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In practice, given the same set of queries, keys, and values we may want our model to combine knowledge from different behaviors of the same attention mechanism, such as capturing dependencies of various ranges (e.g., shorter-range vs. longer-range) within a sequence. Thus, it may be beneficial to allow our attention mechanism to jointly use different representation subspaces of queries, keys, and values.

To this end, instead of performing a single attention pooling, queries, keys, and values can be transformed with $$h$$ independently learned linear projections. Then these $$h$$ projected queries, keys, and values are fed into attention pooling in parallel. In the end, $$h$$ attention pooling outputs are concatenated and transformed with another learned linear projection to produce the final output. This design is called multi-head attention, where each of the $$h$$ attention pooling outputs is a head [Vaswani et al., 2017]. Using fully-connected layers to perform learnable linear transformations, fig_multi-head-attention describes multi-head attention.

## 10.3.1. Model¶

Before providing the implementation of multi-head attention, let us formalize this model mathematically. Given a query $$\mathbf{q} \in \mathbb{R}^{d_q}$$, a key $$\mathbf{k} \in \mathbb{R}^{d_k}$$, and a value $$\mathbf{v} \in \mathbb{R}^{d_v}$$, each attention head $$\mathbf{h}_i$$ ($$i = 1, \ldots, h$$) is computed as

(10.3.1)$\mathbf{h}_i = f(\mathbf W_i^{(q)}\mathbf q, \mathbf W_i^{(k)}\mathbf k,\mathbf W_i^{(v)}\mathbf v) \in \mathbb R^{p_v},$

where learnable parameters $$\mathbf W_i^{(q)}\in\mathbb R^{p_q\times d_q}$$, $$\mathbf W_i^{(k)}\in\mathbb R^{p_k\times d_k}$$ and $$\mathbf W_i^{(v)}\in\mathbb R^{p_v\times d_v}$$, and $$f$$ is attention pooling, such as additive attention and scaled dot-product attention in Section 10.2. The multi-head attention output is another linear transformation via learnable parameters $$\mathbf W_o\in\mathbb R^{p_o\times h p_v}$$ of the concatenation of $$h$$ heads:

(10.3.2)$\begin{split}\mathbf W_o \begin{bmatrix}\mathbf h_1\\\vdots\\\mathbf h_h\end{bmatrix} \in \mathbb{R}^{p_o}.\end{split}$

Based on this design, each head may attend to different parts of the input. More sophisticated functions than the simple weighted average can be expressed.

%load ../utils/djl-imports


NDManager manager = NDManager.newBaseManager();


To allow for parallel computation of multiple heads, the below MultiHeadAttention class uses two transposition functions as defined below. Specifically, the transposeOutput function reverses the operation of the transposeQkv function.

public static NDArray transposeQkv(NDArray X, int numHeads) {
// Shape of input X:
// (batchSize, no. of queries or key-value pairs, numHiddens).
// Shape of output X:
// (batchSize, no. of queries or key-value pairs, numHeads,
// numHiddens / numHeads)
X = X.reshape(X.getShape().get(0), X.getShape().get(1), numHeads, -1);

// Shape of output X:
// (batchSize, numHeads, no. of queries or key-value pairs,
// numHiddens / numHeads)
X = X.transpose(0, 2, 1, 3);

// Shape of output:
// (batchSize * numHeads, no. of queries or key-value pairs,
// numHiddens / numHeads)
return X.reshape(-1, X.getShape().get(2), X.getShape().get(3));
}

public static NDArray transposeOutput(NDArray X, int numHeads) {
X = X.reshape(-1, numHeads, X.getShape().get(1), X.getShape().get(2));
X = X.transpose(0, 2, 1, 3);
return X.reshape(X.getShape().get(0), X.getShape().get(1), -1);
}


## 10.3.2. Implementation¶

In our implementation, we choose the scaled dot-product attention for each head of the multi-head attention. To avoid significant growth of computational cost and parameterization cost, we set $$p_q = p_k = p_v = p_o / h$$. Note that $$h$$ heads can be computed in parallel if we set the number of outputs of linear transformations for the query, key, and value to $$p_q h = p_k h = p_v h = p_o$$. In the following implementation, $$p_o$$ is specified via the argument numHiddens.

public static class MultiHeadAttention extends AbstractBlock {

public DotProductAttention attention;
private Linear W_k;
private Linear W_q;
private Linear W_v;
private Linear W_o;

attention = new DotProductAttention(dropout);

W_q = Linear.builder().setUnits(numHiddens).optBias(useBias).build();

W_k = Linear.builder().setUnits(numHiddens).optBias(useBias).build();

W_v = Linear.builder().setUnits(numHiddens).optBias(useBias).build();

W_o = Linear.builder().setUnits(numHiddens).optBias(useBias).build();

Dropout dropout1 = Dropout.builder().optRate(dropout).build();
}

@Override
protected NDList forwardInternal(
ParameterStore ps,
NDList inputs,
boolean training,
PairList<String, Object> params) {
// Shape of queries, keys, or values:
// (batchSize, no. of queries or key-value pairs, numHiddens)
// Shape of validLens:
// (batchSize,) or (batchSize, no. of queries)
// After transposing, shape of output queries, keys, or values:
// (batchSize * numHeads, no. of queries or key-value pairs,
// numHiddens / numHeads)
NDArray queries = inputs.get(0);
NDArray keys = inputs.get(1);
NDArray values = inputs.get(2);
NDArray validLens = inputs.get(3);
// On axis 0, copy the first item (scalar or vector) for
// numHeads times, then copy the next item, and so on

queries =
transposeQkv(
W_q.forward(ps, new NDList(queries), training, params).get(0),
keys =
transposeQkv(
W_k.forward(ps, new NDList(keys), training, params).get(0), numHeads);
values =
transposeQkv(
W_v.forward(ps, new NDList(values), training, params).get(0), numHeads);

// Shape of output: (batchSize * numHeads, no. of queries,
// numHiddens / numHeads)
NDArray output =
attention
.forward(
ps,
new NDList(queries, keys, values, validLens),
training,
params)
.get(0);

// Shape of outputConcat:
// (batchSize, no. of queries, numHiddens)
return new NDList(W_o.forward(ps, new NDList(outputConcat), training, params).get(0));
}

@Override
public Shape[] getOutputShapes(Shape[] inputShapes) {
throw new UnsupportedOperationException("Not implemented");
}

@Override
public void initializeChildBlocks(
NDManager manager, DataType dataType, Shape... inputShapes) {
try (NDManager sub = manager.newSubManager()) {
NDArray queries = sub.zeros(inputShapes[0], dataType);
NDArray keys = sub.zeros(inputShapes[1], dataType);
NDArray values = sub.zeros(inputShapes[2], dataType);
NDArray validLens = sub.zeros(inputShapes[3], dataType);

ParameterStore ps = new ParameterStore(sub, false);

W_q.initialize(manager, dataType, queries.getShape());
W_k.initialize(manager, dataType, keys.getShape());
W_v.initialize(manager, dataType, values.getShape());

queries =
keys = transposeQkv(W_k.forward(ps, new NDList(keys), false).get(0), numHeads);
values = transposeQkv(W_v.forward(ps, new NDList(values), false).get(0), numHeads);

NDList list = new NDList(queries, keys, values, validLens);
attention.initialize(sub, dataType, list.getShapes());
NDArray output = attention.forward(ps, list, false).head();

W_o.initialize(manager, dataType, outputConcat.getShape());
}
}
}


Let us test our implemented MultiHeadAttention class using a toy example where keys and values are the same. As a result, the shape of the multi-head attention output is (batchSize, numQueries, numHiddens).

int numHiddens = 100;

int batchSize = 2;
int numQueries = 4;
int numKvpairs = 6;
NDArray validLens = manager.create(new float[] {3, 2});
NDArray X = manager.ones(new Shape(batchSize, numQueries, numHiddens));
NDArray Y = manager.ones(new Shape(batchSize, numKvpairs, numHiddens));

ParameterStore ps = new ParameterStore(manager, false);
NDList input = new NDList(X, Y, Y, validLens);
attention.initialize(manager, DataType.FLOAT32, input.getShapes());
NDList result = attention.forward(ps, input, false);
result.get(0).getShape();

(2, 4, 100)


## 10.3.3. Summary¶

• Multi-head attention combines knowledge of the same attention pooling via different representation subspaces of queries, keys, and values.

• To compute multiple heads of multi-head attention in parallel, proper tensor manipulation is needed.

## 10.3.4. Exercises¶

1. Visualize attention weights of multiple heads in this experiment.

2. Suppose that we have a trained model based on multi-head attention and we want to prune least important attention heads to increase the prediction speed. How can we design experiments to measure the importance of an attention head?