#

Long Short-Term Memory (LSTM)

10from typing import Optional, Tuple
11
12import torch
13from torch import nn
14
15from labml_helpers.module import Module

#

Long Short-Term Memory Cell

LSTM Cell computes $c$, and $h$. $c$ is like the long-term memory, and $h$ is like the short term memory. We use the input $x$ and $h$ to update the long term memory. In the update, some features of $c$ are cleared with a forget gate $f$, and some features $i$ are added through a gate $g$.

The new short term memory is the $\tanh$ of the long-term memory multiplied by the output gate $o$.

Note that the cell doesn’t look at long term memory $c$ when doing the update for the update. It only modifies it. Also $c$ never goes through a linear transformation. This is what solves vanishing and exploding gradients.

Here’s the update rule.

$\begin{align} c_t &= \sigma(f_t) \odot c_{t-1} + \sigma(i_t) \odot \tanh(g_t) \\ h_t &= \sigma(o_t) \odot \tanh(c_t) \end{align}$

$\odot$ stands for element-wise multiplication.

Intermediate values and gates are computed as linear transformations of the hidden state and input.

$\begin{align} i_t &= lin_x^i(x_t) + lin_h^i(h_{t-1}) \\ f_t &= lin_x^f(x_t) + lin_h^f(h_{t-1}) \\ g_t &= lin_x^g(x_t) + lin_h^g(h_{t-1}) \\ o_t &= lin_x^o(x_t) + lin_h^o(h_{t-1}) \end{align}$

18class LSTMCell(Module):

#

57    def __init__(self, input_size: int, hidden_size: int, layer_norm: bool = False):
58        super().__init__()

#

These are the linear layer to transform the input and hidden vectors. One of them doesn’t need a bias since we add the transformations.

#

This combines $lin_x^i$, $lin_x^f$, $lin_x^g$, and $lin_x^o$ transformations.

64        self.hidden_lin = nn.Linear(hidden_size, 4 * hidden_size)

#

This combines $lin_h^i$, $lin_h^f$, $lin_h^g$, and $lin_h^o$ transformations.

66        self.input_lin = nn.Linear(input_size, 4 * hidden_size, bias=False)

#

Whether to apply layer normalizations.

Applying layer normalization gives better results. $i$, $f$, $g$ and $o$ embeddings are normalized and $c_t$ is normalized in $h_t = o_t \odot \tanh(\mathop{LN}(c_t))$

73        if layer_norm:
74            self.layer_norm = nn.ModuleList([nn.LayerNorm(hidden_size) for _ in range(4)])
75            self.layer_norm_c = nn.LayerNorm(hidden_size)
76        else:
77            self.layer_norm = nn.ModuleList([nn.Identity() for _ in range(4)])
78            self.layer_norm_c = nn.Identity()

#

80    def __call__(self, x: torch.Tensor, h: torch.Tensor, c: torch.Tensor):

#

We compute the linear transformations for $i_t$, $f_t$, $g_t$ and $o_t$ using the same linear layers.

83        ifgo = self.hidden_lin(h) + self.input_lin(x)

#

Each layer produces an output of 4 times the hidden_size and we split them

85        ifgo = ifgo.chunk(4, dim=-1)

#

Apply layer normalization (not in original paper, but gives better results)

88        ifgo = [self.layer_norm[i](ifgo[i]) for i in range(4)]

#

$i_t, f_t, g_t, o_t$

91        i, f, g, o = ifgo

#

$c_t = \sigma(f_t) \odot c_{t-1} + \sigma(i_t) \odot \tanh(g_t)$

94        c_next = torch.sigmoid(f) * c + torch.sigmoid(i) * torch.tanh(g)

#

$h_t = \sigma(o_t) \odot \tanh(c_t)$ Optionally, apply layer norm to $c_t$

98        h_next = torch.sigmoid(o) * torch.tanh(self.layer_norm_c(c_next))
99
100        return h_next, c_next

#

Multilayer LSTM

103class LSTM(Module):

#

Create a network of n_layers of LSTM.

108    def __init__(self, input_size: int, hidden_size: int, n_layers: int):

#

113        super().__init__()
114        self.n_layers = n_layers
115        self.hidden_size = hidden_size

#

Create cells for each layer. Note that only the first layer gets the input directly. Rest of the layers get the input from the layer below

118        self.cells = nn.ModuleList([LSTMCell(input_size, hidden_size)] +
119                                   [LSTMCell(hidden_size, hidden_size) for _ in range(n_layers - 1)])

#

x has shape [n_steps, batch_size, input_size] and state is a tuple of $h$ and $c$, each with a shape of [batch_size, hidden_size].

121    def __call__(self, x: torch.Tensor, state: Optional[Tuple[torch.Tensor, torch.Tensor]] = None):

#

126        n_steps, batch_size = x.shape[:2]

#

Initialize the state if None

129        if state is None:
130            h = [x.new_zeros(batch_size, self.hidden_size) for _ in range(self.n_layers)]
131            c = [x.new_zeros(batch_size, self.hidden_size) for _ in range(self.n_layers)]
132        else:
133            (h, c) = state

#

Reverse stack the tensors to get the states of each layer
📝 You can just work with the tensor itself but this is easier to debug

136            h, c = list(torch.unbind(h)), list(torch.unbind(c))

#

Array to collect the outputs of the final layer at each time step.

139        out = []
140        for t in range(n_steps):

#

Input to the first layer is the input itself

142            inp = x[t]

#

Loop through the layers

144            for layer in range(self.n_layers):

#

Get the state of the layer

146                h[layer], c[layer] = self.cells[layer](inp, h[layer], c[layer])

#

Input to the next layer is the state of this layer

148                inp = h[layer]

#

Collect the output $h$ of the final layer

150            out.append(h[-1])

#

Stack the outputs and states

153        out = torch.stack(out)
154        h = torch.stack(h)
155        c = torch.stack(c)
156
157        return out, (h, c)