LSTM

In a traditional recurrent neural network, during the gradient back-propagation phase, the gradient signal can end up being multiplied a large number of times (as many as the number of timesteps) by the weight matrix associated with the connections between the neurons of the recurrent hidden layer. This means that, the magnitude of weights in the transition matrix can have a strong impact on the learning process.

If the weights in this matrix are small (or, more formally, if the leading eigenvalue of the weight matrix is smaller than 1.0), it can lead to a situation called vanishing gradients where the gradient signal gets so small that learning either becomes very slow or stops working altogether. It can also make more difficult the task of learning long-term dependencies in the data. Conversely, if the weights in this matrix are large (or, again, more formally, if the leading eigenvalue of the weight matrix is larger than 1.0), it can lead to a situation where the gradient signal is so large that it can cause learning to diverge. This is often referred to as exploding gradients.

These issues are the main motivation behind the LSTM model which introduces a new structure called a memory cell (see Figure 1 below). A memory cell is composed of four main elements: an input gate, a neuron with a self-recurrent connection (a connection to itself), a forget gate and an output gate. The self-recurrent connection has a weight of 1.0 and ensures that, barring any outside interference, the state of a memory cell can remain constant from one timestep to another. The gates serve to modulate the interactions between the memory cell itself and its environment. The input gate can allow incoming signal to alter the state of the memory cell or block it. On the other hand, the output gate can allow the state of the memory cell to have an effect on other neurons or prevent it. Finally, the forget gate can modulate the memory cell’s self-recurrent connection, allowing the cell to remember or forget its previous state, as needed.

Figure 1: Illustration of an LSTM memory cell.

The equations below describe how a layer of memory cells is updated at every timestep . In these equations:

•  is the input to the memory cell layer at time 
• , , , , , , ,  and  are weight matrices
• , ,  and  are bias vectors

First, we compute the values for , the input gate, and  the candidate value for the states of the memory cells at time :

Second, we compute the value for , the activation of the memory cells’ forget gates at time :

Given the value of the input gate activation , the forget gate activation  and the candidate state value , we can compute  the memory cells’ new state at time :

With the new state of the memory cells, we can compute the value of their output gates and, subsequently, their outputs:

Our model

The model we used in this tutorial is a variation of the standard LSTM model. In this variant, the activation of a cell’s output gate does not depend on the memory cell’s state . This allows us to perform part of the computation more efficiently (see the implementation note, below, for details). This means that, in the variant we have implemented, there is no matrix  and equation (5) is replaced by equation (7):

Our model is composed of a single LSTM layer followed by an average pooling and a logistic regression layer as illustrated in Figure 2 below. Thus, from an input sequence , the memory cells in the LSTM layer will produce a representation sequence . This representation sequence is then averaged over all timesteps resulting in representation h. Finally, this representation is fed to a logistic regression layer whose target is the class label associated with the input sequence.

Figure 2 : Illustration of the model used in this tutorial. It is composed of a single LSTM layer followed by mean pooling over time and logistic regression.

Implementation note : In the code included this tutorial, the equations (1), (2), (3) and (7) are performed in parallel to make the computation more efficient. This is possible because none of these equations rely on a result produced by the other ones. It is achieved by concatenating the four matrices  into a single weight matrix  and performing the same concatenation on the weight matrices  to produce the matrix  and the bias vectors  to produce the vector . Then, the pre-nonlinearity activations can be computed with:

The result is then sliced to obtain the pre-nonlinearity activations for , , , and  and the non-linearities are then applied independently for each.

Code

The LSTM implementation can be found in the two following files:

• lstm.py: Main script. Defines and train the model.
• imdb.py: Secondary script. Handles the loading and preprocessing of the IMDB dataset.

After downloading both scripts and putting both in the same folder, the user can run the code by calling:

THEANO_FLAGS="floatX=float32" python lstm.py

The script will automatically download the data and decompress it.

Note: The provided code supports the Stochastic Gradient Descent (SGD), AdaDelta and RMSProp optimization methods. You are advised to use AdaDelta or RMSProp because SGD appears to performs poorly on this task with this particular model.

Papers

If you use this tutorial, please cite the following papers.

Introduction of the LSTM model:

• [pdf] Hochreiter, S., & Schmidhuber, J. (1997). Long short-term memory. Neural computation, 9(8), 1735-1780.

Addition of the forget gate to the LSTM model:

• [pdf] Gers, F. A., Schmidhuber, J., & Cummins, F. (2000). Learning to forget: Continual prediction with LSTM. Neural computation, 12(10), 2451-2471.

More recent LSTM paper:

• [pdf] Graves, Alex. Supervised sequence labelling with recurrent neural networks. Vol. 385. Springer, 2012.

Papers related to Theano:

• [pdf] Bastien, Frédéric, Lamblin, Pascal, Pascanu, Razvan, Bergstra, James, Goodfellow, Ian, Bergeron, Arnaud, Bouchard, Nicolas, and Bengio, Yoshua. Theano: new features and speed improvements. NIPS Workshop on Deep Learning and Unsupervised Feature Learning, 2012.
• [pdf] Bergstra, James, Breuleux, Olivier, Bastien, Frédéric, Lamblin, Pascal, Pascanu, Razvan, Desjardins, Guillaume, Turian, Joseph, Warde-Farley, David, and Bengio, Yoshua. Theano: a CPU and GPU math expression compiler. In Proceedings of the Python for Scientific Computing Conference (SciPy), June 2010.

Thank you!

Contact

Please email Pierre Luc Carrier or Kyunghyun Cho for any problem report or feedback. We will be glad to hear from you.