Reformer
----------------------------------------------------
**DISCLAIMER:** This model is still a work in progress, if you see something strange,
file a `Github Issue <https://github.com/huggingface/transformers/issues/new?assignees=&labels=&template=bug-report.md&title>`_

Overview
~~~~~
The Reformer model was presented in `Reformer: The Efficient Transformer <https://arxiv.org/abs/2001.04451.pdf>`_ by Nikita Kitaev, Łukasz Kaiser, Anselm Levskaya.
Here the abstract: 

*Large Transformer models routinely achieve state-of-the-art results on a number of tasks but training these models can be prohibitively costly, especially on long sequences. We introduce two techniques to improve the efficiency of Transformers. For one, we replace dot-product attention by one that uses locality-sensitive hashing, changing its complexity from O(L^2) to O(Llog(L)), where L is the length of the sequence. Furthermore, we use reversible residual layers instead of the standard residuals, which allows storing activations only once in the training process instead of N times, where N is the number of layers. The resulting model, the Reformer, performs on par with Transformer models while being much more memory-efficient and much faster on long sequences.*

The Authors' code can be found `here <https://github.com/google/trax/tree/master/trax/models/reformer>`_ .

Axial Positional Encodings
~~~~~~~~~~~~~~~~~~~~
Axial Positional Encodings were first implemented in Google's `trax library <https://github.com/google/trax/blob/4d99ad4965bab1deba227539758d59f0df0fef48/trax/layers/research/position_encodings.py#L29>`_ and developed by the authors of this model's paper. In models that are treating very long input sequences, the conventional position id encodings store an embedings vector of size :math:`d` being the ``config.hidden_size`` for every position :math:`i, \ldots, n_s`, with :math:`n_s` being ``config.max_embedding_size``. *E.g.*, having a sequence length of :math:`n_s = 2^{19} \approx 0.5M` and a ``config.hidden_size`` of :math:`d = 2^{10} \approx 1000` would result in a position encoding matrix:

.. math::
    X_{i,j}, \text{ with } i \in \left[1,\ldots, d\right] \text{ and } j \in \left[1,\ldots, n_s\right] 

which alone has over 500M parameters to store. Axial positional encodings factorize :math:`X_{i,j}` into two matrices: 

.. math::
    X^{1}_{i,j}, \text{ with } i \in \left[1,\ldots, d^1\right] \text{ and } j \in \left[1,\ldots, n_s^1\right] 

and 

.. math::
    X^{2}_{i,j}, \text{ with } i \in \left[1,\ldots, d^2\right] \text{ and } j \in \left[1,\ldots, n_s^2\right] 

with:

.. math::
    d = d^1 + d^2 \text{ and } n_s = n_s^1 \times n_s^2 .

Therefore the following holds:

.. math::
    X_{i,j} = \begin{cases}
                X^{1}_{i, k}, & \text{if }\ i < d^1 \text{ with } k = j \mod n_s^1 \\
                X^{2}_{i - d^1, l}, & \text{if } i \ge d^1 \text{ with } l = \lfloor\frac{j}{n_s^1}\rfloor
              \end{cases}

Intuitively, this means that a position embedding vector :math:`x_j \in \mathbb{R}^{d}` is now the composition of two factorized embedding vectors: :math:`x^1_{k, l} + x^2_{l, k}`, where as the ``config.max_embedding_size`` dimension :math:`j` is factorized into :math:`k \text{ and } l`.
This design ensures that each position embedding vector :math:`x_j` is unique.

Using the above example again, axial position encoding with :math:`d^1 = 2^5, d^2 = 2^5, n_s^1 = 2^9, n_s^2 = 2^{10}` can drastically reduced the number of parameters to :math:`2^{14} + 2^{15} \approx 49000` parameters.

In practice, the parameter ``config.axial_pos_embds_dim`` is set to ``list``:math:`(d^1, d^2)` which sum has to be equal to ``config.hidden_size`` and ``config.axial_pos_shape`` is set to ``list``:math:`(n_s^1, n_s^2)` and which product has to be equal to ``config.max_embedding_size`` which during training has to be equal to the ``sequence length`` of the ``input_ids``.



LSH Self Attention
~~~~~~~~~~~~~~~~~~~~
In Locality sensitive hashing (LSH) self attention the key and query projection weights are tied. Therefore, the key query embedding vectors are also tied.
LSH self attention uses the locality sensitive 
hashing mechanism proposed in `Practical and Optimal LSH for Angular Distance <https://arxiv.org/abs/1509.02897>`_ to assign each of the tied key query embedding vectors to one of ``config.num_buckets`` possible buckets. The premise is that the more "similar" key query embedding vectors (in terms of *cosine similarity*) are to each other, the more likely they are assigned to the same bucket. 
The accuracy of the LSH mechanism can be improved by increasing ``config.num_hashes`` or directly the argument ``num_hashes`` of the forward function so that the output of the LSH self attention better approximates the output of the "normal" full self attention.
The buckets are then sorted and chunked into query key embedding vector chunks each of length ``config.lsh_chunk_length``. For each chunk, the query embedding vectors attend to its key vectors (which are tied to themselves) and to the key embedding vectors of ``config.lsh_num_chunks_before`` previous neighboring chunks and ``config.lsh_num_chunks_after`` following neighboring chunks.
For more information, see the `original Paper <https://arxiv.org/abs/2001.04451>`_ or this great `blog post <https://www.pragmatic.ml/reformer-deep-dive/>`_.

Note that ``config.num_buckets`` can also be factorized into a ``list``:math:`(n_{\text{buckets}}^1, n_{\text{buckets}}^2)`. This way instead of assigning the query key embedding vectors to one of :math:`(1,\ldots, n_{\text{buckets}})` they are assigned to one of :math:`(1-1,\ldots, n_{\text{buckets}}^1-1, \ldots, 1-n_{\text{buckets}}^2, \ldots, n_{\text{buckets}}^1-n_{\text{buckets}}^2)`. This is crucial for very long sequences to save memory.

When training a model from scratch, it is recommended to leave ``config.num_buckets=None``, so that depending on the sequence length a good value for ``num_buckets`` is calculated on the fly. This value will then automatically be saved in the config and should be reused for inference.

Using LSH self attention, the memory and time complexity of the query-key matmul operation can be reduced from :math:`\mathcal{O}(n_s \times n_s)` to :math:`\mathcal{O}(n_s \times \log(n_s))`, which usually represents the memory and time bottleneck in a transformer model, with :math:`n_s` being the sequence length.


Local Self Attention
~~~~~~~~~~~~~~~~~~~~
Local self attention is essentially a "normal" self attention layer with 
key, query and value projections, but is chunked so that in each chunk of length ``config.local_chunk_length`` the query embedding vectors only attends to the key embedding vectors in its chunk and to the key embedding vectors of ``config.local_num_chunks_before`` previous neighboring chunks and ``config.local_num_chunks_after`` following neighboring chunks.

Using Local self attention, the memory and time complexity of the query-key matmul operation can be reduced from :math:`\mathcal{O}(n_s \times n_s)` to :math:`\mathcal{O}(n_s \times \log(n_s))`, which usually represents the memory and time bottleneck in a transformer model, with :math:`n_s` being the sequence length.


Training
~~~~~~~~~~~~~~~~~~~~
During training, we must ensure that the sequence length is set to a value that can be divided by the least common multiple of ``config.lsh_chunk_length`` and ``config.local_chunk_length`` and that the parameters of the Axial Positional Encodings are correctly set as described above. Reformer is very memory efficient so that the model can easily be trained on sequences as long as 64000 tokens.
For training, the ``ReformerModelWithLMHead`` should be used as follows: 

::

  input_ids = tokenizer.encode('This is a sentence from the training data', return_tensors='pt')
  loss = model(input_ids, labels=input_ids)[0]


ReformerConfig
~~~~~~~~~~~~~~~~~~~~~

.. autoclass:: transformers.ReformerConfig
    :members:


ReformerTokenizer
~~~~~~~~~~~~~~~~~~~~~

.. autoclass:: transformers.ReformerTokenizer
    :members: 


ReformerModel
~~~~~~~~~~~~~~~~~~~~

.. autoclass:: transformers.ReformerModel
    :members:


ReformerModelWithLMHead
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. autoclass:: transformers.ReformerModelWithLMHead
    :members:
