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extremely small gradients 4
. To counteract this effect, we scale the dot products by √
1
dk
.
3.2.2 Multi-Head Attention
Instead of performing a single attention function with dmodel-dimensional keys, values and queries,
we found it beneficial to linearly project the queries, keys and values h times with different, learned
linear projections to dk, dk and dv dimensions, respectively. On each of these projected versions of
queries, keys and values we then perform the attention function in parallel, yielding dv-dimensional
output values. These are concatenated and once again projected, resulting in the final values, as
depicted in Figure 2.
4To 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 · k =
Pdk
i=1 qiki, has mean 0 and variance dk.
4
Multi-head attention allows the model to jointly attend to information from different representation
subspaces at different positions. With a single attention head, averaging inhibits this.
MultiHead(Q, K, V ) = Concat(head1, ..., headh)WO
where headi = Attention(QWQ
i
, KW K
i
, V WV
i
)
Where the projections are parameter matrices W
Q
i ∈ R
dmodel×dk , W K
i ∈ R
dmodel×dk , WV
i ∈ R
dmodel×dv
and WO ∈ R
hdv×dmodel
.
In this work we employ h = 8 parallel attention layers, or heads. For each of these we use
dk = dv = dmodel/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.
3.2.3 Applications of Attention in our Model
The Transformer uses multi-head attention in three different ways:
• 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
[38, 2, 9].
• 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.
• 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 −∞) all values in the input
of the softmax which correspond to illegal connections. See Figure 2.
3.3 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.
FFN(x) = max(0, xW1 + b1)W2 + b2 (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 dmodel = 512, and the inner-layer has dimensionality
df f = 2048.
3.4 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 dmodel. 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 [30]. In the embedding layers, we multiply those weights by √
dmodel.
3.5 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
5
Table 1: Maximum path lengths, per-layer complexity and minimum number of sequential operations
for different layer types. n is the sequence length, d is the representation dimension, k is the kernel
size of convolutions and r the size of the neighborhood in restricted self-attention.
Layer Type Complexity per Layer Sequential Maximum Path Length
Operations
Self-Attention O(n
2
· d) O(1) O(1)
Recurrent O(n · d
2
) O(n) O(n)
Convolutional O(k · n · d
2
) O(1) O(logk(n))
Self-Attention (restricted) O(r · n · d) O(1) O(n/r)
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 dmodel
as the embeddings, so that the two can be summed. There are many choices of positional encodings,
learned and fixed [9].
In this work, we use sine and cosine functions of different frequencies:
P E(pos,2i) = sin(pos/100002i/dmodel)
P E(pos,2i+1) = cos(pos/100002i/dmodel)
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 2π to 10000 · 2π. We