10.1  Long Short-Term Memory (LSTM)

Shortly after the first Elman-style RNNs were trained using backpropagation (Elman 1990), the problems of learning long-term dependencies (owing to vanishing and exploding gradients) became salient, with Bengio and Hochreiter discussing the problem (Bengio et al. 1994; Hochreiter et al. 2001). Hochreiter had articulated this problem as early as 1991 in his Master’s thesis, although the results were not widely known because the thesis was written in German. While gradient clipping helps with exploding gradients, handling vanishing gradients appears to require a more elaborate solution. One of the first and most successful techniques for addressing vanishing gradients came in the form of the long short-term memory (LSTM) model due to Hochreiter and Schmidhuber (1997) . LSTMs resemble standard recurrent neural networks but here each ordinary recurrent node is replaced by a memory cell. Each memory cell contains an internal state, i.e., a node with a self-connected recurrent edge of fixed weight 1, ensuring that the gradient can pass across many time steps without vanishing or exploding.

The term “long short-term memory” comes from the following intuition. Simple recurrent neural networks have long-term memory in the form of weights. The weights change slowly during training, encoding general knowledge about the data. They also have short-term memory in the form of ephemeral activations, which pass from each node to successive nodes. The LSTM model introduces an intermediate type of storage via the memory cell. A memory cell is a composite unit, built from simpler nodes in a specific connectivity pattern, with the novel inclusion of multiplicative nodes.

from d2l import torch as d2l
import torch
from torch import nn
from d2l import tensorflow as d2l
import tensorflow as tf
from d2l import jax as d2l
from flax import linen as nn
import jax
from jax import numpy as jnp
from d2l import mxnet as d2l
from mxnet import np, npx
from mxnet.gluon import rnn
npx.set_np()

10.1.1 Gated Memory Cell

Each memory cell is equipped with an internal state and a number of multiplicative gates that determine whether (i) a given input should impact the internal state (the input gate), (ii) the internal state should be flushed to \(0\) (the forget gate), and (iii) the internal state of a given neuron should be allowed to impact the cell’s output (the output gate).

10.1.1.1 Gated Hidden State

The key distinction between vanilla RNNs and LSTMs is that the latter support gating of the hidden state. This means that we have dedicated mechanisms for when a hidden state should be updated and also for when it should be reset. These mechanisms are learned and they address the concerns listed above. For instance, if the first token is of great importance we will learn not to update the hidden state after the first observation. Likewise, we will learn to skip irrelevant temporary observations. Last, we will learn to reset the latent state whenever needed. We discuss this in detail below.

10.1.1.2 Input Gate, Forget Gate, and Output Gate

The data feeding into the LSTM gates are the input at the current time step and the hidden state of the previous time step, as illustrated in Figure 10.1.1. Three fully connected layers with sigmoid activation functions compute the values of the input, forget, and output gates. As a result of the sigmoid activation, all values of the three gates are in the range of \((0, 1)\). Additionally, we require an input node, typically computed with a tanh activation function. Intuitively, the input gate determines how much of the input node’s value should be added to the current memory cell internal state. The forget gate determines whether to keep the current value of the memory or flush it. And the output gate determines whether the memory cell should influence the output at the current time step.

Figure 10.1.1: Computing the input gate, the forget gate, and the output gate in an LSTM model.

Mathematically, suppose that there are \(h\) hidden units, the batch size is \(n\), and the number of inputs is \(d\). Thus, the input is \(\mathbf{X}_t \in \mathbb{R}^{n \times d}\) and the hidden state of the previous time step is \(\mathbf{H}_{t-1} \in \mathbb{R}^{n \times h}\). Correspondingly, the gates at time step \(t\) are defined as follows: the input gate is \(\mathbf{I}_t \in \mathbb{R}^{n \times h}\), the forget gate is \(\mathbf{F}_t \in \mathbb{R}^{n \times h}\), and the output gate is \(\mathbf{O}_t \in \mathbb{R}^{n \times h}\). They are calculated as follows:

\[ \begin{aligned} \mathbf{I}_t &= \sigma(\mathbf{X}_t \mathbf{W}_{\textrm{xi}} + \mathbf{H}_{t-1} \mathbf{W}_{\textrm{hi}} + \mathbf{b}_\textrm{i}),\\ \mathbf{F}_t &= \sigma(\mathbf{X}_t \mathbf{W}_{\textrm{xf}} + \mathbf{H}_{t-1} \mathbf{W}_{\textrm{hf}} + \mathbf{b}_\textrm{f}),\\ \mathbf{O}_t &= \sigma(\mathbf{X}_t \mathbf{W}_{\textrm{xo}} + \mathbf{H}_{t-1} \mathbf{W}_{\textrm{ho}} + \mathbf{b}_\textrm{o}), \end{aligned} \]

where \(\mathbf{W}_{\textrm{xi}}, \mathbf{W}_{\textrm{xf}}, \mathbf{W}_{\textrm{xo}} \in \mathbb{R}^{d \times h}\) and \(\mathbf{W}_{\textrm{hi}}, \mathbf{W}_{\textrm{hf}}, \mathbf{W}_{\textrm{ho}} \in \mathbb{R}^{h \times h}\) are weight parameters and \(\mathbf{b}_\textrm{i}, \mathbf{b}_\textrm{f}, \mathbf{b}_\textrm{o} \in \mathbb{R}^{1 \times h}\) are bias parameters. Note that broadcasting (see Section 2.1.4) is triggered during the summation. We use sigmoid functions (as introduced in Section 5.1) to map the input values to the interval \((0, 1)\).

10.1.1.3 Input Node

Next we design the memory cell. Since we have not specified the action of the various gates yet, we first introduce the input node \(\tilde{\mathbf{C}}_t \in \mathbb{R}^{n \times h}\). Its computation is similar to that of the three gates described above, but uses a \(\tanh\) function with a value range for \((-1, 1)\) as the activation function. This leads to the following equation at time step \(t\):

\[\tilde{\mathbf{C}}_t = \textrm{tanh}(\mathbf{X}_t \mathbf{W}_{\textrm{xc}} + \mathbf{H}_{t-1} \mathbf{W}_{\textrm{hc}} + \mathbf{b}_\textrm{c}), \tag{10.1.1}\]

where \(\mathbf{W}_{\textrm{xc}} \in \mathbb{R}^{d \times h}\) and \(\mathbf{W}_{\textrm{hc}} \in \mathbb{R}^{h \times h}\) are weight parameters and \(\mathbf{b}_\textrm{c} \in \mathbb{R}^{1 \times h}\) is a bias parameter.

A quick illustration of the input node is shown in Figure 10.1.2.

Figure 10.1.2: Computing the input node in an LSTM model.

10.1.1.4 Memory Cell Internal State

In LSTMs, the input gate \(\mathbf{I}_t\) governs how much we take new data into account via \(\tilde{\mathbf{C}}_t\) and the forget gate \(\mathbf{F}_t\) addresses how much of the old cell internal state \(\mathbf{C}_{t-1} \in \mathbb{R}^{n \times h}\) we retain. Using the Hadamard (elementwise) product operator \(\odot\) we arrive at the following update equation:

\[\mathbf{C}_t = \mathbf{F}_t \odot \mathbf{C}_{t-1} + \mathbf{I}_t \odot \tilde{\mathbf{C}}_t. \tag{10.1.2}\]

If the forget gate is always 1 and the input gate is always 0, the memory cell internal state \(\mathbf{C}_{t-1}\) will remain constant forever, passing unchanged to each subsequent time step. However, input gates and forget gates give the model the flexibility of being able to learn when to keep this value unchanged and when to perturb it in response to subsequent inputs. In practice, this design alleviates the vanishing gradient problem, resulting in models that are much easier to train, especially when facing datasets with long sequence lengths.

We thus arrive at the flow diagram in Figure 10.1.3.

Figure 10.1.3: Computing the memory cell internal state in an LSTM model.

10.1.1.5 Hidden State

Last, we need to define how to compute the output of the memory cell, i.e., the hidden state \(\mathbf{H}_t \in \mathbb{R}^{n \times h}\), as seen by other layers. This is where the output gate comes into play. In LSTMs, we first apply \(\tanh\) to the memory cell internal state and then apply another point-wise multiplication, this time with the output gate. This ensures that the values of \(\mathbf{H}_t\) are always in the interval \((-1, 1)\):

\[\mathbf{H}_t = \mathbf{O}_t \odot \tanh(\mathbf{C}_t). \tag{10.1.3}\]

Whenever the output gate is close to 1, we allow the memory cell internal state to impact the subsequent layers uninhibited, whereas for output gate values close to 0, we prevent the current memory from impacting other layers of the network at the current time step. Note that a memory cell can accrue information across many time steps without impacting the rest of the network (as long as the output gate takes values close to 0), and then suddenly impact the network at a subsequent time step as soon as the output gate flips from values close to 0 to values close to 1. Figure 10.1.4 has a graphical illustration of the data flow.

Figure 10.1.4: Computing the hidden state in an LSTM model.

10.1.2 Implementation from Scratch

Now let’s implement an LSTM from scratch. As same as the experiments in Section 9.5, we first load The Time Machine dataset.

10.1.2.1 Initializing Model Parameters

Next, we need to define and initialize the model parameters. As previously, the hyperparameter num_hiddens dictates the number of hidden units. We initialize weights following a Gaussian distribution with 0.01 standard deviation, and we set the biases to 0.

class LSTMScratch(d2l.Module):
    def __init__(self, num_inputs, num_hiddens, sigma=0.01):
        super().__init__()
        self.save_hyperparameters()

        init_weight = lambda *shape: nn.Parameter(d2l.randn(*shape) * sigma)
        triple = lambda: (init_weight(num_inputs, num_hiddens),
                          init_weight(num_hiddens, num_hiddens),
                          nn.Parameter(d2l.zeros(num_hiddens)))
        self.W_xi, self.W_hi, self.b_i = triple()  # Input gate
        self.W_xf, self.W_hf, self.b_f = triple()  # Forget gate
        self.W_xo, self.W_ho, self.b_o = triple()  # Output gate
        self.W_xc, self.W_hc, self.b_c = triple()  # Input node
class LSTMScratch(d2l.Module):
    def __init__(self, num_inputs, num_hiddens, sigma=0.01):
        super().__init__()
        self.save_hyperparameters()

        init_weight = lambda *shape: tf.Variable(d2l.normal(shape) * sigma)
        triple = lambda: (init_weight(num_inputs, num_hiddens),
                          init_weight(num_hiddens, num_hiddens),
                          tf.Variable(d2l.zeros(num_hiddens)))

        self.W_xi, self.W_hi, self.b_i = triple()  # Input gate
        self.W_xf, self.W_hf, self.b_f = triple()  # Forget gate
        self.W_xo, self.W_ho, self.b_o = triple()  # Output gate
        self.W_xc, self.W_hc, self.b_c = triple()  # Input node
class LSTMScratch(d2l.Module):
    num_inputs: int
    num_hiddens: int
    sigma: float = 0.01

    def setup(self):
        init_weight = lambda name, shape: self.param(name,
                                                     nn.initializers.normal(self.sigma),
                                                     shape)
        triple = lambda name : (
            init_weight(f'W_x{name}', (self.num_inputs, self.num_hiddens)),
            init_weight(f'W_h{name}', (self.num_hiddens, self.num_hiddens)),
            self.param(f'b_{name}', nn.initializers.zeros, (self.num_hiddens)))

        self.W_xi, self.W_hi, self.b_i = triple('i')  # Input gate
        self.W_xf, self.W_hf, self.b_f = triple('f')  # Forget gate
        self.W_xo, self.W_ho, self.b_o = triple('o')  # Output gate
        self.W_xc, self.W_hc, self.b_c = triple('c')  # Input node
class LSTMScratch(d2l.Module):
    def __init__(self, num_inputs, num_hiddens, sigma=0.01):
        super().__init__()
        self.save_hyperparameters()

        init_weight = lambda *shape: d2l.randn(*shape) * sigma
        triple = lambda: (init_weight(num_inputs, num_hiddens),
                          init_weight(num_hiddens, num_hiddens),
                          d2l.zeros(num_hiddens))
        self.W_xi, self.W_hi, self.b_i = triple()  # Input gate
        self.W_xf, self.W_hf, self.b_f = triple()  # Forget gate
        self.W_xo, self.W_ho, self.b_o = triple()  # Output gate
        self.W_xc, self.W_hc, self.b_c = triple()  # Input node

The actual model is defined as described above, consisting of three gates and an input node. Note that only the hidden state is passed to the output layer.

The actual model is defined as described above, consisting of three gates and an input node. Note that only the hidden state is passed to the output layer.

The actual model is defined as described above, consisting of three gates and an input node. Note that only the hidden state is passed to the output layer. A long for-loop in the forward method will result in an extremely long JIT compilation time for the first run. As a solution to this, instead of using a for-loop to update the state with every time step, JAX has jax.lax.scan utility transformation to achieve the same behavior. It takes in an initial state called carry and an inputs array which is scanned on its leading axis. The scan transformation ultimately returns the final state and the stacked outputs as expected.

The actual model is defined as described above, consisting of three gates and an input node. Note that only the hidden state is passed to the output layer.

@d2l.add_to_class(LSTMScratch)
def forward(self, inputs, H_C=None):
    if H_C is None:
        # Initial state with shape: (batch_size, num_hiddens)
        H = d2l.zeros((inputs.shape[1], self.num_hiddens),
                      device=inputs.device)
        C = d2l.zeros((inputs.shape[1], self.num_hiddens),
                      device=inputs.device)
    else:
        H, C = H_C
    outputs = []
    for X in inputs:
        I = d2l.sigmoid(d2l.matmul(X, self.W_xi) +
                        d2l.matmul(H, self.W_hi) + self.b_i)
        F = d2l.sigmoid(d2l.matmul(X, self.W_xf) +
                        d2l.matmul(H, self.W_hf) + self.b_f)
        O = d2l.sigmoid(d2l.matmul(X, self.W_xo) +
                        d2l.matmul(H, self.W_ho) + self.b_o)
        C_tilde = d2l.tanh(d2l.matmul(X, self.W_xc) +
                           d2l.matmul(H, self.W_hc) + self.b_c)
        C = F * C + I * C_tilde
        H = O * d2l.tanh(C)
        outputs.append(H)
    return outputs, (H, C)
@d2l.add_to_class(LSTMScratch)
def forward(self, inputs, H_C=None):
    if H_C is None:
        # Initial state with shape: (batch_size, num_hiddens)
        H = tf.zeros((tf.shape(inputs)[1], self.num_hiddens))
        C = tf.zeros((tf.shape(inputs)[1], self.num_hiddens))
    else:
        H, C = H_C
    outputs = []
    for X in tf.unstack(inputs):
        I = d2l.sigmoid(d2l.matmul(X, self.W_xi) +
                        d2l.matmul(H, self.W_hi) + self.b_i)
        F = d2l.sigmoid(d2l.matmul(X, self.W_xf) +
                        d2l.matmul(H, self.W_hf) + self.b_f)
        O = d2l.sigmoid(d2l.matmul(X, self.W_xo) +
                        d2l.matmul(H, self.W_ho) + self.b_o)
        C_tilde = d2l.tanh(d2l.matmul(X, self.W_xc) +
                           d2l.matmul(H, self.W_hc) + self.b_c)
        C = F * C + I * C_tilde
        H = O * d2l.tanh(C)
        outputs.append(H)
    return outputs, (H, C)
@d2l.add_to_class(LSTMScratch)
def forward(self, inputs, H_C=None):
    # Use lax.scan primitive instead of looping over the
    # inputs, since scan saves time in jit compilation.
    def scan_fn(carry, X):
        H, C = carry
        I = d2l.sigmoid(d2l.matmul(X, self.W_xi) + (
            d2l.matmul(H, self.W_hi)) + self.b_i)
        F = d2l.sigmoid(d2l.matmul(X, self.W_xf) +
                        d2l.matmul(H, self.W_hf) + self.b_f)
        O = d2l.sigmoid(d2l.matmul(X, self.W_xo) +
                        d2l.matmul(H, self.W_ho) + self.b_o)
        C_tilde = d2l.tanh(d2l.matmul(X, self.W_xc) +
                           d2l.matmul(H, self.W_hc) + self.b_c)
        C = F * C + I * C_tilde
        H = O * d2l.tanh(C)
        return (H, C), H  # return carry, y

    if H_C is None:
        batch_size = inputs.shape[1]
        carry = jnp.zeros((batch_size, self.num_hiddens)), \
                jnp.zeros((batch_size, self.num_hiddens))
    else:
        carry = H_C

    # scan takes the scan_fn, initial carry state, xs with leading axis to be scanned
    carry, outputs = jax.lax.scan(scan_fn, carry, inputs)
    return outputs, carry
@d2l.add_to_class(LSTMScratch)
def forward(self, inputs, H_C=None):
    if H_C is None:
        # Initial state with shape: (batch_size, num_hiddens)
        H = d2l.zeros((inputs.shape[1], self.num_hiddens),
                      ctx=inputs.ctx)
        C = d2l.zeros((inputs.shape[1], self.num_hiddens),
                      ctx=inputs.ctx)
    else:
        H, C = H_C
    outputs = []
    for X in inputs:
        I = d2l.sigmoid(d2l.matmul(X, self.W_xi) +
                        d2l.matmul(H, self.W_hi) + self.b_i)
        F = d2l.sigmoid(d2l.matmul(X, self.W_xf) +
                        d2l.matmul(H, self.W_hf) + self.b_f)
        O = d2l.sigmoid(d2l.matmul(X, self.W_xo) +
                        d2l.matmul(H, self.W_ho) + self.b_o)
        C_tilde = d2l.tanh(d2l.matmul(X, self.W_xc) +
                           d2l.matmul(H, self.W_hc) + self.b_c)
        C = F * C + I * C_tilde
        H = O * d2l.tanh(C)
        outputs.append(H)
    return outputs, (H, C)

10.1.2.2 Training and Prediction

Let’s train an LSTM model by instantiating the RNNLMScratch class from Section 9.5.

data = d2l.TimeMachine(batch_size=1024, num_steps=32)
lstm = LSTMScratch(num_inputs=len(data.vocab), num_hiddens=32)
model = d2l.RNNLMScratch(lstm, vocab_size=len(data.vocab), lr=4)
trainer = d2l.Trainer(max_epochs=50, gradient_clip_val=1, num_gpus=1)
trainer.fit(model, data)

lstm-c3-pytorch
data = d2l.TimeMachine(batch_size=1024, num_steps=32)
with d2l.try_gpu():
    lstm = LSTMScratch(num_inputs=len(data.vocab), num_hiddens=32)
    model = d2l.RNNLMScratch(lstm, vocab_size=len(data.vocab), lr=4)
trainer = d2l.Trainer(max_epochs=50, gradient_clip_val=1)
trainer.fit(model, data)

lstm-c3-tensorflow
data = d2l.TimeMachine(batch_size=1024, num_steps=32)
lstm = LSTMScratch(num_inputs=len(data.vocab), num_hiddens=32)
model = d2l.RNNLMScratch(lstm, vocab_size=len(data.vocab), lr=4)
trainer = d2l.Trainer(max_epochs=50, gradient_clip_val=1, num_gpus=1)
trainer.fit(model, data)

lstm-c3-jax
data = d2l.TimeMachine(batch_size=1024, num_steps=32)
lstm = LSTMScratch(num_inputs=len(data.vocab), num_hiddens=32)
model = d2l.RNNLMScratch(lstm, vocab_size=len(data.vocab), lr=4)
trainer = d2l.Trainer(max_epochs=50, gradient_clip_val=1, num_gpus=1)
trainer.fit(model, data)

lstm-c3-mxnet

10.1.3 Concise Implementation

Using high-level APIs, we can directly instantiate an LSTM model. This encapsulates all the configuration details that we made explicit above. The code is significantly faster as it uses compiled operators rather than Python for many details that we spelled out before.

class LSTM(d2l.RNN):
    def __init__(self, num_inputs, num_hiddens):
        d2l.Module.__init__(self)
        self.save_hyperparameters()
        self.rnn = nn.LSTM(num_inputs, num_hiddens)

    def forward(self, inputs, H_C=None):
        return self.rnn(inputs, H_C)
class LSTM(d2l.RNN):
    def __init__(self, num_hiddens):
        d2l.Module.__init__(self)
        self.save_hyperparameters()
        self.rnn = tf.keras.layers.LSTM(
                num_hiddens, return_sequences=True,
                return_state=True)

    def forward(self, inputs, H_C=None):
        outputs, *H_C = self.rnn(tf.transpose(inputs, perm=[1, 0, 2]), H_C)
        return tf.transpose(outputs, perm=[1, 0, 2]), H_C
class LSTM(d2l.RNN):
    num_hiddens: int

    @nn.compact
    def __call__(self, inputs, H_C=None, training=False):
        if H_C is None:
            batch_size = inputs.shape[1]
            H_C = nn.OptimizedLSTMCell(features=self.num_hiddens).initialize_carry(
                jax.random.PRNGKey(0), (batch_size, self.num_hiddens))

        LSTM = nn.scan(nn.OptimizedLSTMCell, variable_broadcast="params",
                       in_axes=0, out_axes=0, split_rngs={"params": False})

        H_C, outputs = LSTM(features=self.num_hiddens)(H_C, inputs)
        return outputs, H_C
class LSTM(d2l.RNN):
    def __init__(self, num_hiddens):
        d2l.Module.__init__(self)
        self.save_hyperparameters()
        self.rnn = rnn.LSTM(num_hiddens)

    def forward(self, inputs, H_C=None):
        if H_C is None: H_C = self.rnn.begin_state(
            inputs.shape[1], ctx=inputs.ctx)
        return self.rnn(inputs, H_C)
lstm = LSTM(num_inputs=len(data.vocab), num_hiddens=32)
model = d2l.RNNLM(lstm, vocab_size=len(data.vocab), lr=4)
trainer.fit(model, data)

lstm-c5-pytorch
lstm = LSTM(num_hiddens=32)
with d2l.try_gpu():
    model = d2l.RNNLM(lstm, vocab_size=len(data.vocab), lr=4)
trainer.fit(model, data)

lstm-c5-tensorflow
lstm = LSTM(num_hiddens=32)
model = d2l.RNNLM(lstm, vocab_size=len(data.vocab), lr=4)
trainer.fit(model, data)

lstm-c5-jax
lstm = LSTM(num_hiddens=32)
model = d2l.RNNLM(lstm, vocab_size=len(data.vocab), lr=4)
trainer.fit(model, data)

lstm-c5-mxnet
model.predict('it has', 20, data.vocab, d2l.try_gpu())
'it has and the time travel'
model.predict('it has', 20, data.vocab)
'it haster the time travell'
model.predict('it has', 20, data.vocab, trainer.state.params)
'it has istention and the t'
model.predict('it has', 20, data.vocab, d2l.try_gpu())
'it has a dimension and the'

LSTMs are the prototypical latent variable autoregressive model with nontrivial state control. Many variants thereof have been proposed over the years, e.g., multiple layers, residual connections, different types of regularization. However, training LSTMs and other sequence models (such as GRUs) is quite costly because of the long range dependency of the sequence. Later we will encounter alternative models such as Transformers that can be used in some cases.

10.1.4 Summary

While LSTMs were published in 1997, they rose to great prominence with some victories in prediction competitions in the mid-2000s, and became the dominant models for sequence learning from 2011 until the rise of Transformer models, starting in 2017. Even Transformers owe some of their key ideas to architecture design innovations introduced by the LSTM.

LSTMs have three types of gates: input gates, forget gates, and output gates that control the flow of information. The hidden layer output of LSTM includes the hidden state and the memory cell internal state. Only the hidden state is passed into the output layer while the memory cell internal state remains entirely internal. LSTMs can alleviate vanishing and exploding gradients.

10.1.5 Exercises

  1. Adjust the hyperparameters and analyze their influence on running time, perplexity, and the output sequence.
  2. How would you need to change the model to generate proper words rather than just sequences of characters?
  3. Compare the computational cost for GRUs, LSTMs, and regular RNNs for a given hidden dimension. Pay special attention to the training and inference cost.
  4. Since the candidate memory cell ensures that the value range is between \(-1\) and \(1\) by using the \(\tanh\) function, why does the hidden state need to use the \(\tanh\) function again to ensure that the output value range is between \(-1\) and \(1\)?
  5. Implement an LSTM model for time series prediction rather than character sequence prediction.