Added course notes on attention

Author: suxuann

assets/att/attention.png

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+---
+layout: page
+permalink: /attention/
+---
+
+Table of Contents:
+
+- [Motivation](#motivation)
+- [General Attention Layers](#attention)
+    - [Operations](#operations)
+- [Self-Attention](#self)
+    - [Masked Self-Attention Layers](#masked)
+    - [Multi-Head Self-Attention Layers](#multihead)
+- [Summary](#summary)
+- [Additional References](#resources)
+
+## Attention
+
+We discussed fundamental workhorses of modern deep learning such as Convolutional Neural Networks and Recurrent Neural
+Networks in previous sections. This section is devoted to yet another layer -- the attention layer -- that forms a new
+primitive for modern Computer Vision and NLP applications.
+
+<a name='motivation'></a>
+
+### Motivation
+
+To motivate the attention layer, let us look at a sample application -- image captioning, and see what's the problem
+with using plain CNNs and RNNs there.
+
+The figure below shows a pipeline of applying such networks on a given image to generate a caption. It first uses a
+pre-trained CNN feature extractor to summarize the image, resulting in an image feature vector \\(c = h_0\\). It then
+applies a recurrent network to repeatedly generate tokens at each step. After five time steps, the image captioning
+model obtains the sentence: "surfer riding on wave".
+
+<div class="fig figcenter fighighlight">
+  <img src="/assets/att/captioning.png" width="80%">
+</div>
+
+What is the problem here? Notice that the model relies entirely on the context vector \\(c\\) to write the caption --
+everything it wants to say about the image needs to be compressed within this vector. What if we want to be very
+specific, and describe every nitty-gritty detail of the image, e.g. color of the surfer's shirt, facing direction of the
+waves? Obviously, a finite-length vector cannot be used to encode all such possibilities, especially if the desired
+number of tokens goes to the magnitude of hundreds or thousands.
+
+The central idea of the attention layer is borrowed from human's visual attention system: when humans like us are given
+a visual scene and try to understand a specific region of that scene, we focus our eyesight on that region. The
+attention layer simulates this process, and *attends* to different parts of the image while generating words to describe
+it.
+
+With attention in play, a similar diagram showing the pipeline for image captioning is as follows. What's the main
+difference? We incorporate two additional matrices: one for *alignment scores*, and the other for *attention*; and have
+*different context vectors* \\(c_i\\) at different steps. At each step, the model uses a multi-layer perceptron to
+digest the current hidden vector \\(h_i\\) and the input image features, to generate an alignment score matrix of shape
+\\(H \times W\\). This score matrix is then fed into a softmax layer that converts it to an attention matrix with
+weights summing to one. The weights in the attention matrix are next multiplied element-wise with image features,
+allowing the model to focus on regions of the image differently. This entire process is differentiable and enables the
+model to choose its own attention weights.
+
+<div class="fig figcenter fighighlight">
+  <img src="/assets/att/captioning-attention.png" width="60%">
+</div>
+
+<a name='attention'></a>
+
+### General Attention Layers
+
+While the previous section details the application of an attention layer in image captioning, we next present a more
+general and principled formulation of the attention layer, de-contextualizing it from the image captioning and recurrent
+network settings. In a general setting, the attention layer is a layer with input and output vectors, and five major
+operations. These are illustrated in the following diagrams.
+
+<div class="fig figcenter fighighlight">
+  <img src="/assets/att/attention.png" width="70%">
+  <div class="figcaption">Left: A General Attention Layer. Right: A Self-Attention Layer.</div>
+</div>
+
+As illustrated, inputs to an attention layer contain input vectors \\(X\\) and query vectors \\(Q\\). The input vectors,
+\\(X\\), are of shape \\(N \times D_x\\) while the query vectors \\(Q\\) are of shape \\(M \times D_k\\). In the image
+captioning example, input vectors are the image features while query vectors are the hidden states of the recurrent
+network. Outputs of an attention layer are the vectors \\(Y\\) of shape \\(M \times D_k\\), at the top.
+
+The bulk of the attention operations are illustrated as the colorful grids in the middle, and contains two major types
+of operations: linear key-value maps; and align & attend operations that we saw earlier in the image captioning example.
+
+<a name='operations'></a>
+
+#### Operations
+
+**Linear Key and Value Transformations.** These operations are linear transformations that convert the input vectors \\(
+X\\) to two alternative set of vectors:
+
+- Key vectors \\(K\\): These vectors are obtained by using the linear equation \\(K = X W_k\\) where \\(W_k\\) is a
+  learnable weight matrix of shape \\(D_x \times D_k\\), converting from input vector dimension \\(D_x\\) to key
+  dimension \\(D_k\\). The resulting keys have the same dimension as the query vectors, to enable alignment.
+- Value vectors \\(V\\): Similarly, the equation to derive these vectors is the linear rule \\(V = X W_v\\)
+  where \\(W_v\\) is of shape \\(D_x \times D_v\\). The value vectors have the same dimension as the output vectors.
+
+By applying these fully-connected layers on top of the inputs, the attention model achieves additional expressivity.
+
+**Alignment.** Core to the attention layer are two fundamental operations: alignment, and attention. In the alignment
+step, while more complex functions are possible, practitioners often opt for a simple function between vectors: pairwise
+dot products between key and query vectors.
+
+Moreover, for vectors with a larger dimensionality, more terms are multiplied and summed in the dot product and this
+usually implies a larger variance. Vectors with larger magnitude contribute more weights to the resulting softmax
+calculation, and many terms usually receive low attention. To deal with this issue, a scaling factor, the reciprocal of
+\\(\sqrt{D_x}\\), is often incorporated to reduce the alignment scores. This scaling procedure reduces the effect of
+large magnitude terms, so that the resulting attention weights are more spread-out. The alignment computation can be
+summarized as the following equation:
+
+$$ e_{i,j} = \frac{q_j \cdot x_i}{\sqrt{D_x}} $$
+
+**Attention.** The attention matrix is obtained by applying the softmax function column-wise to the alignment matrix.
+
+$$ \mathbf{a} = \text{softmax}(\mathbf{e}) $$
+
+The output vectors are finally calculated as multiplications of the attention matrix and the input vectors:
+
+$$ y_j = \sum_{i} a_{i,j} x_i $$
+
+<a name='self'></a>
+
+### Self-Attention
+
+While we explain the general attention layer above, the self-attention layer refers to the special case where similar to
+the key and value vectors, the query vectors \\(Q\\) are also expressed as a linear transformation of the input vectors:
+\\(Q = X W_q\\) where \\(W_q\\) is of shape \\(D_x \times D_k\\). With this expression of query vectors as a linear
+function of the inputs, the attention layer is self-contained. This is illustrated on the right of the figure above.
+
+**Permutation Invariance.** It is worth noting that the self-attention layer is invariant to the order of the input
+vectors: if we apply a permutation to the input vectors, the outputs will be permuted in exactly the same way. This is
+illustrated in the following diagram.
+
+<div class="fig figcenter fighighlight">
+  <img src="/assets/att/permutation.png" width="55%">
+  <div class="figcaption">Permutation Invariance of Self-Attention Layers.</div>
+</div>
+
+**Positional Encoding.** While the self-attention layer is agnostic to the ordering of inputs, practical applications
+often require some notion of ordering. For example, in natural language sequences, the relative ordering of words often
+plays a pivotal role in differentiating the meaning of the entire sentence. This necessitates the inclusion of a
+positional encoding component into the self-attention module, to endow the model with the ability to determine the
+positions of its inputs. A number of desiderata are needed for this component:
+
+- The positional encodings should be *unique* for each time step.
+- The *distance* between any two consecutive encodings should be the same.
+- The positional encoding function should generalize to arbitrarily *long* sequences.
+- The function should be *deterministic*.
+
+While there exists a number of functions that satisfy the above criteria, a commonly used method makes use of mixed sine
+and cosine values. Concretely, the encoding function looks like the following:
+
+$$ p(t)
+= [\sin(w_1 \cdot t), \cos(w_1 \cdot t), \sin(w_2 \cdot t), \cos(w_2 \cdot t), \cdots, \sin(w_{d/2} \cdot t), \cos(w_{d/2} \cdot t)]
+$$
+
+where the frequency \\(w_k = \frac{1}{10000^{2k/d}}\\). What does this function encode? The following diagram is an
+intuitive explanation of the same phenomenon, but in the binary domain:
+
+<div class="fig figcenter fighighlight">
+  <img src="/assets/att/position-binary.png" width="50%">
+</div>
+
+The frequencies \\(w_k\\) are varied, to represent the relative positions of the inputs, in a similar vein as the 0s and
+1s in the binary case. In practice, the positional encoding component concatenates additional information to the input
+vectors, before they are passed to the self-attention module:
+
+<div class="fig figcenter fighighlight">
+  <img src="/assets/att/position.png" width="15%">
+</div>
+
+**A Comparison Between General Attention vs. Self-Attention**. The general attention layer has access to three sets of
+vectors: key, value, and query vectors. In comparison, the self-attention layer is entirely self-enclosed, and instead
+parameterizes the three sets of vectors as linear functions of the inputs.
+
+<div class="fig figcenter fighighlight">
+  <img src="/assets/att/comparison.png" width="60%">
+</div>
+
+<a name='masked'></a>
+
+#### Masked Self-Attention Layers
+
+While the positional encoding layer integrates some positional information, in more critical applications, it may be
+necessary to distill into the model a clearer idea of relative input orderings and prevent it from *looking-ahead* at
+future vectors. To this end, the *masked* self-attention layer is created: it explicitly sets the lower-triangular part
+of the alignment matrix to negative infinity values, to ignore the corresponding, future vectors while the model
+processes earlier vectors.
+
+<div class="fig figcenter fighighlight">
+  <img src="/assets/att/masked.png" width="30%">
+</div>
+
+<a name='multihead'></a>
+
+#### Multi-Head Self-Attention Layers
+
+Yet another possibility to increase the expressivity of the model is to exploit the notion of a *multi-head* attention.
+Instead of using one single self-attention layer, multi-head attention utilizes multiple, parallel attention layers. In
+some cases, to maintain the total computation, the key and value dimensions \\(D_k, D_v\\) may be reduced accordingly.
+The benefit of using multiple attention heads is to allow the model to focus on different aspects of the input vectors.
+
+<div class="fig figcenter fighighlight">
+  <img src="/assets/att/multihead.png" width="60%">
+</div>
+
+<a name='summary'></a>
+
+### Summary
+
+To summarize this section,
+
+- We motivated and introduced a novel layer popular in deep learning, the **attention** layer.
+- We introduced it in its general formulation and in particular, studied details of the **align and attend** operations.
+- We then specialized to the case of a **self-attention** layer.
+- We learned that self-attention layers are **permutation-invariant** to the input vectors.
+- To retain some positional information, self-attention layers use a **positional-encoding** function.
+- Moreover, we also studied two extensions of the vanilla self-attention layer: the **masked** attention layer, and
+  the **multi-head** attention. While the former layer prevents the model from looking ahead, the latter serves to
+  increase its expressivity.
+
+<a name='resources'></a>
+
+### Additional Resources
+
+- [Show, Attend and Tell: Neural Image Caption Generation with Visual Attention](http://proceedings.mlr.press/v37/xuc15.pdf)
+  presents an application of the attention layer to image captioning.
+- [Women also Snowboard: Overcoming Bias in Captioning Models](https://arxiv.org/pdf/1803.09797.pdf) exploits the
+  attention layer to detect gender bias in image captioning models.
+- [Neural Machine Translation by Jointly Learning to Align and Translate](https://arxiv.org/pdf/1409.0473.pdf) applies
+  attention to natural language translation.
+- [Attention is All You Need](https://arxiv.org/pdf/1706.03762.pdf) is the seminal paper on attention-based
+  Transformers, that took the Vision and NLP communities by storm.