Commentary
My commentary appears in this format. Code and blockquotes are the
work of the Harvard NLP group. The aim of this commentary is to help
complete beginners understand the paper.
The Transformer has been on a lot of
people’s minds over the last year five years.
This post presents an annotated version of the paper in the
form of a line-by-line implementation. It reorders and deletes
some sections from the original paper and adds comments
throughout. This document itself is a working notebook, and should
be a completely usable implementation.
Code is available
here.
import os
from os.path import exists
import torch
import torch.nn as nn
from torch.nn.functional import log_softmax, pad
import math
import copy
import time
from torch.optim.lr_scheduler import LambdaLR
import pandas as pd
import altair as alt
from torchtext.data.functional import to_map_style_dataset
from torch.utils.data import DataLoader
from torchtext.vocab import build_vocab_from_iterator
import torchtext.datasets as datasets
import spacy
import GPUtil
import warnings
from torch.utils.data.distributed import DistributedSampler
import torch.distributed as dist
import torch.multiprocessing as mp
from torch.nn.parallel import DistributedDataParallel as DDP
# Set to False to skip notebook execution (e.g. for debugging)
warnings.filterwarnings("ignore")
RUN_EXAMPLES = True
Commentary
These are some relevant code that needs an explanation.
# From code
import torch # Main PyTorch library for tensors and deep learning
import torch.nn as nn # Neural network layers and model building tools
At this point, it let’s see what a tensor is. It is very good to brush up on Scalars, Vectors and Matrices (Read here). Tensors are mathematical objects describing physical properties like scalars and vectors. In PyTorch, a tensor is a data structure abstracting the idea of a tensor and is used to encode the input, output as well as the model’s parameters. Tensors can run on the CPU, GPU, CUDA or other accelerators.
# From code
from torch.nn.functional import (
log_softmax, # Numerically stable log(softmax(x))
pad # Add padding to tensors
)
Softmax converts raw scores into probabilities.
Study this pieces of code to understand it.
The values of the output tensor sum to 1 and they can be interpreted as probabilities.
You can read a nice explanation about how softmax is calculated. Log softmax is used to stabilise calculations.
pad pads the tensor with values. You can specify where you want to pad.
import torch
import torch.nn.functional as F
x = torch.tensor([[1., 2.],
[3., 4.]])
y = F.pad(x, (1, 1, 1, 1), mode='constant', value=0)
print(y)
Out: tensor([[0., 0., 0., 0.],
[0., 1., 2., 0.],
[0., 3., 4., 0.],
[0., 0., 0., 0.]])
Next we have
# From code
# Learning rate scheduler using a custom lambda function
from torch.optim.lr_scheduler import LambdaLR
It’s to have a mental model of how PyTorch trains a model.
LambdaLR allows you to specify how the learning rate evolves with each training step.
Here is an example where we set the initial learning rate to be 0.05 then it is modified based on epoch 1.0 / (1.0 + 0.01 * epoch:
import torch
import torch.nn as nn
import torch.optim as optim
from torch.optim.lr_scheduler import LambdaLR
X = torch.linspace(-1, 1, 100).unsqueeze(1)
# X = 100 evenly spaced numbers between -1 and 1.
noise = torch.randn_like(X) * 0.1 # small noise (stable learning)
y = 3 * X + 2 + noise # True function, we compare our predictions against this
model = nn.Linear(1, 1)
criterion = nn.MSELoss()
optimizer = optim.SGD(model.parameters(), lr=0.05)
lambda_rule = lambda epoch: 1.0 / (1.0 + 0.01 * epoch)
scheduler = LambdaLR(optimizer, lr_lambda=lambda_rule)
epochs = 100
for epoch in range(epochs):
preds = model(X) # forward pass
loss = criterion(preds, y)
optimizer.zero_grad()
loss.backward() # backward
# gradient clipping (prevents explosion)
torch.nn.utils.clip_grad_norm_(model.parameters(), 1.0)
optimizer.step()
scheduler.step()
lr = optimizer.param_groups[0]["lr"]
if epoch % 10 == 0:
print(f"Epoch {epoch:02d} | Loss: {loss.item():.4f} | LR: {lr:.6f}")
w = model.weight.item()
b = model.bias.item()
print("\nLearned function:")
print(f"y ≈ {w:.3f}x + {b:.3f}")
Glossary:
Step
What it does
Forward pass
Model makes a prediction
Loss computation
Measures how wrong the prediction is
Backpropagation
Computes how each parameter contributed to the error
Zero gradients
Clears old gradient values
Optimizer step
Updates weights and biases to reduce error
Scheduler step
Adjusts learning rate over time
Epoch
One full pass over the dataset
In sum it is called a scheduler as it schedules for the entire run, how the learning rate changes.
import altair as alt
Altair (Vega-Altair) is “a declarative statistical visualization library for Python.” It aims to provide beautiful visualization with little amount of code. It
provides an easy way to provide interactions
In Jupyter
import altair as alt
import pandas as pd
df = pd.DataFrame({
"x": [1, 2, 3, 4],
"y": [10, 20, 15, 25]
})
chart = alt.Chart(df).mark_line().encode(
x="x",
y="y"
)
chart
Next we have TorchText
# TorchText utilities for NLP datasets and vocabularies
from torchtext.data.functional import to_map_style_dataset
# Creates mini-batches and handles dataset loading
from torchtext.vocab import build_vocab_from_iterator
# Builds a vocabulary from tokenized text
import torchtext.datasets as datasets
TorchTextwas primarily used to build the data-loading and preprocessing pipeline for Natural Language Processing (NLP) models in PyTorch.
# Converts iterable datasets into indexable datasets
from torch.utils.data import DataLoader
import spacy # Industrial-strength NLP toolkit for tokenization, parsing, etc.
import GPUtil # Check available GPUs, memory usage, utilization
# Distributed training
from torch.utils.data.distributed import DistributedSampler
# Splits datasets across multiple GPUs/processes
import torch.distributed as dist
# Backend communication for distributed training
import torch.multiprocessing as mp
# Spawn multiple processes for parallel/distributed training
from torch.nn.parallel import DistributedDataParallel as DDP
# Wraps a model for efficient multi-GPU distributed training
Now that we finished the code, on to some interesting reading!
# Some convenience helper functions used throughout the notebook
def is_interactive_notebook():
return __name__ == "__main__"
def show_example(fn, args=[]):
if __name__ == "__main__" and RUN_EXAMPLES:
return fn(*args)
def execute_example(fn, args=[]):
if __name__ == "__main__" and RUN_EXAMPLES:
fn(*args)
class DummyOptimizer(torch.optim.Optimizer):
def __init__(self):
self.param_groups = [{"lr": 0}]
None
def step(self):
None
def zero_grad(self, set_to_none=False):
None
class DummyScheduler:
def step(self):
None
My comments are blockquoted. The main text is all from the paper itself.
The goal of reducing sequential computation also forms the
foundation of the Extended Neural GPU, ByteNet and ConvS2S, all of
which use convolutional neural networks as basic building block,
computing hidden representations in parallel for all input and
output positions. …
Commentary
Term
Simple Explanation
Sequential computation
Doing operations step-by-step in order (one after another), instead of all at once
Convolutional neural networks (CNNs)
Neural networks that process data using filters; often used in images but also in sequences here
Convolutional filters, also called kernels are designed to detect specific patterns or features in the input data. It may sound surprising but CNN can also be used for text. Just like in images they are used to detect edges or objects, in text they are used to find key phrases for example. A convolution is a small window (called a filter or kernel) that scans over input data and computes a weighted sum at each position. A weighted sum is just a normal sum where each number is multiplied by a weight (importance factor) before adding. $w_1 \cdot 1 + w_2 \cdot 2 + w_3 \cdot 3$. Masking with respect to CNN means zeroing some weights of the convolution kernel or doing some computations so that future positions are not connected to influence training.
Extended Neural GPU: The Neural GPUs Learn Algorithms is co-authored by Ilya Sutskever. Neural Turing Machines (NTMs) were used to learn from examples but, due to them not being parallel, they are hard to train. It is based on a type of convolutional gated recurrent unit.
Extensions and Limitations of the Neural GPU: Mentionned in the text, this technique based on the Extended Neural GPU improves binary addition and binary multiplication in a way that generalizes to inputs of arbitrary length. It also fails on some instances.
Recurrent Neural Network: An RNN (Recurrent Neural Network) is a type of neural network designed for sequential data like text, time series, or speech. It processes input one step at a time, keeping a “memory” (hidden state) of what it has seen before.
At each step $t$, it reads input $x_t$. It updates a hidden memory state $h_t$. A rough formula is $h_t = f(W x_t + U h_{t-1})$
Symbol
Meaning
$x_t$
current input word/token
$h_{t-1}$
previous memory (hidden state)
$h_t$
updated memory (new hidden state)
$W, U$
learned weight matrices
$f$
activation function (e.g., $\tanh$, ReLU)
Sine the current $h$ depends on the previous, we cannot compute it in parallel. So, RNN has hidden representations but they are not parallel.
ByteNext: The ByteNet is a one-dimensional convolutional neural network that is composed of two parts, one to encode the source sequence and the other to decode the target sequence. It is a character-level Neural Machine Translation (NMT) approach, which means that it performs translation character by character.
ConvS2S: ConvS2S (Convolutional Sequence-to-Sequence Learning) is a neural network architecture for tasks like machine translation, text summarization, and speech processing, where both input and output are sequences. Contrasting with LSTM which uses RNN, this one uses CNN for the encoder and decoder. RNN processes tokens one by one, this one processes tokens in parallel. It also has an attention step. Notice that it already has multi-attention step, which shows that attention is something that existed well before transformers. We’ll cover a brief history of attention later on!
… In these models, the number of operations required
to relate signals from two arbitrary input or output positions grows
in the distance between positions, linearly for ConvS2S and
logarithmically for ByteNet. This makes it more difficult to learn
dependencies between distant positions. In the Transformer this is
reduced to a constant number of operations, albeit at the cost of
reduced effective resolution due to averaging attention-weighted
positions, an effect we counteract with Multi-Head Attention.
Self-attention, sometimes called intra-attention is an attention
mechanism relating different positions of a single sequence in order
to compute a representation of the sequence. Self-attention has been
used successfully in a variety of tasks including reading
comprehension, abstractive summarization, textual entailment and
learning task-independent sentence representations. End-to-end
memory networks are based on a recurrent attention mechanism instead
of sequencealigned recurrence and have been shown to perform well on
simple-language question answering and language modeling tasks.
To the best of our knowledge, however, the Transformer is the first
transduction model relying entirely on self-attention to compute
representations of its input and output without using sequence
aligned RNNs or convolution.
Part 1: Model Architecture
Model Architecture
Most competitive neural sequence transduction models have an
encoder-decoder structure
(cite). Here, the encoder maps an
input sequence of symbol representations $(x_1, …, x_n)$ to a
sequence of continuous representations $\mathbf{z} = (z_1, …,
z_n)$. Given $\mathbf{z}$, the decoder then generates an output
sequence $(y_1,…,y_m)$ of symbols one element at a time. At each
step the model is auto-regressive
(cite), consuming the previously
generated symbols as additional input when generating the next.
class EncoderDecoder(nn.Module):
"""
A standard Encoder-Decoder architecture. Base for this and many
other models.
"""
def __init__(self, encoder, decoder, src_embed, tgt_embed, generator):
super(EncoderDecoder, self).__init__()
self.encoder = encoder
self.decoder = decoder
self.src_embed = src_embed
self.tgt_embed = tgt_embed
self.generator = generator
def forward(self, src, tgt, src_mask, tgt_mask):
"Take in and process masked src and target sequences."
return self.decode(self.encode(src, src_mask), src_mask, tgt, tgt_mask)
def encode(self, src, src_mask):
return self.encoder(self.src_embed(src), src_mask)
def decode(self, memory, src_mask, tgt, tgt_mask):
return self.decoder(self.tgt_embed(tgt), memory, src_mask, tgt_mask)
class Generator(nn.Module):
"Define standard linear + softmax generation step."
def __init__(self, d_model, vocab):
super(Generator, self).__init__()
self.proj = nn.Linear(d_model, vocab)
def forward(self, x):
return log_softmax(self.proj(x), dim=-1)
The Transformer follows this overall architecture using stacked
self-attention and point-wise, fully connected layers for both the
encoder and decoder, shown in the left and right halves of Figure 1,
respectively.
Encoder and Decoder Stacks
Encoder
The encoder is composed of a stack of $N=6$ identical layers.
def clones(module, N):
"Produce N identical layers."
return nn.ModuleList([copy.deepcopy(module) for _ in range(N)])
class Encoder(nn.Module):
"Core encoder is a stack of N layers"
def __init__(self, layer, N):
super(Encoder, self).__init__()
self.layers = clones(layer, N)
self.norm = LayerNorm(layer.size)
def forward(self, x, mask):
"Pass the input (and mask) through each layer in turn."
for layer in self.layers:
x = layer(x, mask)
return self.norm(x)
We employ a residual connection
(cite) around each of the two
sub-layers, followed by layer normalization
(cite).
class LayerNorm(nn.Module):
"Construct a layernorm module (See citation for details)."
def __init__(self, features, eps=1e-6):
super(LayerNorm, self).__init__()
self.a_2 = nn.Parameter(torch.ones(features))
self.b_2 = nn.Parameter(torch.zeros(features))
self.eps = eps
def forward(self, x):
mean = x.mean(-1, keepdim=True)
std = x.std(-1, keepdim=True)
return self.a_2 * (x - mean) / (std + self.eps) + self.b_2
That is, the output of each sub-layer is $\mathrm{LayerNorm}(x +
\mathrm{Sublayer}(x))$, where $\mathrm{Sublayer}(x)$ is the function
implemented by the sub-layer itself. We apply dropout
(cite) to the
output of each sub-layer, before it is added to the sub-layer input
and normalized.
To facilitate these residual connections, all sub-layers in the
model, as well as the embedding layers, produce outputs of dimension
$d_{\text{model}}=512$.
class SublayerConnection(nn.Module):
"""
A residual connection followed by a layer norm.
Note for code simplicity the norm is first as opposed to last.
"""
def __init__(self, size, dropout):
super(SublayerConnection, self).__init__()
self.norm = LayerNorm(size)
self.dropout = nn.Dropout(dropout)
def forward(self, x, sublayer):
"Apply residual connection to any sublayer with the same size."
return x + self.dropout(sublayer(self.norm(x)))
Each layer has two sub-layers. The first is a multi-head
self-attention mechanism, and the second is a simple, position-wise
fully connected feed-forward network.
class EncoderLayer(nn.Module):
"Encoder is made up of self-attn and feed forward (defined below)"
def __init__(self, size, self_attn, feed_forward, dropout):
super(EncoderLayer, self).__init__()
self.self_attn = self_attn
self.feed_forward = feed_forward
self.sublayer = clones(SublayerConnection(size, dropout), 2)
self.size = size
def forward(self, x, mask):
"Follow Figure 1 (left) for connections."
x = self.sublayer[0](x, lambda x: self.self_attn(x, x, x, mask))
return self.sublayer[1](x, self.feed_forward)
Decoder
The decoder is also composed of a stack of $N=6$ identical layers.
class Decoder(nn.Module):
"Generic N layer decoder with masking."
def __init__(self, layer, N):
super(Decoder, self).__init__()
self.layers = clones(layer, N)
self.norm = LayerNorm(layer.size)
def forward(self, x, memory, src_mask, tgt_mask):
for layer in self.layers:
x = layer(x, memory, src_mask, tgt_mask)
return self.norm(x)
In addition to the two sub-layers in each encoder layer, the decoder
inserts a third sub-layer, which performs multi-head attention over
the output of the encoder stack. Similar to the encoder, we employ
residual connections around each of the sub-layers, followed by
layer normalization.
class DecoderLayer(nn.Module):
"Decoder is made of self-attn, src-attn, and feed forward (defined below)"
def __init__(self, size, self_attn, src_attn, feed_forward, dropout):
super(DecoderLayer, self).__init__()
self.size = size
self.self_attn = self_attn
self.src_attn = src_attn
self.feed_forward = feed_forward
self.sublayer = clones(SublayerConnection(size, dropout), 3)
def forward(self, x, memory, src_mask, tgt_mask):
"Follow Figure 1 (right) for connections."
m = memory
x = self.sublayer[0](x, lambda x: self.self_attn(x, x, x, tgt_mask))
x = self.sublayer[1](x, lambda x: self.src_attn(x, m, m, src_mask))
return self.sublayer[2](x, self.feed_forward)
We also modify the self-attention sub-layer in the decoder stack to
prevent positions from attending to subsequent positions. This
masking, combined with fact that the output embeddings are offset by
one position, ensures that the predictions for position $i$ can
depend only on the known outputs at positions less than $i$.
Below the attention mask shows the position each tgt word (row) is
allowed to look at (column). Words are blocked for attending to
future words during training.
def example_mask():
LS_data = pd.concat(
[
pd.DataFrame(
{
"Subsequent Mask": subsequent_mask(20)[0][x, y].flatten(),
"Window": y,
"Masking": x,
}
)
for y in range(20)
for x in range(20)
]
)
return (
alt.Chart(LS_data)
.mark_rect()
.properties(height=250, width=250)
.encode(
alt.X("Window:O"),
alt.Y("Masking:O"),
alt.Color("Subsequent Mask:Q", scale=alt.Scale(scheme="viridis")),
)
.interactive()
)
show_example(example_mask)
Attention
An attention function can be described as mapping a query and a set
of key-value pairs to an output, where the query, keys, values, and
output are all vectors. The output is computed as a weighted sum of
the values, where the weight assigned to each value is computed by a
compatibility function of the query with the corresponding key.
We call our particular attention “Scaled Dot-Product Attention”.
The input consists of queries and keys of dimension $d_k$, and
values of dimension $d_v$. We compute the dot products of the query
with all keys, divide each by $\sqrt{d_k}$, and apply a softmax
function to obtain the weights on the values.
In practice, we compute the attention function on a set of queries
simultaneously, packed together into a matrix $Q$. The keys and
values are also packed together into matrices $K$ and $V$. We
compute the matrix of outputs as:
def attention(query, key, value, mask=None, dropout=None):
"Compute 'Scaled Dot Product Attention'"
d_k = query.size(-1)
scores = torch.matmul(query, key.transpose(-2, -1)) / math.sqrt(d_k)
if mask is not None:
scores = scores.masked_fill(mask == 0, -1e9)
p_attn = scores.softmax(dim=-1)
if dropout is not None:
p_attn = dropout(p_attn)
return torch.matmul(p_attn, value), p_attn
The two most commonly used attention functions are additive
attention (cite), and dot-product
(multiplicative) attention. Dot-product attention is identical to
our algorithm, except for the scaling factor of
$\frac{1}{\sqrt{d_k}}$. Additive attention computes the
compatibility function using a feed-forward network with a single
hidden layer. While the two are similar in theoretical complexity,
dot-product attention is much faster and more space-efficient in
practice, since it can be implemented using highly optimized matrix
multiplication code.
While for small values of $d_k$ the two mechanisms perform
similarly, additive attention outperforms dot product attention
without scaling for larger values of $d_k$
(cite). We suspect that for
large values of $d_k$, the dot products grow large in magnitude,
pushing the softmax function into regions where it has extremely
small gradients (To illustrate why the dot products get large,
assume that the components of $q$ and $k$ are independent random
variables with mean $0$ and variance $1$. Then their dot product,
$q \cdot k = \sum_{i=1}^{d_k} q_ik_i$, has mean $0$ and variance
$d_k$.). To counteract this effect, we scale the dot products by
$\frac{1}{\sqrt{d_k}}$.
Multi-head attention allows the model to jointly attend to
Deep Dive
The intuition behind multi-head attention: instead of one attention computation,
we project the same input into $h$ different representation subspaces (8 in the paper).
Each head can learn different types of relationships – syntactic, semantic, positional.
Think of it as having 8 different “perspectives” on the same input simultaneously.
The total computation is similar to single-head attention because each head works
in a reduced dimension ($d_k = d_{\text{model}} / h = 64$).
information from different representation subspaces at different
positions. With a single attention head, averaging inhibits this.
Where the projections are parameter matrices $W^Q_i \in
\mathbb{R}^{d_{\text{model}} \times d_k}$, $W^K_i \in
\mathbb{R}^{d_{\text{model}} \times d_k}$, $W^V_i \in
\mathbb{R}^{d_{\text{model}} \times d_v}$ and $W^O \in
\mathbb{R}^{hd_v \times d_{\text{model}}}$.
In this work we employ $h=8$ parallel attention layers, or
heads. For each of these we use $d_k=d_v=d_{\text{model}}/h=64$. Due
to the reduced dimension of each head, the total computational cost
is similar to that of single-head attention with full
dimensionality.
class MultiHeadedAttention(nn.Module):
def __init__(self, h, d_model, dropout=0.1):
"Take in model size and number of heads."
super(MultiHeadedAttention, self).__init__()
assert d_model % h == 0
# We assume d_v always equals d_k
self.d_k = d_model // h
self.h = h
self.linears = clones(nn.Linear(d_model, d_model), 4)
self.attn = None
self.dropout = nn.Dropout(p=dropout)
def forward(self, query, key, value, mask=None):
"Implements Figure 2"
if mask is not None:
# Same mask applied to all h heads.
mask = mask.unsqueeze(1)
nbatches = query.size(0)
# 1) Do all the linear projections in batch from d_model => h x d_k
query, key, value = [
lin(x).view(nbatches, -1, self.h, self.d_k).transpose(1, 2)
for lin, x in zip(self.linears, (query, key, value))
]
# 2) Apply attention on all the projected vectors in batch.
x, self.attn = attention(
query, key, value, mask=mask, dropout=self.dropout
)
# 3) "Concat" using a view and apply a final linear.
x = (
x.transpose(1, 2)
.contiguous()
.view(nbatches, -1, self.h * self.d_k)
)
del query
del key
del value
return self.linears[-1](x)
Applications of Attention in our Model
The Transformer uses multi-head attention in three different ways:
1) In “encoder-decoder attention” layers, the queries come from the
previous decoder layer, and the memory keys and values come from the
output of the encoder. This allows every position in the decoder to
attend over all positions in the input sequence. This mimics the
typical encoder-decoder attention mechanisms in sequence-to-sequence
models such as (cite).
2) The encoder contains self-attention layers. In a self-attention
layer all of the keys, values and queries come from the same place,
in this case, the output of the previous layer in the encoder. Each
position in the encoder can attend to all positions in the previous
layer of the encoder.
3) Similarly, self-attention layers in the decoder allow each
position in the decoder to attend to all positions in the decoder up
to and including that position. We need to prevent leftward
information flow in the decoder to preserve the auto-regressive
property. We implement this inside of scaled dot-product attention
by masking out (setting to $-\infty$) all values in the input of the
softmax which correspond to illegal connections.
Position-wise Feed-Forward Networks
In addition to attention sub-layers, each of the layers in our
encoder and decoder contains a fully connected feed-forward network,
which is applied to each position separately and identically. This
consists of two linear transformations with a ReLU activation in
between.
$$\mathrm{FFN}(x)=\max(0, xW_1 + b_1) W_2 + b_2$$
While the linear transformations are the same across different
positions, they use different parameters from layer to
layer. Another way of describing this is as two convolutions with
kernel size 1. The dimensionality of input and output is
$d_{\text{model}}=512$, and the inner-layer has dimensionality
$d_{ff}=2048$.
Similarly to other sequence transduction models, we use learned
embeddings to convert the input tokens and output tokens to vectors
of dimension $d_{\text{model}}$. We also use the usual learned
linear transformation and softmax function to convert the decoder
output to predicted next-token probabilities. In our model, we
share the same weight matrix between the two embedding layers and
the pre-softmax linear transformation, similar to
(cite). In the embedding layers,
we multiply those weights by $\sqrt{d_{\text{model}}}$.
Since our model contains no recurrence and no convolution, in order
for the model to make use of the order of the sequence, we must
inject some information about the relative or absolute position of
the tokens in the sequence. To this end, we add “positional
encodings” to the input embeddings at the bottoms of the encoder and
decoder stacks. The positional encodings have the same dimension
$d_{\text{model}}$ as the embeddings, so that the two can be summed.
There are many choices of positional encodings, learned and fixed
(cite).
In this work, we use sine and cosine functions of different frequencies:
where $pos$ is the position and $i$ is the dimension. That is, each
dimension of the positional encoding corresponds to a sinusoid. The
wavelengths form a geometric progression from $2\pi$ to $10000 \cdot
2\pi$. We chose this function because we hypothesized it would
allow the model to easily learn to attend by relative positions,
since for any fixed offset $k$, $PE_{pos+k}$ can be represented as a
linear function of $PE_{pos}$.
In addition, we apply dropout to the sums of the embeddings and the
positional encodings in both the encoder and decoder stacks. For
the base model, we use a rate of $P_{drop}=0.1$.
class PositionalEncoding(nn.Module):
"Implement the PE function."
def __init__(self, d_model, dropout, max_len=5000):
super(PositionalEncoding, self).__init__()
self.dropout = nn.Dropout(p=dropout)
# Compute the positional encodings once in log space.
pe = torch.zeros(max_len, d_model)
position = torch.arange(0, max_len).unsqueeze(1)
div_term = torch.exp(
torch.arange(0, d_model, 2) * -(math.log(10000.0) / d_model)
)
pe[:, 0::2] = torch.sin(position * div_term)
pe[:, 1::2] = torch.cos(position * div_term)
pe = pe.unsqueeze(0)
self.register_buffer("pe", pe)
def forward(self, x):
x = x + self.pe[:, : x.size(1)].requires_grad_(False)
return self.dropout(x)
Below the positional encoding will add in a sine wave based on
position. The frequency and offset of the wave is different for
each dimension.
def example_positional():
pe = PositionalEncoding(20, 0)
y = pe.forward(torch.zeros(1, 100, 20))
data = pd.concat(
[
pd.DataFrame(
{
"embedding": y[0, :, dim],
"dimension": dim,
"position": list(range(100)),
}
)
for dim in [4, 5, 6, 7]
]
)
return (
alt.Chart(data)
.mark_line()
.properties(width=800)
.encode(x="position", y="embedding", color="dimension:N")
.interactive()
)
show_example(example_positional)
We also experimented with using learned positional embeddings
(cite) instead, and found
that the two versions produced nearly identical results. We chose
the sinusoidal version because it may allow the model to extrapolate
to sequence lengths longer than the ones encountered during
training.
Full Model
Here we define a function from hyperparameters to a full model.
def make_model(
src_vocab, tgt_vocab, N=6, d_model=512, d_ff=2048, h=8, dropout=0.1
):
"Helper: Construct a model from hyperparameters."
c = copy.deepcopy
attn = MultiHeadedAttention(h, d_model)
ff = PositionwiseFeedForward(d_model, d_ff, dropout)
position = PositionalEncoding(d_model, dropout)
model = EncoderDecoder(
Encoder(EncoderLayer(d_model, c(attn), c(ff), dropout), N),
Decoder(DecoderLayer(d_model, c(attn), c(attn), c(ff), dropout), N),
nn.Sequential(Embeddings(d_model, src_vocab), c(position)),
nn.Sequential(Embeddings(d_model, tgt_vocab), c(position)),
Generator(d_model, tgt_vocab),
)
# This was important from their code.
# Initialize parameters with Glorot / fan_avg.
for p in model.parameters():
if p.dim() > 1:
nn.init.xavier_uniform_(p)
return model
Inference:
Here we make a forward step to generate a prediction of the
model. We try to use our transformer to memorize the input. As you
will see the output is randomly generated due to the fact that the
model is not trained yet. In the next tutorial we will build the
training function and try to train our model to memorize the numbers
from 1 to 10.
This section describes the training regime for our models.
We stop for a quick interlude to introduce some of the tools
needed to train a standard encoder decoder model. First we define a
batch object that holds the src and target sentences for training,
as well as constructing the masks.
Batches and Masking
class Batch:
"""Object for holding a batch of data with mask during training."""
def __init__(self, src, tgt=None, pad=2): # 2 = <blank>
self.src = src
self.src_mask = (src != pad).unsqueeze(-2)
if tgt is not None:
self.tgt = tgt[:, :-1]
self.tgt_y = tgt[:, 1:]
self.tgt_mask = self.make_std_mask(self.tgt, pad)
self.ntokens = (self.tgt_y != pad).data.sum()
@staticmethod
def make_std_mask(tgt, pad):
"Create a mask to hide padding and future words."
tgt_mask = (tgt != pad).unsqueeze(-2)
tgt_mask = tgt_mask & subsequent_mask(tgt.size(-1)).type_as(
tgt_mask.data
)
return tgt_mask
Next we create a generic training and scoring function to keep
track of loss. We pass in a generic loss compute function that
also handles parameter updates.
Training Loop
class TrainState:
"""Track number of steps, examples, and tokens processed"""
step: int = 0 # Steps in the current epoch
accum_step: int = 0 # Number of gradient accumulation steps
samples: int = 0 # total # of examples used
tokens: int = 0 # total # of tokens processed
def run_epoch(
data_iter,
model,
loss_compute,
optimizer,
scheduler,
mode="train",
accum_iter=1,
train_state=TrainState(),
):
"""Train a single epoch"""
start = time.time()
total_tokens = 0
total_loss = 0
tokens = 0
n_accum = 0
for i, batch in enumerate(data_iter):
out = model.forward(
batch.src, batch.tgt, batch.src_mask, batch.tgt_mask
)
loss, loss_node = loss_compute(out, batch.tgt_y, batch.ntokens)
# loss_node = loss_node / accum_iter
if mode == "train" or mode == "train+log":
loss_node.backward()
train_state.step += 1
train_state.samples += batch.src.shape[0]
train_state.tokens += batch.ntokens
if i % accum_iter == 0:
optimizer.step()
optimizer.zero_grad(set_to_none=True)
n_accum += 1
train_state.accum_step += 1
scheduler.step()
total_loss += loss
total_tokens += batch.ntokens
tokens += batch.ntokens
if i % 40 == 1 and (mode == "train" or mode == "train+log"):
lr = optimizer.param_groups[0]["lr"]
elapsed = time.time() - start
print(
(
"Epoch Step: %6d | Accumulation Step: %3d | Loss: %6.2f "
+ "| Tokens / Sec: %7.1f | Learning Rate: %6.1e"
)
% (i, n_accum, loss / batch.ntokens, tokens / elapsed, lr)
)
start = time.time()
tokens = 0
del loss
del loss_node
return total_loss / total_tokens, train_state
Training Data and Batching
We trained on the standard WMT 2014 English-German dataset
consisting of about 4.5 million sentence pairs. Sentences were
encoded using byte-pair encoding, which has a shared source-target
vocabulary of about 37000 tokens. For English-French, we used the
significantly larger WMT 2014 English-French dataset consisting of
36M sentences and split tokens into a 32000 word-piece vocabulary.
Sentence pairs were batched together by approximate sequence length.
Each training batch contained a set of sentence pairs containing
approximately 25000 source tokens and 25000 target tokens.
Hardware and Schedule
We trained our models on one machine with 8 NVIDIA P100 GPUs. For
our base models using the hyperparameters described throughout the
paper, each training step took about 0.4 seconds. We trained the
base models for a total of 100,000 steps or 12 hours. For our big
models, step time was 1.0 seconds. The big models were trained for
300,000 steps (3.5 days).
Optimizer
We used the Adam optimizer (cite)
with $\beta_1=0.9$, $\beta_2=0.98$ and $\epsilon=10^{-9}$. We
varied the learning rate over the course of training, according to
the formula:
This corresponds to increasing the learning rate linearly for the
first $warmup_steps$ training steps, and decreasing it thereafter
proportionally to the inverse square root of the step number. We
used $warmup_steps=4000$.
Note: This part is very important. Need to train with this setup
of the model.
Example of the curves of this model for different model sizes and
for optimization hyperparameters.
def rate(step, model_size, factor, warmup):
"""
we have to default the step to 1 for LambdaLR function
to avoid zero raising to negative power.
"""
if step == 0:
step = 1
return factor * (
model_size ** (-0.5) * min(step ** (-0.5), step * warmup ** (-1.5))
)
def example_learning_schedule():
opts = [
[512, 1, 4000], # example 1
[512, 1, 8000], # example 2
[256, 1, 4000], # example 3
]
dummy_model = torch.nn.Linear(1, 1)
learning_rates = []
# we have 3 examples in opts list.
for idx, example in enumerate(opts):
# run 20000 epoch for each example
optimizer = torch.optim.Adam(
dummy_model.parameters(), lr=1, betas=(0.9, 0.98), eps=1e-9
)
lr_scheduler = LambdaLR(
optimizer=optimizer, lr_lambda=lambda step: rate(step, *example)
)
tmp = []
# take 20K dummy training steps, save the learning rate at each step
for step in range(20000):
tmp.append(optimizer.param_groups[0]["lr"])
optimizer.step()
lr_scheduler.step()
learning_rates.append(tmp)
learning_rates = torch.tensor(learning_rates)
# Enable altair to handle more than 5000 rows
alt.data_transformers.disable_max_rows()
opts_data = pd.concat(
[
pd.DataFrame(
{
"Learning Rate": learning_rates[warmup_idx, :],
"model_size:warmup": ["512:4000", "512:8000", "256:4000"][
warmup_idx
],
"step": range(20000),
}
)
for warmup_idx in [0, 1, 2]
]
)
return (
alt.Chart(opts_data)
.mark_line()
.properties(width=600)
.encode(x="step", y="Learning Rate", color="model_size:warmup:N")
.interactive()
)
example_learning_schedule()
Regularization
Label Smoothing
During training, we employed label smoothing of value
$\epsilon_{ls}=0.1$ (cite).
This hurts perplexity, as the model learns to be more unsure, but
improves accuracy and BLEU score.
We implement label smoothing using the KL div loss. Instead of
using a one-hot target distribution, we create a distribution that
has confidence of the correct word and the rest of the
smoothing mass distributed throughout the vocabulary.
We can begin by trying out a simple copy-task. Given a random set
of input symbols from a small vocabulary, the goal is to generate
back those same symbols.
Synthetic Data
def data_gen(V, batch_size, nbatches):
"Generate random data for a src-tgt copy task."
for i in range(nbatches):
data = torch.randint(1, V, size=(batch_size, 10))
data[:, 0] = 1
src = data.requires_grad_(False).clone().detach()
tgt = data.requires_grad_(False).clone().detach()
yield Batch(src, tgt, 0)
Loss Computation
class SimpleLossCompute:
"A simple loss compute and train function."
def __init__(self, generator, criterion):
self.generator = generator
self.criterion = criterion
def __call__(self, x, y, norm):
x = self.generator(x)
sloss = (
self.criterion(
x.contiguous().view(-1, x.size(-1)), y.contiguous().view(-1)
)
/ norm
)
return sloss.data * norm, sloss
Greedy Decoding
This code predicts a translation using greedy decoding for simplicity.
def tokenize(text, tokenizer):
return [tok.text for tok in tokenizer.tokenizer(text)]
def yield_tokens(data_iter, tokenizer, index):
for from_to_tuple in data_iter:
yield tokenizer(from_to_tuple[index])
def build_vocabulary(spacy_de, spacy_en):
def tokenize_de(text):
return tokenize(text, spacy_de)
def tokenize_en(text):
return tokenize(text, spacy_en)
print("Building German Vocabulary ...")
train, val, test = datasets.Multi30k(language_pair=("de", "en"))
vocab_src = build_vocab_from_iterator(
yield_tokens(train + val + test, tokenize_de, index=0),
min_freq=2,
specials=["<s>", "</s>", "<blank>", "<unk>"],
)
print("Building English Vocabulary ...")
train, val, test = datasets.Multi30k(language_pair=("de", "en"))
vocab_tgt = build_vocab_from_iterator(
yield_tokens(train + val + test, tokenize_en, index=1),
min_freq=2,
specials=["<s>", "</s>", "<blank>", "<unk>"],
)
vocab_src.set_default_index(vocab_src["<unk>"])
vocab_tgt.set_default_index(vocab_tgt["<unk>"])
return vocab_src, vocab_tgt
def load_vocab(spacy_de, spacy_en):
if not exists("vocab.pt"):
vocab_src, vocab_tgt = build_vocabulary(spacy_de, spacy_en)
torch.save((vocab_src, vocab_tgt), "vocab.pt")
else:
vocab_src, vocab_tgt = torch.load("vocab.pt")
print("Finished.\nVocabulary sizes:")
print(len(vocab_src))
print(len(vocab_tgt))
return vocab_src, vocab_tgt
if is_interactive_notebook():
# global variables used later in the script
spacy_de, spacy_en = show_example(load_tokenizers)
vocab_src, vocab_tgt = show_example(load_vocab, args=[spacy_de, spacy_en])
Batching matters a ton for speed. We want to have very evenly
divided batches, with absolutely minimal padding. To do this we
have to hack a bit around the default torchtext batching. This
code patches their default batching to make sure we search over
enough sentences to find tight batches.
Once trained we can decode the model to produce a set of
translations. Here we simply translate the first sentence in the
validation set. This dataset is pretty small so the translations
with greedy search are reasonably accurate.
Additional Components: BPE, Search, Averaging
So this mostly covers the transformer model itself. There are four
aspects that we didn’t cover explicitly. We also have all these
additional features implemented in
OpenNMT-py.
1) BPE/ Word-piece: We can use a library to first preprocess the
data into subword units. See Rico Sennrich’s
subword-nmt
implementation. These models will transform the training data to
look like this:
▁Die ▁Protokoll datei ▁kann ▁ heimlich ▁per ▁E - Mail ▁oder ▁FTP
▁an ▁einen ▁bestimmte n ▁Empfänger ▁gesendet ▁werden .
2) Shared Embeddings: When using BPE with shared vocabulary we can
share the same weight vectors between the source / target /
generator. See the (cite) for
details. To add this to the model simply do this:
if False:
model.src_embed[0].lut.weight = model.tgt_embeddings[0].lut.weight
model.generator.lut.weight = model.tgt_embed[0].lut.weight
3) Beam Search: This is a bit too complicated to cover here. See the
OpenNMT-py
for a pytorch implementation.
4) Model Averaging: The paper averages the last k checkpoints to
create an ensembling effect. We can do this after the fact if we
have a bunch of models:
def average(model, models):
"Average models into model"
for ps in zip(*[m.params() for m in [model] + models]):
ps[0].copy_(torch.sum(*ps[1:]) / len(ps[1:]))
Results
On the WMT 2014 English-to-German translation task, the big
transformer model (Transformer (big) in Table 2) outperforms the
best previously reported models (including ensembles) by more than
2.0 BLEU, establishing a new state-of-the-art BLEU score of
28.4. The configuration of this model is listed in the bottom line
of Table 3. Training took 3.5 days on 8 P100 GPUs. Even our base
model surpasses all previously published models and ensembles, at a
fraction of the training cost of any of the competitive models.
On the WMT 2014 English-to-French translation task, our big model
achieves a BLEU score of 41.0, outperforming all of the previously
published single models, at less than 1/4 the training cost of the
previous state-of-the-art model. The Transformer (big) model trained
for English-to-French used dropout rate Pdrop = 0.1, instead of 0.3.
With the addtional extensions in the last section, the OpenNMT-py
replication gets to 26.9 on EN-DE WMT. Here I have loaded in those
parameters to our reimplemenation.
# Load data and model for output checks
def check_outputs(
valid_dataloader,
model,
vocab_src,
vocab_tgt,
n_examples=15,
pad_idx=2,
eos_string="</s>",
):
results = [()] * n_examples
for idx in range(n_examples):
print("\nExample %d ========\n" % idx)
b = next(iter(valid_dataloader))
rb = Batch(b[0], b[1], pad_idx)
greedy_decode(model, rb.src, rb.src_mask, 64, 0)[0]
src_tokens = [
vocab_src.get_itos()[x] for x in rb.src[0] if x != pad_idx
]
tgt_tokens = [
vocab_tgt.get_itos()[x] for x in rb.tgt[0] if x != pad_idx
]
print(
"Source Text (Input) : "
+ " ".join(src_tokens).replace("\n", "")
)
print(
"Target Text (Ground Truth) : "
+ " ".join(tgt_tokens).replace("\n", "")
)
model_out = greedy_decode(model, rb.src, rb.src_mask, 72, 0)[0]
model_txt = (
" ".join(
[vocab_tgt.get_itos()[x] for x in model_out if x != pad_idx]
).split(eos_string, 1)[0]
+ eos_string
)
print("Model Output : " + model_txt.replace("\n", ""))
results[idx] = (rb, src_tokens, tgt_tokens, model_out, model_txt)
return results
def run_model_example(n_examples=5):
global vocab_src, vocab_tgt, spacy_de, spacy_en
print("Preparing Data ...")
_, valid_dataloader = create_dataloaders(
torch.device("cpu"),
vocab_src,
vocab_tgt,
spacy_de,
spacy_en,
batch_size=1,
is_distributed=False,
)
print("Loading Trained Model ...")
model = make_model(len(vocab_src), len(vocab_tgt), N=6)
model.load_state_dict(
torch.load("multi30k_model_final.pt", map_location=torch.device("cpu"))
)
print("Checking Model Outputs:")
example_data = check_outputs(
valid_dataloader, model, vocab_src, vocab_tgt, n_examples=n_examples
)
return model, example_data
# execute_example(run_model_example)
Attention Visualization
Even with a greedy decoder the translation looks pretty good. We
can further visualize it to see what is happening at each layer of
the attention
def mtx2df(m, max_row, max_col, row_tokens, col_tokens):
"convert a dense matrix to a data frame with row and column indices"
return pd.DataFrame(
[
(
r,
c,
float(m[r, c]),
"%.3d %s"
% (r, row_tokens[r] if len(row_tokens) > r else "<blank>"),
"%.3d %s"
% (c, col_tokens[c] if len(col_tokens) > c else "<blank>"),
)
for r in range(m.shape[0])
for c in range(m.shape[1])
if r < max_row and c < max_col
],
# if float(m[r,c]) != 0 and r < max_row and c < max_col],
columns=["row", "column", "value", "row_token", "col_token"],
)
def attn_map(attn, layer, head, row_tokens, col_tokens, max_dim=30):
df = mtx2df(
attn[0, head].data,
max_dim,
max_dim,
row_tokens,
col_tokens,
)
return (
alt.Chart(data=df)
.mark_rect()
.encode(
x=alt.X("col_token", axis=alt.Axis(title="")),
y=alt.Y("row_token", axis=alt.Axis(title="")),
color="value",
tooltip=["row", "column", "value", "row_token", "col_token"],
)
.properties(height=400, width=400)
.interactive()
)
Commentary of The Annotated Transformer
The Annotated Transformer
Attention is All You Need
The Transformer has been on a lot of people’s minds over the last
yearfive years. This post presents an annotated version of the paper in the form of a line-by-line implementation. It reorders and deletes some sections from the original paper and adds comments throughout. This document itself is a working notebook, and should be a completely usable implementation. Code is available here.Table of Contents
Prelims
Background
Skip
The goal of reducing sequential computation also forms the foundation of the Extended Neural GPU, ByteNet and ConvS2S, all of which use convolutional neural networks as basic building block, computing hidden representations in parallel for all input and output positions. …
… In these models, the number of operations required to relate signals from two arbitrary input or output positions grows in the distance between positions, linearly for ConvS2S and logarithmically for ByteNet. This makes it more difficult to learn dependencies between distant positions. In the Transformer this is reduced to a constant number of operations, albeit at the cost of reduced effective resolution due to averaging attention-weighted positions, an effect we counteract with Multi-Head Attention.
Self-attention, sometimes called intra-attention is an attention mechanism relating different positions of a single sequence in order to compute a representation of the sequence. Self-attention has been used successfully in a variety of tasks including reading comprehension, abstractive summarization, textual entailment and learning task-independent sentence representations. End-to-end memory networks are based on a recurrent attention mechanism instead of sequencealigned recurrence and have been shown to perform well on simple-language question answering and language modeling tasks.
To the best of our knowledge, however, the Transformer is the first transduction model relying entirely on self-attention to compute representations of its input and output without using sequence aligned RNNs or convolution.
Part 1: Model Architecture
Model Architecture
Most competitive neural sequence transduction models have an encoder-decoder structure (cite). Here, the encoder maps an input sequence of symbol representations $(x_1, …, x_n)$ to a sequence of continuous representations $\mathbf{z} = (z_1, …, z_n)$. Given $\mathbf{z}$, the decoder then generates an output sequence $(y_1,…,y_m)$ of symbols one element at a time. At each step the model is auto-regressive (cite), consuming the previously generated symbols as additional input when generating the next.
The Transformer follows this overall architecture using stacked self-attention and point-wise, fully connected layers for both the encoder and decoder, shown in the left and right halves of Figure 1, respectively.
Encoder and Decoder Stacks
Encoder
The encoder is composed of a stack of $N=6$ identical layers.
We employ a residual connection (cite) around each of the two sub-layers, followed by layer normalization (cite).
That is, the output of each sub-layer is $\mathrm{LayerNorm}(x + \mathrm{Sublayer}(x))$, where $\mathrm{Sublayer}(x)$ is the function implemented by the sub-layer itself. We apply dropout (cite) to the output of each sub-layer, before it is added to the sub-layer input and normalized.
To facilitate these residual connections, all sub-layers in the model, as well as the embedding layers, produce outputs of dimension $d_{\text{model}}=512$.
Each layer has two sub-layers. The first is a multi-head self-attention mechanism, and the second is a simple, position-wise fully connected feed-forward network.
Decoder
The decoder is also composed of a stack of $N=6$ identical layers.
In addition to the two sub-layers in each encoder layer, the decoder inserts a third sub-layer, which performs multi-head attention over the output of the encoder stack. Similar to the encoder, we employ residual connections around each of the sub-layers, followed by layer normalization.
We also modify the self-attention sub-layer in the decoder stack to prevent positions from attending to subsequent positions. This masking, combined with fact that the output embeddings are offset by one position, ensures that the predictions for position $i$ can depend only on the known outputs at positions less than $i$.
Attention
An attention function can be described as mapping a query and a set of key-value pairs to an output, where the query, keys, values, and output are all vectors. The output is computed as a weighted sum of the values, where the weight assigned to each value is computed by a compatibility function of the query with the corresponding key.
We call our particular attention “Scaled Dot-Product Attention”. The input consists of queries and keys of dimension $d_k$, and values of dimension $d_v$. We compute the dot products of the query with all keys, divide each by $\sqrt{d_k}$, and apply a softmax function to obtain the weights on the values.
In practice, we compute the attention function on a set of queries simultaneously, packed together into a matrix $Q$. The keys and values are also packed together into matrices $K$ and $V$. We compute the matrix of outputs as:
$$ \mathrm{Attention}(Q, K, V) = \mathrm{softmax}(\frac{QK^T}{\sqrt{d_k}})V $$
The two most commonly used attention functions are additive attention (cite), and dot-product (multiplicative) attention. Dot-product attention is identical to our algorithm, except for the scaling factor of $\frac{1}{\sqrt{d_k}}$. Additive attention computes the compatibility function using a feed-forward network with a single hidden layer. While the two are similar in theoretical complexity, dot-product attention is much faster and more space-efficient in practice, since it can be implemented using highly optimized matrix multiplication code.
While for small values of $d_k$ the two mechanisms perform similarly, additive attention outperforms dot product attention without scaling for larger values of $d_k$ (cite). We suspect that for large values of $d_k$, the dot products grow large in magnitude, pushing the softmax function into regions where it has extremely small gradients (To illustrate why the dot products get large, assume that the components of $q$ and $k$ are independent random variables with mean $0$ and variance $1$. Then their dot product, $q \cdot k = \sum_{i=1}^{d_k} q_ik_i$, has mean $0$ and variance $d_k$.). To counteract this effect, we scale the dot products by $\frac{1}{\sqrt{d_k}}$.
Multi-head attention allows the model to jointly attend to
$$ \mathrm{MultiHead}(Q, K, V) = \mathrm{Concat}(\mathrm{head_1}, …, \mathrm{head_h})W^O \ \text{where}~\mathrm{head_i} = \mathrm{Attention}(QW^Q_i, KW^K_i, VW^V_i) $$
Where the projections are parameter matrices $W^Q_i \in \mathbb{R}^{d_{\text{model}} \times d_k}$, $W^K_i \in \mathbb{R}^{d_{\text{model}} \times d_k}$, $W^V_i \in \mathbb{R}^{d_{\text{model}} \times d_v}$ and $W^O \in \mathbb{R}^{hd_v \times d_{\text{model}}}$.
In this work we employ $h=8$ parallel attention layers, or heads. For each of these we use $d_k=d_v=d_{\text{model}}/h=64$. Due to the reduced dimension of each head, the total computational cost is similar to that of single-head attention with full dimensionality.
Applications of Attention in our Model
The Transformer uses multi-head attention in three different ways: 1) In “encoder-decoder attention” layers, the queries come from the previous decoder layer, and the memory keys and values come from the output of the encoder. This allows every position in the decoder to attend over all positions in the input sequence. This mimics the typical encoder-decoder attention mechanisms in sequence-to-sequence models such as (cite).
2) The encoder contains self-attention layers. In a self-attention layer all of the keys, values and queries come from the same place, in this case, the output of the previous layer in the encoder. Each position in the encoder can attend to all positions in the previous layer of the encoder.
3) Similarly, self-attention layers in the decoder allow each position in the decoder to attend to all positions in the decoder up to and including that position. We need to prevent leftward information flow in the decoder to preserve the auto-regressive property. We implement this inside of scaled dot-product attention by masking out (setting to $-\infty$) all values in the input of the softmax which correspond to illegal connections.
Position-wise Feed-Forward Networks
In addition to attention sub-layers, each of the layers in our encoder and decoder contains a fully connected feed-forward network, which is applied to each position separately and identically. This consists of two linear transformations with a ReLU activation in between.
$$\mathrm{FFN}(x)=\max(0, xW_1 + b_1) W_2 + b_2$$
While the linear transformations are the same across different positions, they use different parameters from layer to layer. Another way of describing this is as two convolutions with kernel size 1. The dimensionality of input and output is $d_{\text{model}}=512$, and the inner-layer has dimensionality $d_{ff}=2048$.
Embeddings and Softmax
Similarly to other sequence transduction models, we use learned embeddings to convert the input tokens and output tokens to vectors of dimension $d_{\text{model}}$. We also use the usual learned linear transformation and softmax function to convert the decoder output to predicted next-token probabilities. In our model, we share the same weight matrix between the two embedding layers and the pre-softmax linear transformation, similar to (cite). In the embedding layers, we multiply those weights by $\sqrt{d_{\text{model}}}$.
Positional Encoding
Since our model contains no recurrence and no convolution, in order for the model to make use of the order of the sequence, we must inject some information about the relative or absolute position of the tokens in the sequence. To this end, we add “positional encodings” to the input embeddings at the bottoms of the encoder and decoder stacks. The positional encodings have the same dimension $d_{\text{model}}$ as the embeddings, so that the two can be summed. There are many choices of positional encodings, learned and fixed (cite).
In this work, we use sine and cosine functions of different frequencies:
$$PE_{(pos,2i)} = \sin(pos / 10000^{2i/d_{\text{model}}})$$
$$PE_{(pos,2i+1)} = \cos(pos / 10000^{2i/d_{\text{model}}})$$
where $pos$ is the position and $i$ is the dimension. That is, each dimension of the positional encoding corresponds to a sinusoid. The wavelengths form a geometric progression from $2\pi$ to $10000 \cdot 2\pi$. We chose this function because we hypothesized it would allow the model to easily learn to attend by relative positions, since for any fixed offset $k$, $PE_{pos+k}$ can be represented as a linear function of $PE_{pos}$.
In addition, we apply dropout to the sums of the embeddings and the positional encodings in both the encoder and decoder stacks. For the base model, we use a rate of $P_{drop}=0.1$.
We also experimented with using learned positional embeddings (cite) instead, and found that the two versions produced nearly identical results. We chose the sinusoidal version because it may allow the model to extrapolate to sequence lengths longer than the ones encountered during training.
Full Model
Inference:
Part 2: Model Training
Training
This section describes the training regime for our models.
Batches and Masking
Training Loop
Training Data and Batching
We trained on the standard WMT 2014 English-German dataset consisting of about 4.5 million sentence pairs. Sentences were encoded using byte-pair encoding, which has a shared source-target vocabulary of about 37000 tokens. For English-French, we used the significantly larger WMT 2014 English-French dataset consisting of 36M sentences and split tokens into a 32000 word-piece vocabulary.
Sentence pairs were batched together by approximate sequence length. Each training batch contained a set of sentence pairs containing approximately 25000 source tokens and 25000 target tokens.
Hardware and Schedule
We trained our models on one machine with 8 NVIDIA P100 GPUs. For our base models using the hyperparameters described throughout the paper, each training step took about 0.4 seconds. We trained the base models for a total of 100,000 steps or 12 hours. For our big models, step time was 1.0 seconds. The big models were trained for 300,000 steps (3.5 days).
Optimizer
We used the Adam optimizer (cite) with $\beta_1=0.9$, $\beta_2=0.98$ and $\epsilon=10^{-9}$. We varied the learning rate over the course of training, according to the formula:
$$ lrate = d_{\text{model}}^{-0.5} \cdot \min({step_num}^{-0.5}, {step_num} \cdot {warmup_steps}^{-1.5}) $$
This corresponds to increasing the learning rate linearly for the first $warmup_steps$ training steps, and decreasing it thereafter proportionally to the inverse square root of the step number. We used $warmup_steps=4000$.
Regularization
Label Smoothing
During training, we employed label smoothing of value $\epsilon_{ls}=0.1$ (cite). This hurts perplexity, as the model learns to be more unsure, but improves accuracy and BLEU score.
A First Example
Synthetic Data
Loss Computation
Greedy Decoding
Part 3: A Real World Example
Data Loading
Iterators
Training the System
Additional Components: BPE, Search, Averaging
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Results
On the WMT 2014 English-to-German translation task, the big transformer model (Transformer (big) in Table 2) outperforms the best previously reported models (including ensembles) by more than 2.0 BLEU, establishing a new state-of-the-art BLEU score of 28.4. The configuration of this model is listed in the bottom line of Table 3. Training took 3.5 days on 8 P100 GPUs. Even our base model surpasses all previously published models and ensembles, at a fraction of the training cost of any of the competitive models.
On the WMT 2014 English-to-French translation task, our big model achieves a BLEU score of 41.0, outperforming all of the previously published single models, at less than 1/4 the training cost of the previous state-of-the-art model. The Transformer (big) model trained for English-to-French used dropout rate Pdrop = 0.1, instead of 0.3.
Attention Visualization
Encoder Self Attention
Decoder Self Attention
Decoder Src Attention
Conclusion
Hopefully this code is useful for future research. Please reach out if you have any issues.
Cheers, Sasha Rush, Austin Huang, Suraj Subramanian, Jonathan Sum, Khalid Almubarak, Stella Biderman