Reformer: Efficient Transformer
Reformer is an efficient version of the transformers proposed by Google Research in 2020 and published in this paper: “Reformer: The efficient Transformer”. The official code for this paper can be found in this GitHub repository: reformerpytorch. In this paper, the authors introduced two techniques to improve the memory efficiency of Transformers while keeping the same great performance.

They replaced the dotproduct attention by one that uses localitysensitive hashing, changing its complexity from $O\left( N^{2} \right)$ to $O\left( N \text{log}\left( N \right) \right)$, where $N$ is the length of the sequence.

They used reversible residual layers instead of the standard residuals, which allows storing activations only once in the training process instead of $L$ times, where $L$ is the number of layers.

They split activations inside feedforward layers and processing them in chunks which removes the effect of the depth and saves memory inside feedforward layers.
Before getting into more details, let’s first see how a transformer model consumes memory assuming that the vocabulary is $V$, the word embedding size is $d$, the input sequence length is $L$, the depth of the feedforward network is $d_{\text{ff}}$ where the $i^{th}$ layer has $N_{i}$ neurons, and the number of batches is $B$:

Word embedding matrix is $V*d$.

Positional Embedding is $d*L$.

Feedforward weights $\prod_{i = 1}^{d_{\text{ff}}}N_{i}$.

The original paper uses 6 heads of attention, each attention head has:

$3*L*d$ parameters for V, Q and K.

$L*L$ attention weights.

This is just for one layer, the memory with N layers is Ntimes larger than that due to the fact that activations need to be stored for backpropagation. And all of that was just for the encoder, the decoder takes even more memory. Also, this was done using one batch, using the batch size will roughly multiply this number by a factor of B. And the standard way of saving parameters is float32 bit. So, all this needs to be multiplied by 4 to get the number of bytes needed for saving this model. Which is a lot!
Nowadays, researchers tend to use largescale longsequence transformers since they yield great results. But this strains resources to the point that this trend is breaking NLP research as these large Transformer models can only realistically be trained in large industrial research laboratories. And that’s hurting the field since only the big corporations are the ones who can afford such a thing. So, there was a need for a more efficient Transformer.
LSH Attention
LSH Attention stands for “LocalitySensitive Hashing Attention” which is a memory and computationally efficient way to perform the attention mechanism. A hashing scheme that assigns each vector $x$ to a hash $h(x)$ is called “localitysensitive” if nearby vectors get the same hash with high probability and distant ones do not.
Dotproduct Attention
Before getting into the First, let’s recap the standard attention used in the original Transformer paper which was the scaled dotproduct attention:
\[\text{Attention}\left( Q,\ K,\ V \right) = softmax\left( \frac{QK^{T}}{\sqrt{d_{k}}} \right)V\]As we can see from the previous formula, the input consists of queries $Q$, keys $K$ and values $V$. These matrices are obtained by projecting the contextualized word embedding (Word embedding + position embedding) into $Q$, $K$, and $V$ (each of $B*L*d$ dimension) using three different feedforward networks.
Modifications
LSH attention can be seen as a more optimized form of the scaleddot product attention which was obtains using the following modifications:

It uses the same feedforward network to obtain both $Q$ and $K$ as it turns out that this does not affect the performance of the Transformer model.

It doesn’t calculate the whole $QK^{T}$ term. It only computes the attention for each query $q_{i}$ separately and then recomputes it on the backward pass when needed for gradients.

According to the equation, the model actually is interested in the most similar q and k represented in the $\text{softmax}\left( QK^{T} \right)$ term. That’s why LSH attention only considers a subset of $K$ that are close to $q_{i}$ rather than the whole $K$. The problem of finding nearest neighbors quickly in highdimensional spaces can be solved by Localsensitivity hashing scheme, hence the name of the mechanism.
So, now let’s rewrite the same scaleddot product attention formula after putting in mind the mentioned modifications:
\[o_{i} = \sum_{j \in \mathcal{P}_{i}}^{}{\exp\left( q_{i}.k_{j}  z\left( i,\ \mathcal{P}_{i} \right) \right)v_{j}}\] \[\mathcal{P}_{i} = \left\{ j:\ i \geq j \right\}\]As you can see, now we are considering just a single query position i at a time (denoted $q_{i}$). We introduced the notion $\mathcal{P}_{i}$ to represent the set that the query at position i attends to. And we replaced the normalizing term in the softmax $\sum_{i \in L}^{}{\exp\left( q_{i}.k_{i} \right)}$ with a function $z$.
For batching purposes, we typically perform attention over a larger set ${\widetilde{\mathcal{P}}}_{i} = \left[ 0,\ 1,\ …,\ l \right]$ while masking out elements not in $\mathcal{P}_{i}$ using a masking function $m()$. So, the formula becomes:
\[o_{i} = \sum_{j \in {\widetilde{\mathcal{P}}}_{i}}^{}{\exp\left( q_{i}.k_{j}  m\left( j,\ \mathcal{P}_{i} \right)  z\left( i,\ \mathcal{P}_{i} \right) \right)v_{j}}\] \[\text{m}\left( j,\ \mathcal{P}_{i} \right) = \left\{ \begin{matrix} \infty\ \ \ \ \ \ \text{if}\text{j} \notin \mathcal{P}_{i} \\ 0\ \ \ \ \ \ \text{otherwise} \\ \end{matrix} \right.\]LSH Attention Mechanism
Now, let’s turn to LSH attention, which we can think of in terms of restricting the set $\mathcal{P}_{i}$ by only allowing attention within a single hash bucket $\mathcal{P}_{i} = \left( j:h\left( q_{i} \right) = h\left( k_{i} \right) \right)$. And this can be done by the following steps:

First, apply the LSH scheme over the given queries Q and keys K by employing random projections. To get $b$ hashes, we first fix a random matrix $R$ of size $\left\lbrack b_{k},\ \frac{b}{2} \right\rbrack$. And then define a hashing function $h(x) = \text{argmax}(\lbrack xR;  xR\rbrack)$ where $\lbrack u;\ v\rbrack$ denotes the concatenation of two vectors knowing that they ensured that $h\left( k_{j} \right) = h\left( q_{j} \right)$ by setting $\frac{q_{j}}{\left q_{j} \right}$.

Then, sort the queries and keys according to their hash bucket and then allow attention within each bucket (right graph) rather than the full matrix (left graph).
 Next, they sorted the queries by bucket number and, within each bucket, by sequence position; this defines a permutation where $i \rightarrow s_{i}$ after sorting. In the sorted attention matrix, pairs from the same bucket will cluster near the diagonal.
 Hash buckets in this formulation tend to have different number of queries and keys. In fact, it is possible for a bucket to contain many queries but no keys. To alleviate these issues, they process $m$ consecutive queries where they attend to each other, and one chunk back. In the following graph $m = 2$.
 Following our earlier notation, this corresponds to setting where $m = \frac{2*l}{n_{\text{buckets}}}$ where $l$ is the sequence length and the average bucket size is $\frac{l}{n_{\text{buckets}}}$:
Multiround LSH attention
With hashing, there is always a small probability that similar items nevertheless fall in different buckets. This probability can be reduced by doing multiple rounds of hashing with $n_{\text{rounds}}$ distinct hash functions $\left( h^{\left( 1 \right)},\ h^{\left( 2 \right)},\ …\ \right)$ such that:
\[\mathcal{P}_{i} = \bigcup_{r = 1}^{n_{\text{rounds}}}\mathcal{P}_{i}^{\left( r \right)}\ \ \ \ \ \ \ \ \mathcal{P}_{i}^{\left( r \right)} = \left\{ j:\ h^{\left( r \right)}\left( q_{i} \right) = h^{\left( r \right)}\left( q_{j} \right) \right\}\]The multiround case essentially involves performing LSH attention $n_{\text{rounds}}$ times in parallel. So, starting with the original formula:
\[o_{i} = \sum_{j \in {\widetilde{\mathcal{P}}}_{i}}^{}{\exp\left( q_{i}.k_{j}  m\left( j,\ \mathcal{P}_{i} \right)  z\left( i,\ \mathcal{P}_{i} \right) \right)v_{j}}\]We can combine it with the multiple rounds hashing like so:
\[o_{i} = \sum_{r = 1}^{n_{\text{rounds}}}{\exp\left( z\left( i,\ \mathcal{P}_{i}^{\left( r \right)} \right)  z\left( i,\ \mathcal{P}_{i} \right) \right)}\sum_{j \in {\widetilde{\mathcal{P}}}_{i}}^{}{\frac{1}{N_{i,j}}\exp\left( q_{i}.k_{j}  m\left( j,\ \mathcal{P}_{i}^{\left( r \right)} \right)  z\left( i,\ \mathcal{P}_{i}^{\left( r \right)} \right) \right)v_{j}} = \sum_{r = 1}^{n_{\text{rounds}}}{\exp\left( z\left( i,\ \mathcal{P}_{i}^{\left( r \right)} \right)  z\left( i,\ \mathcal{P}_{i} \right) \right)}o_{i}^{\left( r \right)}\]Each round of LSH attention produces a vector $o_{i}^{\left( r \right)}$ that can be computed independently from other rounds, except for the inclusion of a term $N_{i,j}$ to avoid that they folded the $N_{i,j}$ factor into the masking term . So, the becomes:
\[o_{i}^{\left( r \right)} = \sum_{j \in {\widetilde{\mathcal{P}}}_{i}}^{}{\exp\left( q_{i}.k_{j}  m_{i,j}^{\left( r \right)}  z\left( i,\ \mathcal{P}_{i}^{\left( r \right)} \right) \right)v_{j}}\]Where;
\[m_{i,j}^{\left( r \right)} = \left\{ \begin{matrix} \ \ \ \infty\ \ \ \ \ \ \text{if}\ \ \text{j} \notin \mathcal{P}_{i} \\ 10^{5}\ \ \ \ \text{if}\ \ i = j \\ \log N_{i,j}\ \ \ \text{otherwise} \\ \end{matrix} \right.\] \[N_{i,j} = \left \left\{ r':\ j \in \mathcal{P}_{i}^{\left( r' \right)} \right\} \right\]They introduced a special case for $m_{i,j}^{\left( r \right)}$ for when $i = j$. This case is added because causal masking in a standard Transformer allows position $i$ to attend to itself, which is not desirable in a sharedQK formulation. They set the mask to a large but finite value to disallow attentioninplace, except in the situation where a token has no other valid attention targets.
Reversible Transformer
TO BE CONTINUED