#

Feedback Transformer

This is a PyTorch implementation of the paper Accessing Higher-level Representations in Sequential Transformers with Feedback Memory.

Normal transformers process tokens in parallel. Each transformer layer pays attention to the outputs of the previous layer. Feedback transformer pays attention to the output of all layers in previous steps. So this adds recurrence, and we need to process token-by-token. This slows down the training significantly (about 5X - 10X depending on the sequence length). However, when predicting Feedback Transformer is faster because you can predict the next token if you cache the memory vectors.

In order to speed up the training, the paper discusses starting with a short sequence length and gradually increasing it. They also discuss using a pretrained parallel transformer as the starting point.

The original feedback transformer doesn't keep the outputs of all layers. Instead it keeps weighted sum of the output of all layers. This reduces the memory used for caching during prediction. The first half of this file implements this.

The updated feedback transformer shares weights $W_{k}^{l}$ and $W_{v}^{l}$ used to calculate keys and values among the layers. We then calculate the keys and values for each step only once and keep them cached. The second half of this file implements this. We implemented a custom PyTorch function to improve performance.

Here's the training code and a notebook for training a feedback transformer on Tiny Shakespeare dataset.

42import math
43from typing import Optional
44
45import torch
46from torch import nn
47
48from labml_nn.transformers.feed_forward import FeedForward
49from labml_nn.transformers.mha import PrepareForMultiHeadAttention
50from labml_nn.utils import clone_module_list

#

Feedback Attention

This module computes recurrent attention similar to attention from original transformers paper.

$A tt e n t i o n (Q, K, V) = se q so f t ma x (\frac{Q ^{⊤} K}{d _{k}}) V$

53class FeedbackAttention(nn.Module):

#

'heads' is the number of attention heads
d_model is the number of features in the transformer
dropout_prob is the attention dropout probability
is_kv_precomputed is whether key, value tensors are already calculated

64    def __init__(self, heads: int, d_model: int, dropout_prob: float = 0.1, *,
65                 is_kv_precomputed: bool = False):

#

73        super().__init__()

#

Number of features per head

76        self.d_k = d_model // heads

#

78        self.heads = heads

#

These transform the query multi-headed attention.

81        self.query = PrepareForMultiHeadAttention(d_model, heads, self.d_k, bias=False)

#

These transform the key and value for multi-headed attention.

83        if not is_kv_precomputed:
84            self.key = PrepareForMultiHeadAttention(d_model, heads, self.d_k, bias=False)
85            self.value = PrepareForMultiHeadAttention(d_model, heads, self.d_k, bias=True)

#

Keys and values are already calculated

87        else:
88            self.key = None
89            self.value = None

#

Output layer

92        self.output = nn.Linear(d_model, d_model)

#

Dropout

94        self.dropout = nn.Dropout(dropout_prob)

#

Scaling factor before the softmax

96        self.scale = 1 / math.sqrt(self.d_k)

#

Softmax for attention along the time dimension of key

99        self.softmax = nn.Softmax(dim=0)

#

Number of relative positions

102        self.P = 2 ** 12

#

Relative positional embeddings for key relative to the query.

105        self.key_pos_embeddings = nn.Parameter(torch.zeros((self.P, heads, self.d_k)), requires_grad=True)

#

Relative positional embedding bias for key relative to the query.

107        self.key_pos_bias = nn.Parameter(torch.zeros((self.P, heads)), requires_grad=True)

#

Positional embeddings for the query is independent of the position of the query

109        self.query_pos_bias = nn.Parameter(torch.zeros((heads, self.d_k)), requires_grad=True)

#

We store attentions so that it can be used for logging, or other computations if needed

112        self.attn = None

#

Get attention scores

We use relative positional encodings for attention, similar to relative multi-head attention form Transformer-XL paper.

Attention from current step's query to key in step $j$ (relative to current step) is,

A_{j} = Q^{⊤} K_{j} = l i n_{q} (X^{q} + P_{q})^{⊤} l i n_{k} (X_{j}^{k} + P_{j}) = (Q + U^{Q})^{⊤} (K_{j} + U_{j}^{K}) = A Q^{⊤} K_{j} + B Q^{⊤} U_{j}^{K} + C U^{Q}^{⊤} K_{j} + D U^{Q}^{⊤} U_{j}^{K}

where $Q, K_{j}$ , are linear transformations of original embeddings $X^{q}, X_{j}^{k}$ and $U^{Q}, U_{j}^{K}$ are linear transformations of positional encodings $P_{q}, P_{j}$ .

We replace term $D$ with $S_{j}$ .

114    def get_scores(self, query: torch.Tensor, key: torch.Tensor):

#

$U_{j}^{K}$

142        key_pos_emb = self.key_pos_embeddings[-key.shape[0]:]

#

$U^{Q}$

144        query_pos_bias = self.query_pos_bias[None, :, :]

#

$S_{j}$

146        key_pos_bias = self.key_pos_bias[-key.shape[0]:]

#

$A Q^{⊤} K_{j} + C U^{Q}^{⊤} K_{j}$

149        ac = torch.einsum('bhd,jbhd->jbh', query + query_pos_bias, key)

#

$B Q^{⊤} U_{j}^{K} + D S_{j}$

151        bd = torch.einsum('bhd,jhd->jbh', query, key_pos_emb) + key_pos_bias[:, None, :]

#

$A_{j}$

154        return ac + bd

#

query has shape [batch_size, d_model]
key and value has shape [seq_len, batch_size, d_model]

156    def forward(self, *,
157                query: torch.Tensor,
158                key: torch.Tensor,
159                value: torch.Tensor):

#

Prepare query , key and value for attention computation key and value will then have shape [seq_len, batch_size, heads, d_k] and query will have shape [batch_size, heads, d_k]

168        query = self.query(query)
169        if self.key:
170            key = self.key(key)
171        if self.value:
172            value = self.value(value)

#

Compute attention scores. Results in a tensor of shape [seq_len, batch_size, heads]

176        scores = self.get_scores(query, key)

#

Scale scores $\frac{1}{d _{k}}$

179        scores *= self.scale

#

Softmax

182        attn = self.softmax(scores)

#

Apply dropout

185        attn = self.dropout(attn)

#

Multiply by the values

188        x = torch.einsum("jbh,jbhd->bhd", attn, value)

#

Concatenate multiple heads

191        x = x.reshape(x.shape[0], -1)

#

Output layer

194        return self.output(x)

#

Feedback Transformer Layer

This implements a single transformer layer in the feedback transformer.

197class FeedbackTransformerLayer(nn.Module):

#

d_model is the number of features in the transformer
attn is the feedback attention module
feed_forward is the position-wise feed forward layer
dropout_prob is the dropout probability for dropout layers after attention and feed-forward

204    def __init__(self, *,
205                 d_model: int,
206                 attn: FeedbackAttention,
207                 feed_forward: FeedForward,
208                 dropout_prob: float):

#

215        super().__init__()

#

Transformer size $d_{m o d e l}$

217        self.size = d_model

#

219        self.attn = attn
220        self.feed_forward = feed_forward
221        self.dropout = nn.Dropout(dropout_prob)

#

Normalization layers

224        self.norm_self_attn = nn.LayerNorm([d_model])
225        self.norm_ff = nn.LayerNorm([d_model])

#

227    def forward(self, *,
228                x: torch.Tensor,
229                key: Optional[torch.Tensor],
230                value: Optional[torch.Tensor]):

#

If there is memory

232        if key is not None:

#

Normalize the vectors before doing self attention

234            z = self.norm_self_attn(x)

#

Run through self attention, i.e. keys and values are from self

236            self_attn = self.attn(query=z, key=key, value=value)

#

Add the self attention results

238            x = x + self.dropout(self_attn)

#

Normalize for feed-forward

241        z = self.norm_ff(x)

#

Pass through the feed-forward network

243        ff = self.feed_forward(z)

#

Add the feed-forward results back

245        x = x + self.dropout(ff)

#

248        return x

#

Feedback Transformer Module

251class FeedbackTransformer(nn.Module):

#

layer is the feedback transformer layer, which we clone for each layer
n_layers is the number of layers in the transformer

256    def __init__(self, layer: FeedbackTransformerLayer, n_layers: int):

#

262        super().__init__()

#

Make copies of the transformer layer

264        self.layers = clone_module_list(layer, n_layers)

#

Final normalization layer

266        self.norm = nn.LayerNorm([layer.size])

#

Memory vectors are computed as a weighted sum of representations of each layer. This is the weights parameter for that.

269        self.weights = nn.Parameter(torch.ones(n_layers + 1), requires_grad=True)

#

Softmax for weights before taking the weighted sum

271        self.softmax = nn.Softmax(0)

#

x_seq is the input with shape [seq_len, batch_size, d_model]

273    def forward(self, x_seq: torch.Tensor):

#

Split the input to a list along the sequence axis

279        x_seq = torch.unbind(x_seq, dim=0)

#

List to store the outputs

281        res = []

#

List to store the memory vectors

283        mem = []

#

For each input step

285        for x in x_seq:

#

List to store layer outputs

287            layer_outputs = [x]

#

If there is memory, stack them into a vector

290            mem_tensor = torch.stack(mem) if mem else None

#

Run through each layer

293            for layer in self.layers:

#

Get layer output

295                x = layer(x=x, key=mem_tensor, value=mem_tensor)

#

Append them to the list of layer outputs

297                layer_outputs.append(x)

#

Stack the layer outputs to a tensor

300            layer_outputs = torch.stack(layer_outputs)

#

Calculate the memory vector as a weighted sum of layer outputs

302            mem.append(torch.einsum('lbd,l->bd', layer_outputs, self.softmax(self.weights)))

#

Append the output to results

304            res.append(x)

#

Stack the output tensors

307        res = torch.stack(res)

#

Normalize the output

309        return self.norm(res)

#

Shared keys and values among layers

#

Stack Function implementation

We implement a custom function instead of appending to a python list and then doing torch.stack . This greatly improves the performance over calling torch.stack at each step along the sequence. Everytime torch.stack is called, it creates a new tensor, while this method and the accompanying class Stack share memory for each step.

316class StackFunction(torch.autograd.Function):

#

ctx is the context of the function (which lets us cache stuff)
memory is the shared memory tensor where we stack and store the values of each step (keys & values)
memory_grad is the shared memory tensor to store and accumulate gradients of each step
last is the last value stacked
n is the number of steps (i.e. size of the stack)

This returns the stacked tensor for steps upto n .

328    @staticmethod
329    def forward(ctx, memory, memory_grad, last, n):

#

Cache accumulated gradients

341        ctx._mem_grad = memory_grad

#

Cache the size of the stack

343        ctx._n = n

#

Return the stack

345        return memory[:n + 1]

#

grad_output is the gradient with respect to the output of about forward function

This accumulates the gradients in the shared memory tensor and return the gradients with respect to the last result in the stack.

347    @staticmethod
348    def backward(ctx, grad_output):

#

Get the current size of the stack

356        n = ctx._n

#

Get the accumulated gradients

358        memory_grad = ctx._mem_grad

#

Add the gradients

360        memory_grad[:n + 1] += grad_output

#

Return the gradients w.r.t to last value in the stack

362        return None, None, memory_grad[n], None

#

Stack Module

This uses the stack function defined above, and does the necessary initializations.

365class Stack:

#

max_len is the maximum size of the stack

372    def __init__(self, max_len: int):

#

376        self.max_len = max_len
377        self.memory = None
378        self.memory_grad = None
379        self.last = None
380        self.n = -1
381        self.last_get_n = -1

#

n is the size of the stack
value is the tensor that needs to be added to the stack

383    def append(self, n: int, value: torch.Tensor):

#

You need to get (use) the stack after adding a value. Otherwise this implementation fails

391        assert n == 0 or self.last_get_n == n - 1, f"{n}, {self.last_get_n}"

#

Do this without gradients

394        with torch.no_grad():

#

Initialize the shared memory tensor to keep the stack

396            if self.memory is None or self.memory.shape[1:] != value.shape:

#

This should only happen when the stack is empty

398                assert n == 0

#

Create a tensor for the stack

400                self.memory = value.new_zeros(self.max_len, *value.shape, requires_grad=False)

#

Create a tensor to accumulate the gradients

402                self.memory_grad = value.new_zeros(self.memory.shape, requires_grad=False)

#

The memory is already initialized but we are resetting the stack.

This could have been another function like reset , but we found this easier to use.

407            elif n == 0:

#

Reset accumulated gradients

409                self.memory_grad.fill_(0.)

#

Set the value in the correct position of the stack

412            self.memory.data[n] = value.detach()

#

Keep track of the stack (for debugging)

414            self.n = n

#

Keep track of the last value added to the stack. We need this to be passed on to StackFunction in order to get the gradients propagated backwards.

419        self.last = value

#

Returns the stack

421    def get(self):

#

Keep track of the size of the stack when it was used. This is used for a sanity check in append .

428        self.last_get_n = self.n

#

Take it all through StackFunction so that StackFunction.backwards is called by PyTorch during backpropagation.

431        return StackFunction.apply(self.memory, self.memory_grad, self.last, self.n)

#

To release memory

433    def free(self):

#

438        self.memory = None
439        self.memory_grad = None
440        self.last = None

#

Updated Feedback Transformer Module

This is the updated feedback transformer module that caches the keys and values.

443class FeedbackTransformerKV(nn.Module):

#

layer is the feedback transformer layer, which we clone for each layer
n_layers is the number of layers in the transformer
d_model is the number of features in the transformer
'heads' is the number of attention heads

450    def __init__(self, layer: FeedbackTransformerLayer, n_layers: int, d_model: int, heads: int):

#

458        super().__init__()

#

Make copies of the transformer layer

460        self.layers = clone_module_list(layer, n_layers)

#

Final normalization layer

462        self.norm = nn.LayerNorm([layer.size])

#

Memory vectors are computed as a weighted sum of representations of each layer. This is the weights parameter for that.

465        self.weights = nn.Parameter(torch.ones(n_layers + 1), requires_grad=True)

#

Softmax for weights before taking the weighted sum

467        self.softmax = nn.Softmax(0)

#

Number of features in a head

470        d_k = d_model // heads

#

Module to transform embeddings (memory) to get keys

472        self.key = PrepareForMultiHeadAttention(d_model, heads, d_k, bias=False)

#

Module to transform embeddings (memory) to get keys

474        self.value = PrepareForMultiHeadAttention(d_model, heads, d_k, bias=False)

#

Memory for stacked keys

477        self.mem_key = Stack(512)

#

Memory for stacked values

479        self.mem_value = Stack(512)

#

x_seq is the input with shape [seq_len, batch_size, d_model]

481    def forward(self, x_seq: torch.Tensor):

#