"""
.. _model-transformer:

Transformer tutorial
====================

**Author**: Zihao Ye, Jinjing Zhou, Qipeng Guo, Quan Gan, Zheng Zhang
"""
################################################################################################
# In this tutorial, you learn about a simplified implementation of the Transformer model.
# You can see highlights of the most important design points. For instance, there is
# only single-head attention. The complete code can be found
# `here <https://github.com/dmlc/dgl/tree/master/examples/pytorch/transformer>`__.
#
# The overall structure is similar to the one from the research papaer `Annotated
# Transformer <http://nlp.seas.harvard.edu/2018/04/03/attention.html>`__.
#
# The Transformer model, as a replacement of CNN/RNN architecture for
# sequence modeling, was introduced in the research paper: `Attention is All
# You Need <https://arxiv.org/pdf/1706.03762.pdf>`__. It improved the
# state of the art for machine translation as well as natural language
# inference task
# (`GPT <https://s3-us-west-2.amazonaws.com/openai-assets/research-covers/language-unsupervised/language_understanding_paper.pdf>`__).
# Recent work on pre-training Transformer with large scale corpus
# (`BERT <https://arxiv.org/pdf/1810.04805.pdf>`__) supports that it is
# capable of learning high-quality semantic representation.
#
# The interesting part of Transformer is its extensive employment of
# attention. The classic use of attention comes from machine translation
# model, where the output token attends to all input tokens.
#
# Transformer additionally applies *self-attention* in both decoder and
# encoder. This process forces words relate to each other to combine
# together, irrespective of their positions in the sequence. This is
# different from RNN-based model, where words (in the source sentence) are
# combined along the chain, which is thought to be too constrained.
#
# Attention layer of Transformer
# ------------------------------
#
# In the attention layer of Transformer, for each node the module learns to
# assign weights on its in-coming edges. For node pair :math:`(i, j)`
# (from :math:`i` to :math:`j`) with node
# :math:`x_i, x_j \in \mathbb{R}^n`, the score of their connection is
# defined as follows:
#
# .. math::
#
#
#    q_j = W_q\cdot x_j \\
#    k_i = W_k\cdot x_i\\
#    v_i = W_v\cdot x_i\\
#    \textrm{score} = q_j^T k_i
#
# where :math:`W_q, W_k, W_v \in \mathbb{R}^{n\times d_k}` map the
# representations :math:`x` to “query”, “key”, and “value” space
# respectively.
#
# There are other possibilities to implement the score function. The dot
# product measures the similarity of a given query :math:`q_j` and a key
# :math:`k_i`: if :math:`j` needs the information stored in :math:`i`, the
# query vector at position :math:`j` (:math:`q_j`) is supposed to be close
# to key vector at position :math:`i` (:math:`k_i`).
#
# The score is then used to compute the sum of the incoming values,
# normalized over the weights of edges, stored in :math:`\textrm{wv}`.
# Then apply an affine layer to :math:`\textrm{wv}` to get the output
# :math:`o`:
#
# .. math::
#
#
#    w_{ji} = \frac{\exp\{\textrm{score}_{ji} \}}{\sum\limits_{(k, i)\in E}\exp\{\textrm{score}_{ki} \}} \\
#    \textrm{wv}_i = \sum_{(k, i)\in E} w_{ki} v_k \\
#    o = W_o\cdot \textrm{wv} \\
#
# Multi-head attention layer
# ~~~~~~~~~~~~~~~~~~~~~~~~~~
#
# In Transformer, attention is *multi-headed*. A head is very much like a
# channel in a convolutional network. The multi-head attention consists of
# multiple attention heads, in which each head refers to a single
# attention module. :math:`\textrm{wv}^{(i)}` for all the heads are
# concatenated and mapped to output :math:`o` with an affine layer:
#
# .. math::
#
#
#    o = W_o \cdot \textrm{concat}\left(\left[\textrm{wv}^{(0)}, \textrm{wv}^{(1)}, \cdots, \textrm{wv}^{(h)}\right]\right)
#
# The code below wraps necessary components for multi-head attention, and
# provides two interfaces.
#
# -  ``get`` maps state ‘x’, to query, key and value, which is required by
#    following steps(\ ``propagate_attention``).
# -  ``get_o`` maps the updated value after attention to the output
#    :math:`o` for post-processing.
#
# .. code:: 
#
#    class MultiHeadAttention(nn.Module):
#        "Multi-Head Attention"
#        def __init__(self, h, dim_model):
#            "h: number of heads; dim_model: hidden dimension"
#            super(MultiHeadAttention, self).__init__()
#            self.d_k = dim_model // h
#            self.h = h
#            # W_q, W_k, W_v, W_o
#            self.linears = clones(nn.Linear(dim_model, dim_model), 4)
#
#        def get(self, x, fields='qkv'):
#            "Return a dict of queries / keys / values."
#            batch_size = x.shape[0]
#            ret = {}
#            if 'q' in fields:
#                ret['q'] = self.linears[0](x).view(batch_size, self.h, self.d_k)
#            if 'k' in fields:
#                ret['k'] = self.linears[1](x).view(batch_size, self.h, self.d_k)
#            if 'v' in fields:
#                ret['v'] = self.linears[2](x).view(batch_size, self.h, self.d_k)
#            return ret
#
#        def get_o(self, x):
#            "get output of the multi-head attention"
#            batch_size = x.shape[0]
#            return self.linears[3](x.view(batch_size, -1))
#
#
# How DGL implements Transformer with a graph neural network
# ----------------------------------------------------------
#
# You get a different perspective of Transformer by treating the
# attention as edges in a graph and adopt message passing on the edges to
# induce the appropriate processing.
#
# Graph structure
# ~~~~~~~~~~~~~~~
#
# Construct the graph by mapping tokens of the source and target
# sentence to nodes. The complete Transformer graph is made up of three
# subgraphs:   
# 
# **Source language graph**. This is a complete graph, each
# token :math:`s_i` can attend to any other token :math:`s_j` (including
# self-loops). |image0|    
# **Target language graph**. The graph is
# half-complete, in that :math:`t_i` attends only to :math:`t_j` if
# :math:`i > j` (an output token can not depend on future words). |image1|    
# **Cross-language graph**. This is a bi-partitie graph, where there is
# an edge from every source token :math:`s_i` to every target token
# :math:`t_j`, meaning every target token can attend on source tokens.
# |image2|
#
# The full picture looks like this: |image3|
#
# Pre-build the graphs in dataset preparation stage.
#
# Message passing
# ~~~~~~~~~~~~~~~
#
# Once you define the graph structure, move on to defining the
# computation for message passing.
#
# Assuming that you have already computed all the queries :math:`q_i`, keys
# :math:`k_i` and values :math:`v_i`. For each node :math:`i` (no matter
# whether it is a source token or target token), you can decompose the
# attention computation into two steps:
#
# 1. **Message computation:** Compute attention score
#    :math:`\mathrm{score}_{ij}` between :math:`i` and all nodes :math:`j`
#    to be attended over, by taking the scaled-dot product between
#    :math:`q_i` and :math:`k_j`. The message sent from :math:`j` to
#    :math:`i` will consist of the score :math:`\mathrm{score}_{ij}` and
#    the value :math:`v_j`.
# 2. **Message aggregation:** Aggregate the values :math:`v_j` from all
#    :math:`j` according to the scores :math:`\mathrm{score}_{ij}`.
#
# Simple implementation
# ^^^^^^^^^^^^^^^^^^^^
#
# Message computation
# '''''''''''''''''''
#
# Compute ``score`` and send source node’s ``v`` to destination’s mailbox
#
# .. code:: 
#
#    def message_func(edges):
#        return {'score': ((edges.src['k'] * edges.dst['q'])
#                          .sum(-1, keepdim=True)),
#                'v': edges.src['v']}
#
# Message aggregation
# '''''''''''''''''''
#
# Normalize over all in-edges and weighted sum to get output
#
# .. code:: 
#
#    import torch as th
#    import torch.nn.functional as F
#
#    def reduce_func(nodes, d_k=64):
#        v = nodes.mailbox['v']
#        att = F.softmax(nodes.mailbox['score'] / th.sqrt(d_k), 1)
#        return {'dx': (att * v).sum(1)}
#
# Execute on specific edges
# '''''''''''''''''''''''''
#
# .. code:: 
#
#    import functools.partial as partial
#    def naive_propagate_attention(self, g, eids):
#        g.send_and_recv(eids, message_func, partial(reduce_func, d_k=self.d_k))
#
# Speeding up with built-in functions
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# To speed up the message passing process, use DGL’s built-in
# functions, including:
#
# - ``fn.src_mul_egdes(src_field, edges_field, out_field)`` multiplies
#   source’s attribute and edges attribute, and send the result to the
#   destination node’s mailbox keyed by ``out_field``.
# - ``fn.copy_edge(edges_field, out_field)`` copies edge’s attribute to
#   destination node’s mailbox.
# - ``fn.sum(edges_field, out_field)`` sums up
#   edge’s attribute and sends aggregation to destination node’s mailbox.
#
# Here, you assemble those built-in functions into ``propagate_attention``,
# which is also the main graph operation function in the final
# implementation. To accelerate it, break the ``softmax`` operation into
# the following steps. Recall that for each head there are two phases.
#
# 1. Compute attention score by multiply src node’s ``k`` and dst node’s
#    ``q``
#
#    -  ``g.apply_edges(src_dot_dst('k', 'q', 'score'), eids)``
#
# 2. Scaled Softmax over all dst nodes’ in-coming edges
#
#    -  Step 1: Exponentialize score with scale normalize constant
#
#       -  ``g.apply_edges(scaled_exp('score', np.sqrt(self.d_k)))``
#
#          .. math:: \textrm{score}_{ij}\leftarrow\exp{\left(\frac{\textrm{score}_{ij}}{ \sqrt{d_k}}\right)}
#
#    -  Step 2: Get the “values” on associated nodes weighted by “scores”
#       on in-coming edges of each node; get the sum of “scores” on
#       in-coming edges of each node for normalization. Note that here
#       :math:`\textrm{wv}` is not normalized.
#
#       -  ``msg: fn.src_mul_edge('v', 'score', 'v'), reduce: fn.sum('v', 'wv')``
#
#          .. math:: \textrm{wv}_j=\sum_{i=1}^{N} \textrm{score}_{ij} \cdot v_i
#
#       -  ``msg: fn.copy_edge('score', 'score'), reduce: fn.sum('score', 'z')``
#
#          .. math:: \textrm{z}_j=\sum_{i=1}^{N} \textrm{score}_{ij}
#
# The normalization of :math:`\textrm{wv}` is left to post processing.
#
# .. code:: 
#
#    def src_dot_dst(src_field, dst_field, out_field):
#        def func(edges):
#            return {out_field: (edges.src[src_field] * edges.dst[dst_field]).sum(-1, keepdim=True)}
#
#        return func
#
#    def scaled_exp(field, scale_constant):
#        def func(edges):
#            # clamp for softmax numerical stability
#            return {field: th.exp((edges.data[field] / scale_constant).clamp(-5, 5))}
#
#        return func
#
#
#    def propagate_attention(self, g, eids):
#        # Compute attention score
#        g.apply_edges(src_dot_dst('k', 'q', 'score'), eids)
#        g.apply_edges(scaled_exp('score', np.sqrt(self.d_k)))
#        # Update node state
#        g.send_and_recv(eids,
#                        [fn.src_mul_edge('v', 'score', 'v'), fn.copy_edge('score', 'score')],
#                        [fn.sum('v', 'wv'), fn.sum('score', 'z')])
#
# Preprocessing and postprocessing
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#
# In Transformer, data needs to be pre- and post-processed before and
# after the ``propagate_attention`` function.
#
# **Preprocessing** The preprocessing function ``pre_func`` first
# normalizes the node representations and then map them to a set of
# queries, keys and values, using self-attention as an example:
#
# .. math::
#
#
#    x \leftarrow \textrm{LayerNorm}(x) \\
#    [q, k, v] \leftarrow [W_q, W_k, W_v ]\cdot x
#
# **Postprocessing** The postprocessing function ``post_funcs`` completes
# the whole computation correspond to one layer of the transformer: 1.
# Normalize :math:`\textrm{wv}` and get the output of Multi-Head Attention
# Layer :math:`o`.
#
# .. math::
#
#
#    \textrm{wv} \leftarrow \frac{\textrm{wv}}{z} \\
#    o \leftarrow W_o\cdot \textrm{wv} + b_o
#
# add residual connection:
#
# .. math::
#
#
#    x \leftarrow x + o
#
# 2. Applying a two layer position-wise feed forward layer on :math:`x`
#    then add residual connection:
#
#    .. math::
#
#
#       x \leftarrow x + \textrm{LayerNorm}(\textrm{FFN}(x))
#
#    where :math:`\textrm{FFN}` refers to the feed forward function.
#
# .. code:: 
#
#    class Encoder(nn.Module):
#        def __init__(self, layer, N):
#            super(Encoder, self).__init__()
#            self.N = N
#            self.layers = clones(layer, N)
#            self.norm = LayerNorm(layer.size)
#
#        def pre_func(self, i, fields='qkv'):
#            layer = self.layers[i]
#            def func(nodes):
#                x = nodes.data['x']
#                norm_x = layer.sublayer[0].norm(x)
#                return layer.self_attn.get(norm_x, fields=fields)
#            return func
#
#        def post_func(self, i):
#            layer = self.layers[i]
#            def func(nodes):
#                x, wv, z = nodes.data['x'], nodes.data['wv'], nodes.data['z']
#                o = layer.self_attn.get_o(wv / z)
#                x = x + layer.sublayer[0].dropout(o)
#                x = layer.sublayer[1](x, layer.feed_forward)
#                return {'x': x if i < self.N - 1 else self.norm(x)}
#            return func
#
#    class Decoder(nn.Module):
#        def __init__(self, layer, N):
#            super(Decoder, self).__init__()
#            self.N = N
#            self.layers = clones(layer, N)
#            self.norm = LayerNorm(layer.size)
#
#        def pre_func(self, i, fields='qkv', l=0):
#            layer = self.layers[i]
#            def func(nodes):
#                x = nodes.data['x']
#                if fields == 'kv':
#                    norm_x = x # In enc-dec attention, x has already been normalized.
#                else:
#                    norm_x = layer.sublayer[l].norm(x)
#                return layer.self_attn.get(norm_x, fields)
#            return func
#
#        def post_func(self, i, l=0):
#            layer = self.layers[i]
#            def func(nodes):
#                x, wv, z = nodes.data['x'], nodes.data['wv'], nodes.data['z']
#                o = layer.self_attn.get_o(wv / z)
#                x = x + layer.sublayer[l].dropout(o)
#                if l == 1:
#                    x = layer.sublayer[2](x, layer.feed_forward)
#                return {'x': x if i < self.N - 1 else self.norm(x)}
#            return func
#
# This completes all procedures of one layer of encoder and decoder in
# Transformer.
#
# .. note::
#
#    The sublayer connection part is little bit different from the
#    original paper. However, this implementation is the same as `The Annotated
#    Transformer <http://nlp.seas.harvard.edu/2018/04/03/attention.html>`__
#    and
#    `OpenNMT <https://github.com/OpenNMT/OpenNMT-py/blob/cd29c1dbfb35f4a2701ff52a1bf4e5bdcf02802e/onmt/encoders/transformer.py>`__.
#
# Main class of Transformer graph
# -------------------------------
#
# The processing flow of Transformer can be seen as a 2-stage
# message-passing within the complete graph (adding pre- and post-
# processing appropriately): 1) self-attention in encoder, 2)
# self-attention in decoder followed by cross-attention between encoder
# and decoder, as shown below. |image4|
#
# .. code:: python
#
#    class Transformer(nn.Module):
#        def __init__(self, encoder, decoder, src_embed, tgt_embed, pos_enc, generator, h, d_k):
#            super(Transformer, self).__init__()
#            self.encoder, self.decoder = encoder, decoder
#            self.src_embed, self.tgt_embed = src_embed, tgt_embed
#            self.pos_enc = pos_enc
#            self.generator = generator
#            self.h, self.d_k = h, d_k
#
#        def propagate_attention(self, g, eids):
#            # Compute attention score
#            g.apply_edges(src_dot_dst('k', 'q', 'score'), eids)
#            g.apply_edges(scaled_exp('score', np.sqrt(self.d_k)))
#            # Send weighted values to target nodes
#            g.send_and_recv(eids,
#                            [fn.src_mul_edge('v', 'score', 'v'), fn.copy_edge('score', 'score')],
#                            [fn.sum('v', 'wv'), fn.sum('score', 'z')])
#
#        def update_graph(self, g, eids, pre_pairs, post_pairs):
#            "Update the node states and edge states of the graph."
#
#            # Pre-compute queries and key-value pairs.
#            for pre_func, nids in pre_pairs:
#                g.apply_nodes(pre_func, nids)
#            self.propagate_attention(g, eids)
#            # Further calculation after attention mechanism
#            for post_func, nids in post_pairs:
#                g.apply_nodes(post_func, nids)
#
#        def forward(self, graph):
#            g = graph.g
#            nids, eids = graph.nids, graph.eids
#
#            # Word Embedding and Position Embedding
#            src_embed, src_pos = self.src_embed(graph.src[0]), self.pos_enc(graph.src[1])
#            tgt_embed, tgt_pos = self.tgt_embed(graph.tgt[0]), self.pos_enc(graph.tgt[1])
#            g.nodes[nids['enc']].data['x'] = self.pos_enc.dropout(src_embed + src_pos)
#            g.nodes[nids['dec']].data['x'] = self.pos_enc.dropout(tgt_embed + tgt_pos)
#
#            for i in range(self.encoder.N):
#                # Step 1: Encoder Self-attention
#                pre_func = self.encoder.pre_func(i, 'qkv')
#                post_func = self.encoder.post_func(i)
#                nodes, edges = nids['enc'], eids['ee']
#                self.update_graph(g, edges, [(pre_func, nodes)], [(post_func, nodes)])
#
#            for i in range(self.decoder.N):
#                # Step 2: Dncoder Self-attention
#                pre_func = self.decoder.pre_func(i, 'qkv')
#                post_func = self.decoder.post_func(i)
#                nodes, edges = nids['dec'], eids['dd']
#                self.update_graph(g, edges, [(pre_func, nodes)], [(post_func, nodes)])
#                # Step 3: Encoder-Decoder attention
#                pre_q = self.decoder.pre_func(i, 'q', 1)
#                pre_kv = self.decoder.pre_func(i, 'kv', 1)
#                post_func = self.decoder.post_func(i, 1)
#                nodes_e, nodes_d, edges = nids['enc'], nids['dec'], eids['ed']
#                self.update_graph(g, edges, [(pre_q, nodes_d), (pre_kv, nodes_e)], [(post_func, nodes_d)])
#
#            return self.generator(g.ndata['x'][nids['dec']])
#
#
# .. note::
#
#    By calling ``update_graph`` function, you can create your own
#    Transformer on any subgraphs with nearly the same code. This
#    flexibility enables us to discover new, sparse structures (c.f. local attention
#    mentioned `here <https://arxiv.org/pdf/1508.04025.pdf>`__). Note in this
#    implementation you don't use mask or padding, which makes the logic
#    more clear and saves memory. The trade-off is that the implementation is
#    slower.
#
# Training
# --------
#
# This tutorial does not cover several other techniques such as Label
# Smoothing and Noam Optimizations mentioned in the original paper. For
# detailed description about these modules, read `The
# Annotated
# Transformer <http://nlp.seas.harvard.edu/2018/04/03/attention.html>`__
# written by Harvard NLP team.
#
# Task and the dataset
# ~~~~~~~~~~~~~~~~~~~~
#
# The Transformer is a general framework for a variety of NLP tasks. This tutorial focuses 
# on the sequence to sequence learning: it’s a typical case to illustrate how it works.
#
# As for the dataset, there are two example tasks: copy and sort, together
# with two real-world translation tasks: multi30k en-de task and wmt14
# en-de task.
#
# -  **copy dataset**: copy input sequences to output. (train/valid/test:
#    9000, 1000, 1000)
# -  **sort dataset**: sort input sequences as output. (train/valid/test:
#    9000, 1000, 1000)
# -  **Multi30k en-de**, translate sentences from En to De.
#    (train/valid/test: 29000, 1000, 1000)
# -  **WMT14 en-de**, translate sentences from En to De.
#    (Train/Valid/Test: 4500966/3000/3003)
#
# .. note::
#    Training with wmt14 requires multi-GPU support and is not available. Contributions are welcome!
#
# Graph building
# ~~~~~~~~~~~~~~
#
# **Batching** This is similar to the way you handle Tree-LSTM. Build a graph pool in
# advance, including all possible combination of input lengths and output
# lengths. Then for each sample in a batch, call ``dgl.batch`` to batch
# graphs of their sizes together in to a single large graph.
#
# You can wrap the process of creating graph pool and building
# BatchedGraph in ``dataset.GraphPool`` and
# ``dataset.TranslationDataset``.
#
# .. code:: python
#
#    graph_pool = GraphPool()
#
#    data_iter = dataset(graph_pool, mode='train', batch_size=1, devices=devices)
#    for graph in data_iter:
#        print(graph.nids['enc']) # encoder node ids
#        print(graph.nids['dec']) # decoder node ids
#        print(graph.eids['ee']) # encoder-encoder edge ids
#        print(graph.eids['ed']) # encoder-decoder edge ids
#        print(graph.eids['dd']) # decoder-decoder edge ids
#        print(graph.src[0]) # Input word index list
#        print(graph.src[1]) # Input positions
#        print(graph.tgt[0]) # Output word index list
#        print(graph.tgt[1]) # Ouptut positions
#        break
#
# Output:
#
# .. code::
#
#    tensor([0, 1, 2, 3, 4, 5, 6, 7, 8], device='cuda:0')
#    tensor([ 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], device='cuda:0')
#    tensor([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
#            18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
#            36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
#            54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
#            72, 73, 74, 75, 76, 77, 78, 79, 80], device='cuda:0')
#    tensor([ 81,  82,  83,  84,  85,  86,  87,  88,  89,  90,  91,  92,  93,  94,
#             95,  96,  97,  98,  99, 100, 101, 102, 103, 104, 105, 106, 107, 108,
#            109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122,
#            123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136,
#            137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150,
#            151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164,
#            165, 166, 167, 168, 169, 170], device='cuda:0')
#    tensor([171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184,
#            185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198,
#            199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212,
#            213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225],
#           device='cuda:0')
#    tensor([28, 25,  7, 26,  6,  4,  5,  9, 18], device='cuda:0')
#    tensor([0, 1, 2, 3, 4, 5, 6, 7, 8], device='cuda:0')
#    tensor([ 0, 28, 25,  7, 26,  6,  4,  5,  9, 18], device='cuda:0')
#    tensor([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], device='cuda:0')
#
# Put it all together
# -------------------
#
# Train a one-head transformer with one layer, 128 dimension on copy
# task. Set other parameters to the default.
#
# Inference module is not included in this tutorial. It
# requires beam search. For a full implementation, see the `GitHub
# repo <https://github.com/dmlc/dgl/tree/master/examples/pytorch/transformer>`__.
#
# .. code:: python
#
#    from tqdm import tqdm
#    import torch as th
#    import numpy as np
#
#    from loss import LabelSmoothing, SimpleLossCompute
#    from modules import make_model
#    from optims import NoamOpt
#    from dgl.contrib.transformer import get_dataset, GraphPool
#
#    def run_epoch(data_iter, model, loss_compute, is_train=True):
#        for i, g in tqdm(enumerate(data_iter)):
#            with th.set_grad_enabled(is_train):
#                output = model(g)
#                loss = loss_compute(output, g.tgt_y, g.n_tokens)
#        print('average loss: {}'.format(loss_compute.avg_loss))
#        print('accuracy: {}'.format(loss_compute.accuracy))
#
#    N = 1
#    batch_size = 128
#    devices = ['cuda' if th.cuda.is_available() else 'cpu']
#
#    dataset = get_dataset("copy")
#    V = dataset.vocab_size
#    criterion = LabelSmoothing(V, padding_idx=dataset.pad_id, smoothing=0.1)
#    dim_model = 128
#
#    # Create model
#    model = make_model(V, V, N=N, dim_model=128, dim_ff=128, h=1)
#
#    # Sharing weights between Encoder & Decoder
#    model.src_embed.lut.weight = model.tgt_embed.lut.weight
#    model.generator.proj.weight = model.tgt_embed.lut.weight
#
#    model, criterion = model.to(devices[0]), criterion.to(devices[0])
#    model_opt = NoamOpt(dim_model, 1, 400,
#                        th.optim.Adam(model.parameters(), lr=1e-3, betas=(0.9, 0.98), eps=1e-9))
#    loss_compute = SimpleLossCompute
#
#    att_maps = []
#    for epoch in range(4):
#        train_iter = dataset(graph_pool, mode='train', batch_size=batch_size, devices=devices)
#        valid_iter = dataset(graph_pool, mode='valid', batch_size=batch_size, devices=devices)
#        print('Epoch: {} Training...'.format(epoch))
#        model.train(True)
#        run_epoch(train_iter, model,
#                  loss_compute(criterion, model_opt), is_train=True)
#        print('Epoch: {} Evaluating...'.format(epoch))
#        model.att_weight_map = None
#        model.eval()
#        run_epoch(valid_iter, model,
#                  loss_compute(criterion, None), is_train=False)
#        att_maps.append(model.att_weight_map)
#
# Visualization
# -------------
#
# After training, you can visualize the attention that the Transformer generates
# on copy task.
#
# .. code:: python
#
#    src_seq = dataset.get_seq_by_id(VIZ_IDX, mode='valid', field='src')
#    tgt_seq = dataset.get_seq_by_id(VIZ_IDX, mode='valid', field='tgt')[:-1]
#    # visualize head 0 of encoder-decoder attention
#    att_animation(att_maps, 'e2d', src_seq, tgt_seq, 0)
#
# |image5| from the figure you see the decoder nodes gradually learns to
# attend to corresponding nodes in input sequence, which is the expected
# behavior.
#
# Multi-head attention
# ~~~~~~~~~~~~~~~~~~~~
#
# Besides the attention of a one-head attention trained on toy task. We
# also visualize the attention scores of Encoder’s Self Attention,
# Decoder’s Self Attention and the Encoder-Decoder attention of an
# one-Layer Transformer network trained on multi-30k dataset.
#
# From the visualization you see the diversity of different heads, which is what you would
# expect. Different heads learn different relations between word pairs.
#
# -  **Encoder Self-Attention** |image6|
#
# -  **Encoder-Decoder Attention** Most words in target sequence attend on
#    their related words in source sequence, for example: when generating
#    “See” (in De), several heads attend on “lake”; when generating
#    “Eisfischerhütte”, several heads attend on “ice”. |image7|
#
# -  **Decoder Self-Attention** Most words attend on their previous few
#    words. |image8|
#
# Adaptive Universal Transformer
# ------------------------------
#
# A recent research paper by Google, `Universal
# Transformer <https://arxiv.org/pdf/1807.03819.pdf>`__, is an example to
# show how ``update_graph`` adapts to more complex updating rules.
#
# The Universal Transformer was proposed to address the problem that
# vanilla Transformer is not computationally universal by introducing
# recurrence in Transformer:
#
# -  The basic idea of Universal Transformer is to repeatedly revise its
#    representations of all symbols in the sequence with each recurrent
#    step by applying a Transformer layer on the representations.
# -  Compared to vanilla Transformer, Universal Transformer shares weights
#    among its layers, and it does not fix the recurrence time (which
#    means the number of layers in Transformer).
#
# A further optimization employs an `adaptive computation time
# (ACT) <https://arxiv.org/pdf/1603.08983.pdf>`__ mechanism to allow the
# model to dynamically adjust the number of times the representation of
# each position in a sequence is revised (refereed to as **step**
# hereafter). This model is also known as the Adaptive Universal
# Transformer (AUT).
#
# In AUT, you maintain an active nodes list. In each step :math:`t`, we
# compute a halting probability: :math:`h (0<h<1)` for all nodes in this
# list by:
#
# .. math::  h^t_i = \sigma(W_h x^t_i + b_h)
#
# then dynamically decide which nodes are still active. A node is halted
# at time :math:`T` if and only if
# :math:`\sum_{t=1}^{T-1} h_t < 1 - \varepsilon \leq \sum_{t=1}^{T}h_t`.
# Halted nodes are removed from the list. The procedure proceeds until the
# list is empty or a pre-defined maximum step is reached. From DGL’s
# perspective, this means that the “active” graph becomes sparser over
# time.
#
# The final state of a node :math:`s_i` is a weighted average of
# :math:`x_i^t` by :math:`h_i^t`:
#
# .. math::  s_i = \sum_{t=1}^{T} h_i^t\cdot x_i^t
#
# In DGL, implement an algorithm by calling
# ``update_graph`` on nodes that are still active and edges associated
# with this nodes. The following code shows the Universal Transformer
# class in DGL:
#
# .. code:: 
#
#    class UTransformer(nn.Module):
#        "Universal Transformer(https://arxiv.org/pdf/1807.03819.pdf) with ACT(https://arxiv.org/pdf/1603.08983.pdf)."
#        MAX_DEPTH = 8
#        thres = 0.99
#        act_loss_weight = 0.01
#        def __init__(self, encoder, decoder, src_embed, tgt_embed, pos_enc, time_enc, generator, h, d_k):
#            super(UTransformer, self).__init__()
#            self.encoder,  self.decoder = encoder, decoder
#            self.src_embed, self.tgt_embed = src_embed, tgt_embed
#            self.pos_enc, self.time_enc = pos_enc, time_enc
#            self.halt_enc = HaltingUnit(h * d_k)
#            self.halt_dec = HaltingUnit(h * d_k)
#            self.generator = generator
#            self.h, self.d_k = h, d_k
#
#        def step_forward(self, nodes):
#            # add positional encoding and time encoding, increment step by one
#            x = nodes.data['x']
#            step = nodes.data['step']
#            pos = nodes.data['pos']
#            return {'x': self.pos_enc.dropout(x + self.pos_enc(pos.view(-1)) + self.time_enc(step.view(-1))),
#                    'step': step + 1}
#
#        def halt_and_accum(self, name, end=False):
#            "field: 'enc' or 'dec'"
#            halt = self.halt_enc if name == 'enc' else self.halt_dec
#            thres = self.thres
#            def func(nodes):
#                p = halt(nodes.data['x'])
#                sum_p = nodes.data['sum_p'] + p
#                active = (sum_p < thres) & (1 - end)
#                _continue = active.float()
#                r = nodes.data['r'] * (1 - _continue) + (1 - sum_p) * _continue
#                s = nodes.data['s'] + ((1 - _continue) * r + _continue * p) * nodes.data['x']
#                return {'p': p, 'sum_p': sum_p, 'r': r, 's': s, 'active': active}
#            return func
#
#        def propagate_attention(self, g, eids):
#            # Compute attention score
#            g.apply_edges(src_dot_dst('k', 'q', 'score'), eids)
#            g.apply_edges(scaled_exp('score', np.sqrt(self.d_k)), eids)
#            # Send weighted values to target nodes
#            g.send_and_recv(eids,
#                            [fn.src_mul_edge('v', 'score', 'v'), fn.copy_edge('score', 'score')],
#                            [fn.sum('v', 'wv'), fn.sum('score', 'z')])
#
#        def update_graph(self, g, eids, pre_pairs, post_pairs):
#            "Update the node states and edge states of the graph."
#            # Pre-compute queries and key-value pairs.
#            for pre_func, nids in pre_pairs:
#                g.apply_nodes(pre_func, nids)
#            self.propagate_attention(g, eids)
#            # Further calculation after attention mechanism
#            for post_func, nids in post_pairs:
#                g.apply_nodes(post_func, nids)
#
#        def forward(self, graph):
#            g = graph.g
#            N, E = graph.n_nodes, graph.n_edges
#            nids, eids = graph.nids, graph.eids
#
#            # embed & pos
#            g.nodes[nids['enc']].data['x'] = self.src_embed(graph.src[0])
#            g.nodes[nids['dec']].data['x'] = self.tgt_embed(graph.tgt[0])
#            g.nodes[nids['enc']].data['pos'] = graph.src[1]
#            g.nodes[nids['dec']].data['pos'] = graph.tgt[1]
#
#            # init step
#            device = next(self.parameters()).device
#            g.ndata['s'] = th.zeros(N, self.h * self.d_k, dtype=th.float, device=device)    # accumulated state
#            g.ndata['p'] = th.zeros(N, 1, dtype=th.float, device=device)                    # halting prob
#            g.ndata['r'] = th.ones(N, 1, dtype=th.float, device=device)                     # remainder
#            g.ndata['sum_p'] = th.zeros(N, 1, dtype=th.float, device=device)                # sum of pondering values
#            g.ndata['step'] = th.zeros(N, 1, dtype=th.long, device=device)                  # step
#            g.ndata['active'] = th.ones(N, 1, dtype=th.uint8, device=device)                # active
#
#            for step in range(self.MAX_DEPTH):
#                pre_func = self.encoder.pre_func('qkv')
#                post_func = self.encoder.post_func()
#                nodes = g.filter_nodes(lambda v: v.data['active'].view(-1), nids['enc'])
#                if len(nodes) == 0: break
#                edges = g.filter_edges(lambda e: e.dst['active'].view(-1), eids['ee'])
#                end = step == self.MAX_DEPTH - 1
#                self.update_graph(g, edges,
#                                  [(self.step_forward, nodes), (pre_func, nodes)],
#                                  [(post_func, nodes), (self.halt_and_accum('enc', end), nodes)])
#
#            g.nodes[nids['enc']].data['x'] = self.encoder.norm(g.nodes[nids['enc']].data['s'])
#
#            for step in range(self.MAX_DEPTH):
#                pre_func = self.decoder.pre_func('qkv')
#                post_func = self.decoder.post_func()
#                nodes = g.filter_nodes(lambda v: v.data['active'].view(-1), nids['dec'])
#                if len(nodes) == 0: break
#                edges = g.filter_edges(lambda e: e.dst['active'].view(-1), eids['dd'])
#                self.update_graph(g, edges,
#                                  [(self.step_forward, nodes), (pre_func, nodes)],
#                                  [(post_func, nodes)])
#
#                pre_q = self.decoder.pre_func('q', 1)
#                pre_kv = self.decoder.pre_func('kv', 1)
#                post_func = self.decoder.post_func(1)
#                nodes_e = nids['enc']
#                edges = g.filter_edges(lambda e: e.dst['active'].view(-1), eids['ed'])
#                end = step == self.MAX_DEPTH - 1
#                self.update_graph(g, edges,
#                                  [(pre_q, nodes), (pre_kv, nodes_e)],
#                                  [(post_func, nodes), (self.halt_and_accum('dec', end), nodes)])
#
#            g.nodes[nids['dec']].data['x'] = self.decoder.norm(g.nodes[nids['dec']].data['s'])
#            act_loss = th.mean(g.ndata['r']) # ACT loss
#
#            return self.generator(g.ndata['x'][nids['dec']]), act_loss * self.act_loss_weight
#
# Call ``filter_nodes`` and ``filter_edge`` to find nodes/edges
# that are still active:
#
# .. note::
# 
#    - :func:`~dgl.DGLGraph.filter_nodes` takes a predicate and a node
#      ID list/tensor as input, then returns a tensor of node IDs that satisfy
#      the given predicate.     
#    - :func:`~dgl.DGLGraph.filter_edges` takes a predicate
#      and an edge ID list/tensor as input, then returns a tensor of edge IDs
#      that satisfy the given predicate.
#
# For the full implementation, see the `GitHub
# repo <https://github.com/dmlc/dgl/tree/master/examples/pytorch/transformer/modules/act.py>`__.
#
# The figure below shows the effect of Adaptive Computational
# Time. Different positions of a sentence were revised different times.
#
# |image9|
#
# You can also visualize the dynamics of step distribution on nodes during the
# training of AUT on sort task(reach 99.7% accuracy), which demonstrates
# how AUT learns to reduce recurrence steps during training. |image10|
#
# .. |image0| image:: https://i.imgur.com/zV5LmTX.png
# .. |image1| image:: https://i.imgur.com/dETQMMx.png
# .. |image2| image:: https://i.imgur.com/hnGP229.png
# .. |image3| image:: https://i.imgur.com/Hj2rRGT.png
# .. |image4| image:: https://i.imgur.com/zlUpJ41.png
# .. |image5| image:: https://s1.ax1x.com/2018/12/06/F126xI.gif
# .. |image6| image:: https://i.imgur.com/HjYb7F2.png
# .. |image7| image:: https://i.imgur.com/383J5O5.png
# .. |image8| image:: https://i.imgur.com/c0UWB1V.png
# .. |image9| image:: https://s1.ax1x.com/2018/12/06/F1sGod.png
# .. |image10| image:: https://s1.ax1x.com/2018/12/06/F1r8Cq.gif
#
# .. note::
#     The notebook itself is not executable due to many dependencies.
#     Download `7_transformer.py <https://s3.us-east-2.amazonaws.com/dgl.ai/tutorial/7_transformer.py>`__, 
#     and copy the python script to directory ``examples/pytorch/transformer`` 
#     then run ``python 7_transformer.py`` to see how it works.
