rnn: recurrent neural networks

Note: this repository is deprecated in favor of https://github.com/torch/rnn.

This is a Recurrent Neural Network library that extends Torch's nn. You can use it to build RNNs, LSTMs, GRUs, BRNNs, BLSTMs, and so forth and so on. This library includes documentation for the following objects:

Modules that consider successive calls to forward as different time-steps in a sequence :

AbstractRecurrent : an abstract class inherited by Recurrent and LSTM;
Recurrent : a generalized recurrent neural network container;
LSTM : a vanilla Long-Short Term Memory module;
FastLSTM : a faster LSTM with optional support for batch normalization;
GRU : Gated Recurrent Units module;
MuFuRu : Multi-function Recurrent Unit module;
Recursor : decorates a module to make it conform to the AbstractRecurrent interface;
Recurrence : decorates a module that outputs output(t) given {input(t), output(t-1)};
NormStabilizer : implements norm-stabilization criterion (add this module between RNNs);

Modules that forward entire sequences through a decorated AbstractRecurrent instance :

AbstractSequencer : an abstract class inherited by Sequencer, Repeater, RecurrentAttention, etc.;
Sequencer : applies an encapsulated module to all elements in an input sequence (Tensor or Table);
SeqLSTM : a very fast version of nn.Sequencer(nn.FastLSTM) where the input and output are tensors;
SeqLSTMP : SeqLSTM with a projection layer;
SeqGRU : a very fast version of nn.Sequencer(nn.GRU) where the input and output are tensors;
SeqBRNN : Bidirectional RNN based on SeqLSTM;
BiSequencer : used for implementing Bidirectional RNNs and LSTMs;
BiSequencerLM : used for implementing Bidirectional RNNs and LSTMs for language models;
Repeater : repeatedly applies the same input to an AbstractRecurrent instance;
RecurrentAttention : a generalized attention model for REINFORCE modules;

Miscellaneous modules and criterions :

MaskZero : zeroes the output and gradOutput rows of the decorated module for commensurate input rows which are tensors of zeros;
TrimZero : same behavior as MaskZero, but more efficient when input contains lots zero-masked rows;
LookupTableMaskZero : extends nn.LookupTable to support zero indexes for padding. Zero indexes are forwarded as tensors of zeros;
MaskZeroCriterion : zeros the gradInput and err rows of the decorated criterion for commensurate input rows which are tensors of zeros;
SeqReverseSequence : reverses an input sequence on a specific dimension;

Criterions used for handling sequential inputs and targets :

SequencerCriterion : sequentially applies the same criterion to a sequence of inputs and targets (Tensor or Table).
RepeaterCriterion : repeatedly applies the same criterion with the same target on a sequence.

To install this repository:

git clone git@github.com:Element-Research/rnn.git
cd rnn
luarocks make rocks/rnn-scm-1.rockspec

Note that luarocks intall rnn now installs https://github.com/torch/rnn instead.

Examples

The following are example training scripts using this package :

RNN/LSTM/GRU for Penn Tree Bank dataset;
Noise Contrastive Estimate for training multi-layer SeqLSTM language models on the Google Billion Words dataset. The example uses MaskZero to train independent variable length sequences using the NCEModule and NCECriterion. This script is our fastest yet boasting speeds of 20,000 words/second (on NVIDIA Titan X) with a 2-layer LSTM having 250 hidden units, a batchsize of 128 and sequence length of a 100. Note that you will need to have Torch installed with Lua instead of LuaJIT;
Recurrent Model for Visual Attention for the MNIST dataset;
Encoder-Decoder LSTM shows you how to couple encoder and decoder LSTMs for sequence-to-sequence networks;
Simple Recurrent Network shows a simple example for building and training a simple recurrent neural network;
Simple Sequencer Network is a version of the above script that uses the Sequencer to decorate the rnn instead;
Sequence to One demonstrates how to do many to one sequence learning as is the case for sentiment analysis;
Multivariate Time Series demonstrates how train a simple RNN to do multi-variate time-series predication.

External Resources

rnn-benchmarks : benchmarks comparing Torch (using this library), Theano and TensorFlow.
Harvard Jupyter Notebook Tutorial : an in-depth tutorial for how to use the Element-Research rnn package by Harvard University;
dpnn : this is a dependency of the rnn package. It contains useful nn extensions, modules and criterions;
dataload : a collection of torch dataset loaders;
RNN/LSTM/BRNN/BLSTM training script for Penn Tree Bank or Google Billion Words datasets;
A brief (1 hours) overview of Torch7, which includes some details about the rnn packages (at the end), is available via this NVIDIA GTC Webinar video. In any case, this presentation gives a nice overview of Logistic Regression, Multi-Layer Perceptrons, Convolutional Neural Networks and Recurrent Neural Networks using Torch7;
Sequence to Sequence mapping using encoder-decoder RNNs : a complete training example using synthetic data.
ConvLSTM is a repository for training a Spatio-temporal video autoencoder with differentiable memory.
An time series example for univariate timeseries prediction.

Citation

If you use rnn in your work, we'd really appreciate it if you could cite the following paper:

Léonard, Nicholas, Sagar Waghmare, Yang Wang, and Jin-Hwa Kim. rnn: Recurrent Library for Torch. arXiv preprint arXiv:1511.07889 (2015).

Any significant contributor to the library will also get added as an author to the paper. A significant contributor is anyone who added at least 300 lines of code to the library.

Troubleshooting

Most issues can be resolved by updating the various dependencies:

luarocks install torch
luarocks install nn
luarocks install dpnn
luarocks install torchx

If you are using CUDA :

luarocks install cutorch
luarocks install cunn
luarocks install cunnx

And don't forget to update this package :

git clone git@github.com:Element-Research/rnn.git
cd rnn
luarocks make rocks/rnn-scm-1.rockspec

If that doesn't fix it, open and issue on github.

AbstractRecurrent

An abstract class inherited by Recurrent, LSTM and GRU. The constructor takes a single argument :

rnn = nn.AbstractRecurrent([rho])

Argument rho is the maximum number of steps to backpropagate through time (BPTT). Sub-classes can set this to a large number like 99999 (the default) if they want to backpropagate through the entire sequence whatever its length. Setting lower values of rho are useful when long sequences are forward propagated, but we only whish to backpropagate through the last rho steps, which means that the remainder of the sequence doesn't need to be stored (so no additional cost).

[recurrentModule] getStepModule(step)

Returns a module for time-step step. This is used internally by sub-classes to obtain copies of the internal recurrentModule. These copies share parameters and gradParameters but each have their own output, gradInput and any other intermediate states.

setOutputStep(step)

This is a method reserved for internal use by Recursor when doing backward propagation. It sets the object's output attribute to point to the output at time-step step. This method was introduced to solve a very annoying bug.

maskZero(nInputDim)

Decorates the internal recurrentModule with MaskZero. The output Tensor (or table thereof) of the recurrentModule will have each row (i.e. samples) zeroed when the commensurate row of the input is a tensor of zeros.

The nInputDim argument must specify the number of non-batch dims in the first Tensor of the input. In the case of an input table, the first Tensor is the first one encountered when doing a depth-first search.

Calling this method makes it possible to pad sequences with different lengths in the same batch with zero vectors.

When a sample time-step is masked (i.e. input is a row of zeros), then the hidden state is effectively reset (i.e. forgotten) for the next non-mask time-step. In other words, it is possible seperate unrelated sequences with a masked element.

trimZero(nInputDim)

Decorates the internal recurrentModule with TrimZero.

[output] updateOutput(input)

Forward propagates the input for the current step. The outputs or intermediate states of the previous steps are used recurrently. This is transparent to the caller as the previous outputs and intermediate states are memorized. This method also increments the step attribute by 1.

updateGradInput(input, gradOutput)

Like backward, this method should be called in the reverse order of forward calls used to propagate a sequence. So for example :

rnn = nn.LSTM(10, 10) -- AbstractRecurrent instance
local outputs = {}
for i=1,nStep do -- forward propagate sequence
   outputs[i] = rnn:forward(inputs[i])
end

for i=nStep,1,-1 do -- backward propagate sequence in reverse order
   gradInputs[i] = rnn:backward(inputs[i], gradOutputs[i])
end

rnn:forget()

The reverse order implements backpropagation through time (BPTT).

accGradParameters(input, gradOutput, scale)

Like updateGradInput, but for accumulating gradients w.r.t. parameters.

recycle(offset)

This method goes hand in hand with forget. It is useful when the current time-step is greater than rho, at which point it starts recycling the oldest recurrentModule sharedClones, such that they can be reused for storing the next step. This offset is used for modules like nn.Recurrent that use a different module for the first step. Default offset is 0.

forget(offset)

This method brings back all states to the start of the sequence buffers, i.e. it forgets the current sequence. It also resets the step attribute to 1. It is highly recommended to call forget after each parameter update. Otherwise, the previous state will be used to activate the next, which will often lead to instability. This is caused by the previous state being the result of now changed parameters. It is also good practice to call forget at the start of each new sequence.

maxBPTTstep(rho)

This method sets the maximum number of time-steps for which to perform backpropagation through time (BPTT). So say you set this to rho = 3 time-steps, feed-forward for 4 steps, and then backpropgate, only the last 3 steps will be used for the backpropagation. If your AbstractRecurrent instance is wrapped by a Sequencer, this will be handled auto-magically by the Sequencer. Otherwise, setting this value to a large value (i.e. 9999999), is good for most, if not all, cases.

backwardOnline()

This method was deprecated Jan 6, 2016. Since then, by default, AbstractRecurrent instances use the backwardOnline behaviour. See updateGradInput for details.

training()

In training mode, the network remembers all previous rho (number of time-steps) states. This is necessary for BPTT.

evaluate()

During evaluation, since their is no need to perform BPTT at a later time, only the previous step is remembered. This is very efficient memory-wise, such that evaluation can be performed using potentially infinite-length sequence.

Recurrent

References :

A composite Module for implementing Recurrent Neural Networks (RNN), excluding the output layer.

The nn.Recurrent(start, input, feedback, [transfer, rho, merge]) constructor takes 6 arguments:

start : the size of the output (excluding the batch dimension), or a Module that will be inserted between the input Module and transfer module during the first step of the propagation. When start is a size (a number or torch.LongTensor), then this start Module will be initialized as nn.Add(start) (see Ref. A).
input : a Module that processes input Tensors (or Tables). Output must be of same size as start (or its output in the case of a start Module), and same size as the output of the feedback Module.
feedback : a Module that feedbacks the previous output Tensor (or Tables) up to the merge module.
merge : a table Module that merges the outputs of the input and feedback Module before being forwarded through the transfer Module.
transfer : a non-linear Module used to process the output of the merge module, or in the case of the first step, the output of the start Module.
rho : the maximum amount of backpropagation steps to take back in time. Limits the number of previous steps kept in memory. Due to the vanishing gradients effect, references A and B recommend rho = 5 (or lower). Defaults to 99999.

An RNN is used to process a sequence of inputs. Each step in the sequence should be propagated by its own forward (and backward), one input (and gradOutput) at a time. Each call to forward keeps a log of the intermediate states (the input and many Module.outputs) and increments the step attribute by 1. Method backward must be called in reverse order of the sequence of calls to forward in order to backpropgate through time (BPTT). This reverse order is necessary to return a gradInput for each call to forward.

The step attribute is only reset to 1 when a call to the forget method is made. In which case, the Module is ready to process the next sequence (or batch thereof). Note that the longer the sequence, the more memory that will be required to store all the output and gradInput states (one for each time step).

To use this module with batches, we suggest using different sequences of the same size within a batch and calling updateParameters every rho steps and forget at the end of the sequence.

Note that calling the evaluate method turns off long-term memory; the RNN will only remember the previous output. This allows the RNN to handle long sequences without allocating any additional memory.

For a simple concise example of how to make use of this module, please consult the simple-recurrent-network.lua training script.

Decorate it with a Sequencer

Note that any AbstractRecurrent instance can be decorated with a Sequencer such that an entire sequence (a table) can be presented with a single forward/backward call. This is actually the recommended approach as it allows RNNs to be stacked and makes the rnn conform to the Module interface, i.e. each call to forward can be followed by its own immediate call to backward as each input to the model is an entire sequence, i.e. a table of tensors where each tensor represents a time-step.

seq = nn.Sequencer(module)

The simple-sequencer-network.lua training script is equivalent to the above mentionned simple-recurrent-network.lua script, except that it decorates the rnn with a Sequencer which takes a table of inputs and gradOutputs (the sequence for that batch). This lets the Sequencer handle the looping over the sequence.

You should only think about using the AbstractRecurrent modules without a Sequencer if you intend to use it for real-time prediction. Actually, you can even use an AbstractRecurrent instance decorated by a Sequencer for real time prediction by calling Sequencer:remember() and presenting each time-step input as {input}.

Other decorators can be used such as the Repeater or RecurrentAttention. The Sequencer is only the most common one.

LSTM

References :

This is an implementation of a vanilla Long-Short Term Memory module. We used Ref. A's LSTM as a blueprint for this module as it was the most concise. Yet it is also the vanilla LSTM described in Ref. C.

The nn.LSTM(inputSize, outputSize, [rho]) constructor takes 3 arguments:

inputSize : a number specifying the size of the input;
outputSize : a number specifying the size of the output;
rho : the maximum amount of backpropagation steps to take back in time. Limits the number of previous steps kept in memory. Defaults to 9999.

The actual implementation corresponds to the following algorithm:

i[t] = σ(W[x->i]x[t] + W[h->i]h[t−1] + W[c->i]c[t−1] + b[1->i])      (1)
f[t] = σ(W[x->f]x[t] + W[h->f]h[t−1] + W[c->f]c[t−1] + b[1->f])      (2)
z[t] = tanh(W[x->c]x[t] + W[h->c]h[t−1] + b[1->c])                   (3)
c[t] = f[t]c[t−1] + i[t]z[t]                                         (4)
o[t] = σ(W[x->o]x[t] + W[h->o]h[t−1] + W[c->o]c[t] + b[1->o])        (5)
h[t] = o[t]tanh(c[t])                                                (6)

where W[s->q] is the weight matrix from s to q, t indexes the time-step, b[1->q] are the biases leading into q, σ() is Sigmoid, x[t] is the input, i[t] is the input gate (eq. 1), f[t] is the forget gate (eq. 2), z[t] is the input to the cell (which we call the hidden) (eq. 3), c[t] is the cell (eq. 4), o[t] is the output gate (eq. 5), and h[t] is the output of this module (eq. 6). Also note that the weight matrices from cell to gate vectors are diagonal W[c->s], where s is i,f, or o.

As you can see, unlike Recurrent, this implementation isn't generic enough that it can take arbitrary component Module definitions at construction. However, the LSTM module can easily be adapted through inheritance by overriding the different factory methods :

buildGate : builds generic gate that is used to implement the input, forget and output gates;
buildInputGate : builds the input gate (eq. 1). Currently calls buildGate;
buildForgetGate : builds the forget gate (eq. 2). Currently calls buildGate;
buildHidden : builds the hidden (eq. 3);
buildCell : builds the cell (eq. 4);
buildOutputGate : builds the output gate (eq. 5). Currently calls buildGate;
buildModel : builds the actual LSTM model which is used internally (eq. 6).

Note that we recommend decorating the LSTM with a Sequencer (refer to this for details).

FastLSTM

A faster version of the LSTM. Basically, the input, forget and output gates, as well as the hidden state are computed at one fellswoop.

Note that FastLSTM does not use peephole connections between cell and gates. The algorithm from LSTM changes as follows:

i[t] = σ(W[x->i]x[t] + W[h->i]h[t−1] + b[1->i])                      (1)
f[t] = σ(W[x->f]x[t] + W[h->f]h[t−1] + b[1->f])                      (2)
z[t] = tanh(W[x->c]x[t] + W[h->c]h[t−1] + b[1->c])                   (3)
c[t] = f[t]c[t−1] + i[t]z[t]                                         (4)
o[t] = σ(W[x->o]x[t] + W[h->o]h[t−1] + b[1->o])                      (5)
h[t] = o[t]tanh(c[t])                                                (6)

i.e. omitting the summands W[c->i]c[t−1] (eq. 1), W[c->f]c[t−1] (eq. 2), and W[c->o]c[t] (eq. 5).

usenngraph

This is a static attribute of the FastLSTM class. The default value is false. Setting usenngraph = true will force all new instantiated instances of FastLSTM to use nngraph's nn.gModule to build the internal recurrentModule which is cloned for each time-step.

Recurrent Batch Normalization

This extends the FastLSTM class to enable faster convergence during training by zero-centering the input-to-hidden and hidden-to-hidden transformations. It reduces the internal covariate shift between time steps. It is an implementation of Cooijmans et. al.'s Recurrent Batch Normalization. The hidden-to-hidden transition of each LSTM cell is normalized according to

i[t] = σ(BN(W[x->i]x[t]) + BN(W[h->i]h[t−1]) + b[1->i])                      (1)
f[t] = σ(BN(W[x->f]x[t]) + BN(W[h->f]h[t−1]) + b[1->f])                      (2)
z[t] = tanh(BN(W[x->c]x[t]) + BN(W[h->c]h[t−1]) + b[1->c])                   (3)
c[t] = f[t]c[t−1] + i[t]z[t]                                                 (4)
o[t] = σ(BN(W[x->o]x[t]) + BN(W[h->o]h[t−1]) + b[1->o])                      (5)
h[t] = o[t]tanh(c[t])                                                        (6)

where the batch normalizing transform is:

  BN(h; gamma, beta) = beta + gamma *      hd - E(hd)
                                      ------------------
                                       sqrt(E(σ(hd) + eps))

where hd is a vector of (pre)activations to be normalized, gamma, and beta are model parameters that determine the mean and standard deviation of the normalized activation. eps is a regularization hyperparameter to keep the division numerically stable and E(hd) and E(σ(hd)) are the estimates of the mean and variance in the mini-batch respectively. The authors recommend initializing gamma to a small value and found 0.1 to be the value that did not cause vanishing gradients. beta, the shift parameter, is null by default.

To turn on batch normalization during training, do:

nn.FastLSTM.bn = true
lstm = nn.FastLSTM(inputsize, outputsize, [rho, eps, momentum, affine]

where momentum is same as gamma in the equation above (defaults to 0.1), eps is defined above and affine is a boolean whose state determines if the learnable affine transform is turned off (false) or on (true, the default).

GRU

References :

This is an implementation of Gated Recurrent Units module.

The nn.GRU(inputSize, outputSize [,rho [,p [, mono]]]) constructor takes 3 arguments likewise nn.LSTM or 4 arguments for dropout:

inputSize : a number specifying the size of the input;
outputSize : a number specifying the size of the output;
rho : the maximum amount of backpropagation steps to take back in time. Limits the number of previous steps kept in memory. Defaults to 9999;
p : dropout probability for inner connections of GRUs.
mono : Monotonic sample for dropouts inside GRUs. Only needed in a TrimZero + BGRU(p>0) situation.

The actual implementation corresponds to the following algorithm:

z[t] = σ(W[x->z]x[t] + W[s->z]s[t−1] + b[1->z])            (1)
r[t] = σ(W[x->r]x[t] + W[s->r]s[t−1] + b[1->r])            (2)
h[t] = tanh(W[x->h]x[t] + W[hr->c](s[t−1]r[t]) + b[1->h])  (3)
s[t] = (1-z[t])h[t] + z[t]s[t-1]                           (4)

where W[s->q] is the weight matrix from s to q, t indexes the time-step, b[1->q] are the biases leading into q, σ() is Sigmoid, x[t] is the input and s[t] is the output of the module (eq. 4). Note that unlike the LSTM, the GRU has no cells.

The GRU was benchmark on PennTreeBank dataset using recurrent-language-model.lua script. It slightly outperfomed FastLSTM, however, since LSTMs have more parameters than GRUs, the dataset larger than PennTreeBank might change the performance result. Don't be too hasty to judge on which one is the better of the two (see Ref. C and D).

                Memory   examples/s
    FastLSTM      176M        16.5K 
    GRU            92M        15.8K

Memory is measured by the size of dp.Experiment save file. examples/s is measured by the training speed at 1 epoch, so, it may have a disk IO bias.

RNN dropout (see Ref. E and F) was benchmark on PennTreeBank dataset using recurrent-language-model.lua script, too. The details can be found in the script. In the benchmark, GRU utilizes a dropout after LookupTable, while BGRU, stands for Bayesian GRUs, uses dropouts on inner connections (naming as Ref. F), but not after LookupTable.

As Yarin Gal (Ref. F) mentioned, it is recommended that one may use p = 0.25 for the first attempt.

SAdd

To implement GRU, a simple module is added, which cannot be possible to build only using nn modules.

module = nn.SAdd(addend, negate)

Applies a single scalar addition to the incoming data, i.e. y_i = x_i + b, then negate all components if negate is true. Which is used to implement s[t] = (1-z[t])h[t] + z[t]s[t-1] of GRU (see above Equation (4)).

nn.SAdd(-1, true)

Here, if the incoming data is z[t], then the output becomes -(z[t]-1)=1-z[t]. Notice that nn.Mul() multiplies a scalar which is a learnable parameter.

MuFuRu

References :

This is an implementation of the Multi-Function Recurrent Unit module.

The nn.MuFuRu(inputSize, outputSize [,ops [,rho]]) constructor takes 2 required arguments, plus optional arguments:

inputSize : a number specifying the dimension of the input;
outputSize : a number specifying the dimension of the output;
ops: a table of strings, representing which composition operations should be used. The table can be any subset of {'keep', 'replace', 'mul', 'diff', 'forget', 'sqrt_diff', 'max', 'min'}. By default, all composition operations are enabled.
rho : the maximum amount of backpropagation steps to take back in time. Limits the number of previous steps kept in memory. Defaults to 9999;

The Multi-Function Recurrent Unit generalizes the GRU by allowing weightings of arbitrary composition operators to be learned. As in the GRU, the reset gate is computed based on the current input and previous hidden state, and used to compute a new feature vector:

r[t] = σ(W[x->r]x[t] + W[s->r]s[t−1] + b[1->r])            (1)
v[t] = tanh(W[x->v]x[t] + W[sr->v](s[t−1]r[t]) + b[1->v])  (2)

where W[a->b] denotes the weight matrix from activation a to b, t denotes the time step, b[1->a] is the bias for activation a, and s[t-1]r[t] is the element-wise multiplication of the two vectors.

Unlike in the GRU, rather than computing a single update gate (z[t] in GRU), MuFuRU computes a weighting over an arbitrary number of composition operators.

A composition operator is any differentiable operator which takes two vectors of the same size, the previous hidden state, and a new feature vector, and returns a new vector representing the new hidden state. The GRU implicitly defines two such operations, keep and replace, defined as keep(s[t-1], v[t]) = s[t-1] and replace(s[t-1], v[t]) = v[t].

Ref. A proposes 6 additional operators, which all operate element-wise:

mul(x,y) = x * y
diff(x,y) = x - y
forget(x,y) = 0
sqrt_diff(x,y) = 0.25 * sqrt(|x - y|)
max(x,y)
min(x,y)

The weightings of each operation are computed via a softmax from the current input and previous hidden state, similar to the update gate in the GRU. The produced hidden state is then the element-wise weighted sum of the output of each operation.

p^[t][j] = W[x->pj]x[t] + W[s->pj]s[t−1] + b[1->pj])         (3)
(p[t][1], ... p[t][J])  = softmax (p^[t][1], ..., p^[t][J])  (4)
s[t] = sum(p[t][j] * op[j](s[t-1], v[t]))                    (5)

where p[t][j] is the weightings for operation j at time step t, and sum in equation 5 is over all operators J.

Recursor

This module decorates a module to be used within an AbstractSequencer instance. It does this by making the decorated module conform to the AbstractRecurrent interface, which like the LSTM and Recurrent classes, this class inherits.

rec = nn.Recursor(module[, rho])

For each successive call to updateOutput (i.e. forward), this decorator will create a stepClone() of the decorated module. So for each time-step, it clones the module. Both the clone and original share parameters and gradients w.r.t. parameters. However, for modules that already conform to the AbstractRecurrent interface, the clone and original module are one and the same (i.e. no clone).

Examples :

Let's assume I want to stack two LSTMs. I could use two sequencers :

lstm = nn.Sequential()
   :add(nn.Sequencer(nn.LSTM(100,100)))
   :add(nn.Sequencer(nn.LSTM(100,100)))

Using a Recursor, I make the same model with a single Sequencer :

lstm = nn.Sequencer(
   nn.Recursor(
      nn.Sequential()
         :add(nn.LSTM(100,100))
         :add(nn.LSTM(100,100))
      )
   )

Actually, the Sequencer will wrap any non-AbstractRecurrent module automatically, so I could simplify this further to :

lstm = nn.Sequencer(
   nn.Sequential()
      :add(nn.LSTM(100,100))
      :add(nn.LSTM(100,100))
   )

I can also add a Linear between the two LSTMs. In this case, a Linear will be cloned (and have its parameters shared) for each time-step, while the LSTMs will do whatever cloning internally :

lstm = nn.Sequencer(
   nn.Sequential()
      :add(nn.LSTM(100,100))
      :add(nn.Linear(100,100))
      :add(nn.LSTM(100,100))
   )

AbstractRecurrent instances like Recursor, Recurrent and LSTM are expcted to manage time-steps internally. Non-AbstractRecurrent instances can be wrapped by a Recursor to have the same behavior.

Every call to forward on an AbstractRecurrent instance like Recursor will increment the self.step attribute by 1, using a shared parameter clone for each successive time-step (for a maximum of rho time-steps, which defaults to 9999999). In this way, backward can be called in reverse order of the forward calls to perform backpropagation through time (BPTT). Which is exactly what AbstractSequencer instances do internally. The backward call, which is actually divided into calls to updateGradInput and accGradParameters, decrements by 1 the self.udpateGradInputStep and self.accGradParametersStep respectively, starting at self.step. Successive calls to backward will decrement these counters and use them to backpropagate through the appropriate internall step-wise shared-parameter clones.

Anyway, in most cases, you will not have to deal with the Recursor object directly as AbstractSequencer instances automatically decorate non-AbstractRecurrent instances with a Recursor in their constructors.

For a concrete example of its use, please consult the simple-recurrent-network.lua training script for an example of its use.

Recurrence

A extremely general container for implementing pretty much any type of recurrence.

rnn = nn.Recurrence(recurrentModule, outputSize, nInputDim, [rho])

Unlike Recurrent, this module doesn't manage a separate modules like inputModule, startModule, mergeModule and the like. Instead, it only manages a single recurrentModule, which should output a Tensor or table : output(t) given an input table : {input(t), output(t-1)}. Using a mix of Recursor (say, via Sequencer) with Recurrence, one can implement pretty much any type of recurrent neural network, including LSTMs and RNNs.

For the first step, the Recurrence forwards a Tensor (or table thereof) of zeros through the recurrent layer (like LSTM, unlike Recurrent). So it needs to know the outputSize, which is either a number or torch.LongStorage, or table thereof. The batch dimension should be excluded from the outputSize. Instead, the size of the batch dimension (i.e. number of samples) will be extrapolated from the input using the nInputDim argument. For example, say that our input is a Tensor of size 4 x 3 where 4 is the number of samples, then nInputDim should be 1. As another example, if our input is a table of table [...] of tensors where the first tensor (depth first) is the same as in the previous example, then our nInputDim is also 1.

As an example, let's use Sequencer and Recurrence to build a Simple RNN for language modeling :

rho = 5
hiddenSize = 10
outputSize = 5 -- num classes
nIndex = 10000

-- recurrent module
rm = nn.Sequential()
   :add(nn.ParallelTable()
      :add(nn.LookupTable(nIndex, hiddenSize))
      :add(nn.Linear(hiddenSize, hiddenSize)))
   :add(nn.CAddTable())
   :add(nn.Sigmoid())

rnn = nn.Sequencer(
   nn.Sequential()
      :add(nn.Recurrence(rm, hiddenSize, 1))
      :add(nn.Linear(hiddenSize, outputSize))
      :add(nn.LogSoftMax())
)

Note : We could very well reimplement the LSTM module using the newer Recursor and Recurrent modules, but that would mean breaking backwards compatibility for existing models saved on disk.

NormStabilizer

Ref. A : Regularizing RNNs by Stabilizing Activations

This module implements the norm-stabilization criterion:

ns = nn.NormStabilizer([beta])

This module regularizes the hidden states of RNNs by minimizing the difference between the L2-norms of consecutive steps. The cost function is defined as :

loss = beta * 1/T sum_t( ||h[t]|| - ||h[t-1]|| )^2

where T is the number of time-steps. Note that we do not divide the gradient by T such that the chosen beta can scale to different sequence sizes without being changed.

The sole argument beta is defined in ref. A. Since we don't divide the gradients by the number of time-steps, the default value of beta=1 should be valid for most cases.

This module should be added between RNNs (or LSTMs or GRUs) to provide better regularization of the hidden states. For example :

local stepmodule = nn.Sequential()
   :add(nn.FastLSTM(10,10))
   :add(nn.NormStabilizer())
   :add(nn.FastLSTM(10,10))
   :add(nn.NormStabilizer())
local rnn = nn.Sequencer(stepmodule)

To use it with SeqLSTM you can do something like this :

local rnn = nn.Sequential()
   :add(nn.SeqLSTM(10,10))
   :add(nn.Sequencer(nn.NormStabilizer()))
   :add(nn.SeqLSTM(10,10))
   :add(nn.Sequencer(nn.NormStabilizer()))

AbstractSequencer

This abstract class implements a light interface shared by subclasses like : Sequencer, Repeater, RecurrentAttention, BiSequencer and so on.

Sequencer

The nn.Sequencer(module) constructor takes a single argument, module, which is the module to be applied from left to right, on each element of the input sequence.

seq = nn.Sequencer(module)

This Module is a kind of decorator used to abstract away the intricacies of AbstractRecurrent modules. While an AbstractRecurrent instance requires that a sequence to be presented one input at a time, each with its own call to forward (and backward), the Sequencer forwards an input sequence (a table) into an output sequence (a table of the same length). It also takes care of calling forget on AbstractRecurrent instances.

Input/Output Format

The Sequencer requires inputs and outputs to be of shape seqlen x batchsize x featsize :

seqlen is the number of time-steps that will be fed into the Sequencer.
batchsize is the number of examples in the batch. Each example is its own independent sequence.
featsize is the size of the remaining non-batch dimensions. So this could be 1 for language models, or c x h x w for convolutional models, etc.

Above is an example input sequence for a character level language model. It has seqlen is 5 which means that it contains sequences of 5 time-steps. The openning { and closing } illustrate that the time-steps are elements of a Lua table, although it also accepts full Tensors of shape seqlen x batchsize x featsize. The batchsize is 2 as their are two independent sequences : { H, E, L, L, O } and { F, U, Z, Z, Y, }. The featsize is 1 as their is only one feature dimension per character and each such character is of size 1. So the input in this case is a table of seqlen time-steps where each time-step is represented by a batchsize x featsize Tensor.

Above is another example of a sequence (input or output). It has a seqlen of 4 time-steps. The batchsize is again 2 which means there are two sequences. The featsize is 3 as each time-step of each sequence has 3 variables. So each time-step (element of the table) is represented again as a tensor of size batchsize x featsize. Note that while in both examples the featsize encodes one dimension, it could encode more.

Example

For example, rnn : an instance of nn.AbstractRecurrent, can forward an input sequence one forward at a time:

input = {torch.randn(3,4), torch.randn(3,4), torch.randn(3,4)}
rnn:forward(input[1])
rnn:forward(input[2])
rnn:forward(input[3])

Equivalently, we can use a Sequencer to forward the entire input sequence at once:

seq = nn.Sequencer(rnn)
seq:forward(input)

We can also forward Tensors instead of Tables :

-- seqlen x batchsize x featsize
input = torch.randn(3,3,4)
seq:forward(input)

Details

The Sequencer can also take non-recurrent Modules (i.e. non-AbstractRecurrent instances) and apply it to each input to produce an output table of the same length. This is especially useful for processing variable length sequences (tables).

Internally, the Sequencer expects the decorated module to be an AbstractRecurrent instance. When this is not the case, the module is automatically decorated with a Recursor module, which makes it conform to the AbstractRecurrent interface.

Note : this is due a recent update (27 Oct 2015), as before this AbstractRecurrent and and non-AbstractRecurrent instances needed to be decorated by their own Sequencer. The recent update, which introduced the Recursor decorator, allows a single Sequencer to wrap any type of module, AbstractRecurrent, non-AbstractRecurrent or a composite structure of both types. Nevertheless, existing code shouldn't be affected by the change.

For a concise example of its use, please consult the simple-sequencer-network.lua training script.

remember([mode])

When mode='neither' (the default behavior of the class), the Sequencer will additionally call forget before each call to forward. When mode='both' (the default when calling this function), the Sequencer will never call forget. In which case, it is up to the user to call forget between independent sequences. This behavior is only applicable to decorated AbstractRecurrent modules. Accepted values for argument mode are as follows :

'eval' only affects evaluation (recommended for RNNs)
'train' only affects training
'neither' affects neither training nor evaluation (default behavior of the class)
'both' affects both training and evaluation (recommended for LSTMs)

forget()

Calls the decorated AbstractRecurrent module's forget method.

SeqLSTM

This module is a faster version of nn.Sequencer(nn.FastLSTM(inputsize, outputsize)) :

seqlstm = nn.SeqLSTM(inputsize, outputsize)

Each time-step is computed as follows (same as FastLSTM):

i[t] = σ(W[x->i]x[t] + W[h->i]h[t−1] + b[1->i])                      (1)
f[t] = σ(W[x->f]x[t] + W[h->f]h[t−1] + b[1->f])                      (2)
z[t] = tanh(W[x->c]x[t] + W[h->c]h[t−1] + b[1->c])                   (3)
c[t] = f[t]c[t−1] + i[t]z[t]                                         (4)
o[t] = σ(W[x->o]x[t] + W[h->o]h[t−1] + b[1->o])                      (5)
h[t] = o[t]tanh(c[t])                                                (6)

A notable difference is that this module expects the input and gradOutput to be tensors instead of tables. The default shape is seqlen x batchsize x inputsize for the input and seqlen x batchsize x outputsize for the output :

input = torch.randn(seqlen, batchsize, inputsize)
gradOutput = torch.randn(seqlen, batchsize, outputsize)

output = seqlstm:forward(input)
gradInput = seqlstm:backward(input, gradOutput)

Note that if you prefer to transpose the first two dimension (i.e. batchsize x seqlen instead of the default seqlen x batchsize) you can set seqlstm.batchfirst = true following initialization.

For variable length sequences, set seqlstm.maskzero = true. This is equivalent to calling maskZero(1) on a FastLSTM wrapped by a Sequencer:

fastlstm = nn.FastLSTM(inputsize, outputsize)
fastlstm:maskZero(1)
seqfastlstm = nn.Sequencer(fastlstm)

For maskzero = true, input sequences are expected to be seperated by tensor of zeros for a time step.

The seqlstm:toFastLSTM() method generates a FastLSTM instance initialized with the parameters of the seqlstm instance. Note however that the resulting parameters will not be shared (nor can they ever be).

Like the FastLSTM, the SeqLSTM does not use peephole connections between cell and gates (see FastLSTM for details).

Like the Sequencer, the SeqLSTM provides a remember method.

Note that a SeqLSTM cannot replace FastLSTM in code that decorates it with a AbstractSequencer or Recursor as this would be equivalent to Sequencer(Sequencer(FastLSTM)). You have been warned.

SeqLSTMP

References:

lstmp = nn.SeqLSTMP(inputsize, hiddensize, outputsize)

The SeqLSTMP is a subclass of SeqLSTM. It differs in that after computing the hidden state h[t] (eq. 6), it is projected onto r[t] using a simple linear transform (eq. 7). The computation of the gates also uses the previous such projection r[t-1] (eq. 1, 2, 3, 5). This differs from SeqLSTM which uses h[t-1] instead of r[t-1].

The computation of a time-step outlined in SeqLSTM is replaced with the following:

i[t] = σ(W[x->i]x[t] + W[r->i]r[t−1] + b[1->i])                      (1)
f[t] = σ(W[x->f]x[t] + W[r->f]r[t−1] + b[1->f])                      (2)
z[t] = tanh(W[x->c]x[t] + W[h->c]r[t−1] + b[1->c])                   (3)
c[t] = f[t]c[t−1] + i[t]z[t]                                         (4)
o[t] = σ(W[x->o]x[t] + W[r->o]r[t−1] + b[1->o])                      (5)
h[t] = o[t]tanh(c[t])                                                (6)
r[t] = W[h->r]h[t]                                                   (7)

The algorithm is outlined in ref. A and benchmarked with state of the art results on the Google billion words dataset in ref. B. SeqLSTMP can be used with an hiddensize >> outputsize such that the effective size of the memory cells c[t] and gates i[t], f[t] and o[t] can be much larger than the actual input x[t] and output r[t]. For fixed inputsize and outputsize, the SeqLSTMP will be able to remember much more information than the SeqLSTM.

SeqGRU

This module is a faster version of nn.Sequencer(nn.GRU(inputsize, outputsize)) :

seqGRU = nn.SeqGRU(inputsize, outputsize)

Usage of SeqGRU differs from GRU in the same manner as SeqLSTM differs from LSTM. Therefore see SeqLSTM for more details.

SeqBRNN

brnn = nn.SeqBRNN(inputSize, outputSize, [batchFirst], [merge])

A bi-directional RNN that uses SeqLSTM. Internally contains a 'fwd' and 'bwd' module of SeqLSTM. Expects an input shape of seqlen x batchsize x inputsize. By setting [batchFirst] to true, the input shape can be batchsize x seqLen x inputsize. Merge module defaults to CAddTable(), summing the outputs from each output layer.

Example:

input = torch.rand(1, 1, 5)
brnn = nn.SeqBRNN(5, 5)
print(brnn:forward(input))

Prints an output of a 1x1x5 tensor.

BiSequencer

Applies encapsulated fwd and bwd rnns to an input sequence in forward and reverse order. It is used for implementing Bidirectional RNNs and LSTMs.

brnn = nn.BiSequencer(fwd, [bwd, merge])

The input to the module is a sequence (a table) of tensors and the output is a sequence (a table) of tensors of the same length. Applies a fwd rnn (an AbstractRecurrent instance) to each element in the sequence in forward order and applies the bwd rnn in reverse order (from last element to first element). The bwd rnn defaults to:

bwd = fwd:clone()
bwd:reset()

For each step (in the original sequence), the outputs of both rnns are merged together using the merge module (defaults to nn.JoinTable(1,1)). If merge is a number, it specifies the JoinTable constructor's nInputDim argument. Such that the merge module is then initialized as :

merge = nn.JoinTable(1,merge)

Internally, the BiSequencer is implemented by decorating a structure of modules that makes use of 3 Sequencers for the forward, backward and merge modules.

Similarly to a Sequencer, the sequences in a batch must have the same size. But the sequence length of each batch can vary.

Note : make sure you call brnn:forget() after each call to updateParameters(). Alternatively, one could call brnn.bwdSeq:forget() so that only bwd rnn forgets. This is the minimum requirement, as it would not make sense for the bwd rnn to remember future sequences.

BiSequencerLM

Applies encapsulated fwd and bwd rnns to an input sequence in forward and reverse order. It is used for implementing Bidirectional RNNs and LSTMs for Language Models (LM).

brnn = nn.BiSequencerLM(fwd, [bwd, merge])

The input to the module is a sequence (a table) of tensors and the output is a sequence (a table) of tensors of the same length. Applies a fwd rnn (an AbstractRecurrent instance to the first N-1 elements in the sequence in forward order. Applies the bwd rnn in reverse order to the last N-1 elements (from second-to-last element to first element). This is the main difference of this module with the BiSequencer. The latter cannot be used for language modeling because the bwd rnn would be trained to predict the input it had just be fed as input.

The bwd rnn defaults to:

bwd = fwd:clone()
bwd:reset()

While the fwd rnn will output representations for the last N-1 steps, the bwd rnn will output representations for the first N-1 steps. The missing outputs for each rnn ( the first step for the fwd, the last step for the bwd) will be filled with zero Tensors of the same size the commensure rnn's outputs. This way they can be merged. If nn.JoinTable is used (the default), then the first and last output elements will be padded with zeros for the missing fwd and bwd rnn outputs, respectively.

For each step (in the original sequence), the outputs of both rnns are merged together using the merge module (defaults to nn.JoinTable(1,1)). If merge is a number, it specifies the JoinTable constructor's nInputDim argument. Such that the merge module is then initialized as :

merge = nn.JoinTable(1,merge)

Similarly to a Sequencer, the sequences in a batch must have the same size. But the sequence length of each batch can vary.

Note that LMs implemented with this module will not be classical LMs as they won't measure the probability of a word given the previous words. Instead, they measure the probabiliy of a word given the surrounding words, i.e. context. While for mathematical reasons you may not be able to use this to measure the probability of a sequence of words (like a sentence), you can still measure the pseudo-likeliness of such a sequence (see this for a discussion).

Repeater

This Module is a decorator similar to Sequencer. It differs in that the sequence length is fixed before hand and the input is repeatedly forwarded through the wrapped module to produce an output table of length nStep:

r = nn.Repeater(module, nStep)

Argument module should be an AbstractRecurrent instance. This is useful for implementing models like RCNNs, which are repeatedly presented with the same input.

RecurrentAttention

References :

This module can be used to implement the Recurrent Attention Model (RAM) presented in Ref. A :

ram = nn.RecurrentAttention(rnn, action, nStep, hiddenSize)

rnn is an AbstractRecurrent instance. Its input is {x, z} where x is the input to the ram and z is an action sampled from the action module. The output size of the rnn must be equal to hiddenSize.

action is a Module that uses a REINFORCE module (ref. B) like ReinforceNormal, ReinforceCategorical, or ReinforceBernoulli to sample actions given the previous time-step's output of the rnn. During the first time-step, the action module is fed with a Tensor of zeros of size input:size(1) x hiddenSize. It is important to understand that the sampled actions do not receive gradients backpropagated from the training criterion. Instead, a reward is broadcast from a Reward Criterion like VRClassReward Criterion to the action's REINFORCE module, which will backprogate graidents computed from the output samples and the reward. Therefore, the action module's outputs are only used internally, within the RecurrentAttention module.

nStep is the number of actions to sample, i.e. the number of elements in the output table.

hiddenSize is the output size of the rnn. This variable is necessary to generate the zero Tensor to sample an action for the first step (see above).

A complete implementation of Ref. A is available here.

MaskZero

This module zeroes the output rows of the decorated module for commensurate input rows which are tensors of zeros.

mz = nn.MaskZero(module, nInputDim)

The output Tensor (or table thereof) of the decorated module will have each row (samples) zeroed when the commensurate row of the input is a tensor of zeros.

The nInputDim argument must specify the number of non-batch dims in the first Tensor of the input. In the case of an input table, the first Tensor is the first one encountered when doing a depth-first search.

This decorator makes it possible to pad sequences with different lengths in the same batch with zero vectors.

Caveat: MaskZero not guarantee that the output and gradInput tensors of the internal modules of the decorated module will be zeroed as well when the input is zero as well. MaskZero only affects the immediate gradInput and output of the module that it encapsulates. However, for most modules, the gradient update for that time-step will be zero because backpropagating a gradient of zeros will typically yield zeros all the way to the input. In this respect, modules to avoid in encapsulating inside a MaskZero are AbsractRecurrent instances as the flow of gradients between different time-steps internally. Instead, call the AbstractRecurrent.maskZero method to encapsulate the internal recurrentModule.

TrimZero

WARNING : only use this module if your input contains lots of zeros. In almost all cases, MaskZero will be faster, especially with CUDA.

Ref. A : TrimZero: A Torch Recurrent Module for Efficient Natural Language Processing

The usage is the same with MaskZero.

mz = nn.TrimZero(module, nInputDim)

The only difference from MaskZero is that it reduces computational costs by varying a batch size, if any, for the case that varying lengths are provided in the input. Notice that when the lengths are consistent, MaskZero will be faster, because TrimZero has an operational cost.

In short, the result is the same with MaskZero's, however, TrimZero is faster than MaskZero only when sentence lengths is costly vary.

In practice, e.g. language model, TrimZero is expected to be faster than MaskZero about 30%. (You can test with it using test/test_trimzero.lua.)

LookupTableMaskZero

This module extends nn.LookupTable to support zero indexes. Zero indexes are forwarded as zero tensors.

lt = nn.LookupTableMaskZero(nIndex, nOutput)

The output Tensor will have each row zeroed when the commensurate row of the input is a zero index.

This lookup table makes it possible to pad sequences with different lengths in the same batch with zero vectors.

MaskZeroCriterion

This criterion zeroes the err and gradInput rows of the decorated criterion for commensurate input rows which are tensors of zeros.

mzc = nn.MaskZeroCriterion(criterion, nInputDim)

The gradInput Tensor (or table thereof) of the decorated criterion will have each row (samples) zeroed when the commensurate row of the input is a tensor of zeros. The err will also disregard such zero rows.

The nInputDim argument must specify the number of non-batch dims in the first Tensor of the input. In the case of an input table, the first Tensor is the first one encountered when doing a depth-first search.

This decorator makes it possible to pad sequences with different lengths in the same batch with zero vectors.

SeqReverseSequence

reverseSeq = nn.SeqReverseSequence(dim)

Reverses an input tensor on a specified dimension. The reversal dimension can be no larger than three.

Example:

input = torch.Tensor({{1,2,3,4,5}, {6,7,8,9,10}})
reverseSeq = nn.SeqReverseSequence(1)
print(reverseSeq:forward(input))

Gives us an output of torch.Tensor({{6,7,8,9,10},{1,2,3,4,5}})

SequencerCriterion

This Criterion is a decorator:

c = nn.SequencerCriterion(criterion, [sizeAverage])

Both the input and target are expected to be a sequence, either as a table or Tensor. For each step in the sequence, the corresponding elements of the input and target will be applied to the criterion. The output of forward is the sum of all individual losses in the sequence. This is useful when used in conjunction with a Sequencer.

If sizeAverage is true (default is false), the output loss and gradInput is averaged over each time-step.

RepeaterCriterion

This Criterion is a decorator:

c = nn.RepeaterCriterion(criterion)

The input is expected to be a sequence (table or Tensor). A single target is repeatedly applied using the same criterion to each element in the input sequence. The output of forward is the sum of all individual losses in the sequence. This is useful for implementing models like RCNNs, which are repeatedly presented with the same target.

Name		Name	Last commit message	Last commit date
Latest commit History 622 Commits
doc		doc
examples		examples
rocks		rocks
scripts		scripts
test		test
.gitignore		.gitignore
AbstractRecurrent.lua		AbstractRecurrent.lua
AbstractSequencer.lua		AbstractSequencer.lua
BiSequencer.lua		BiSequencer.lua
BiSequencerLM.lua		BiSequencerLM.lua
CMakeLists.txt		CMakeLists.txt
CopyGrad.lua		CopyGrad.lua
Dropout.lua		Dropout.lua
ExpandAs.lua		ExpandAs.lua
FastLSTM.lua		FastLSTM.lua
GRU.lua		GRU.lua
LICENSE.2nd.txt		LICENSE.2nd.txt
LICENSE.txt		LICENSE.txt
LSTM.lua		LSTM.lua
LinearNoBias.lua		LinearNoBias.lua
LookupTableMaskZero.lua		LookupTableMaskZero.lua
MaskZero.lua		MaskZero.lua
MaskZeroCriterion.lua		MaskZeroCriterion.lua
Module.lua		Module.lua
Mufuru.lua		Mufuru.lua
NormStabilizer.lua		NormStabilizer.lua
Padding.lua		Padding.lua
README.md		README.md
Recurrence.lua		Recurrence.lua
Recurrent.lua		Recurrent.lua
RecurrentAttention.lua		RecurrentAttention.lua
Recursor.lua		Recursor.lua
Repeater.lua		Repeater.lua
RepeaterCriterion.lua		RepeaterCriterion.lua
SAdd.lua		SAdd.lua
SeqBRNN.lua		SeqBRNN.lua
SeqGRU.lua		SeqGRU.lua
SeqLSTM.lua		SeqLSTM.lua
SeqLSTMP.lua		SeqLSTMP.lua
SeqReverseSequence.lua		SeqReverseSequence.lua
Sequencer.lua		Sequencer.lua
SequencerCriterion.lua		SequencerCriterion.lua
TrimZero.lua		TrimZero.lua
ZeroGrad.lua		ZeroGrad.lua
init.lua		init.lua
recursiveUtils.lua		recursiveUtils.lua

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rnn: recurrent neural networks

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Troubleshooting

AbstractRecurrent

[recurrentModule] getStepModule(step)

setOutputStep(step)

maskZero(nInputDim)

trimZero(nInputDim)

[output] updateOutput(input)

updateGradInput(input, gradOutput)

accGradParameters(input, gradOutput, scale)

recycle(offset)

forget(offset)

maxBPTTstep(rho)

backwardOnline()

training()

evaluate()

Recurrent

Decorate it with a Sequencer

LSTM

FastLSTM

usenngraph

Recurrent Batch Normalization

GRU

SAdd

MuFuRu

Recursor

Recurrence

NormStabilizer

AbstractSequencer

Sequencer

Input/Output Format

Example

Details

remember([mode])

forget()

SeqLSTM

SeqLSTMP

SeqGRU

SeqBRNN

BiSequencer

BiSequencerLM

Repeater

RecurrentAttention

MaskZero

TrimZero

LookupTableMaskZero

MaskZeroCriterion

SeqReverseSequence

SequencerCriterion

RepeaterCriterion

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