes the gradients for all model parameters and sends these gradients back
to the CPU when the GPU is done with its mini-batch. The CPU then asynchronously
updates the model parameters whenever it recieves a set of gradients from a GPU.
Asynchronous parallel optimization has several advantages and several
disadvantages. One large advantage is throughput. No GPU will every be waiting
idle. When a GPU is done processing a mini-batch, it can immediately obtain the
next mini-batch to process. It never has to wait on other GPU's to finish their
mini-batch. However, this means that the model updates will also be asynchronous
which can have problems.
For example, one may have model parameters <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//84c95f91a742c9ceb460a83f9b5090bf.svg?invert_in_darkmode" align=middle width=17.74179pt height=22.38192pt/> on the CPU and send mini-batch
<img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//55a049b8f161ae7cfeb0197d75aff967.svg?invert_in_darkmode" align=middle width=9.83004pt height=14.10255pt/> to GPU 1 and send mini-batch <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//3f18d8f60c110e865571bba5ba67dcc6.svg?invert_in_darkmode" align=middle width=38.062035pt height=21.10812pt/> to GPU 2. As processing is asynchronous,
GPU 2 may finish before GPU 1 and thus update the CPU's model parameters <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//84c95f91a742c9ceb460a83f9b5090bf.svg?invert_in_darkmode" align=middle width=17.74179pt height=22.38192pt/>
with its gradients <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//d945d0d19d93957738ab47285585af55.svg?invert_in_darkmode" align=middle width=85.19445pt height=24.56553pt/>, where the subscript <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//3f18d8f60c110e865571bba5ba67dcc6.svg?invert_in_darkmode" align=middle width=38.062035pt height=21.10812pt/> identifies the
mini-batch and the argument <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//84c95f91a742c9ceb460a83f9b5090bf.svg?invert_in_darkmode" align=middle width=17.74179pt height=22.38192pt/> the location at which the gradient was evaluated.
This results in the new model parameters
<p align="center"><img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//821c06a38f428b87e586f7b26800c3af.svg?invert_in_darkmode" align=middle width=127.537245pt height=16.376943pt/></p>
Next GPU 1 could finish with its mini-batch and update the parameters to
<p align="center"><img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//9f326591601cc1a07f0590bbed3768d8.svg?invert_in_darkmode" align=middle width=216.13185pt height=16.376943pt/></p>
The problem with this is that <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//f8a131f8ae97558cd5a81df2db955b0f.svg?invert_in_darkmode" align=middle width=68.55057pt height=24.56553pt/> is evaluated at <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//84c95f91a742c9ceb460a83f9b5090bf.svg?invert_in_darkmode" align=middle width=17.74179pt height=22.38192pt/> and not
<img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//866ca4413312fc41c943f3c4fbca73c1.svg?invert_in_darkmode" align=middle width=122.988195pt height=24.56553pt/>. Hence, the direction of the gradient <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//f8a131f8ae97558cd5a81df2db955b0f.svg?invert_in_darkmode" align=middle width=68.55057pt height=24.56553pt/>
is slightly incorrect as it is evaluated at the wrong location. This can be
counteracted through synchronous updates of model, but this is also problematic.
## Synchronous Optimization
Synchronous optimization solves the problem we saw above. In synchronous
optimization, one places the model initially in CPU memory. Then one of the <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//5201385589993766eea584cd3aa6fa13.svg?invert_in_darkmode" align=middle width=12.876435pt height=22.38192pt/>
GPU's is given a mini-batch of data along with the current model parameters.
Using the mini-batch the GPU computes the gradients for all model parameters and
sends the gradients back to the CPU. The CPU then updates the model parameters
and starts the process of sending out the next mini-batch.
As on can readily see, synchronous optimization does not have the problem we
found in the last section, that of incorrect gradients. However, synchronous
optimization can only make use of a single GPU at a time. So, when we have a
multi-GPU setup, <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//f1484f55b33675c60d5cc8a5689ba737.svg?invert_in_darkmode" align=middle width=42.934815pt height=22.38192pt/>, all but one of the GPU's will remain idle, which is
unacceptable. However, there is a third alternative which is combines the
advantages of asynchronous and synchronous optimization.
## Hybrid Parallel Optimization
Hybrid parallel optimization combines most of the benifits of asynchronous and
synchronous optimization. It allows for multiple GPU's to be used, but does not
suffer from the incorrect gradient problem exhibited by asynchronous
optimization.
In hybrid parallel optimization one places the model initially in CPU memory.
Then, as in asynchronous optimization, each of the <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//5201385589993766eea584cd3aa6fa13.svg?invert_in_darkmode" align=middle width=12.876435pt height=22.38192pt/> GPU'S obtains a
mini-batch of data along with the current model parameters. Using the mini-batch
each of the GPU's then computes the gradients for all model parameters and sends
these gradients back to the CPU. Now, in contrast to asynchronous optimization,
the CPU waits until each GPU is finished with its mini-batch then takes the mean
of all the gradients from the <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//5201385589993766eea584cd3aa6fa13.svg?invert_in_darkmode" align=middle width=12.876435pt height=22.38192pt/> GPU's and updates the model with this mean
gradient.
<img src="images/Parallelism.png" alt="LSTM" width="600">
Hybrid parallel optimization has several advantages and few disadvantages. As in
asynchronous parallel optimization, hybrid parallel optimization allows for one
to use multiple GPU's in parallel. Furthermore, unlike asynchronous parallel
optimization, the incorrect gradient problem is not present here. In fact,
hybrid parallel optimization performs as if one is working with a single
mini-batch which is <img src="https://rawgit.com/mozilla/DeepSpeech/issue313/doc/svgs//5201385589993766eea584cd3aa6fa13.svg?invert_in_darkmode" align=middle width=12.876435pt height=22.38192pt/> times the size of a mini-batch handled by a single GPU.
Hoewever, hybrid parallel optimization is not perfect. If one GPU is slower than
all the others in completing its mini-batch, all other GPU's will have to sit
idle until this straggler finishes with its mini-batch. This hurts throughput.
But, if all GPU'S are of the same make and model, this problem should be
minimized.
So, relatively speaking, hybrid parallel optimization seems the have more
advantages and fewer disadvantages as compared to both asynchronous and
synchronous optimization. So, we will, for our work, use this hybrid model.
## Adam Optimization
In constrast to
[Deep Speech: Scaling up end-to-end speech recognition](http://arxiv.org/abs/1412.5567),
in which
[Nesterov’s Accelerated Gradient Descent](www.cs.toronto.edu/~fritz/absps/momentum.pdf)
was used, we will use the Adam method for optimization
[[3](http://arxiv.org/abs/1412.6980)],
because, generally, it requires less fine-tuning.

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This directory contains further documentation.
Some of the files contain `TeX` code and therefore need to be generated. So if you want to edit a markdown file that has a `.raw` counterpart, please edit the `raw` one and call `doc/gendocs.sh` from the project root. This will (re)generate the corresponding `.md` file. Please make sure to always commit both files.
Note that you have to follow [this](https://github.com/leegao/readme2tex#installation) installation instructions first.

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#!/bin/sh
set -xe
python -m readme2tex --svgdir doc/svgs/ --username mozilla --project DeepSpeech --output doc/DeepSpeech.md doc/DeepSpeech.raw
python -m readme2tex --svgdir doc/svgs/ --username mozilla --project DeepSpeech --output doc/Geometry.md doc/Geometry.raw
python -m readme2tex --svgdir doc/svgs/ --username mozilla --project DeepSpeech --output doc/ParallelOptimization.md doc/ParallelOptimization.raw

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