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+# Design: Sequence Decoder Generating LoDTensors
+In tasks such as machine translation and image to text, 
+a [sequence decoder](https://github.com/PaddlePaddle/book/blob/develop/08.machine_translation/README.md) is necessary to generate sequences.
+
+This documentation describes how to implement the sequence decoder as an operator.
+
+## Beam Search based Decoder
+The [beam search algorithm](https://en.wikipedia.org/wiki/Beam_search) is necessary when generating sequences, 
+it is a heuristic search algorithm that explores the paths by expanding the most promising node in a limited set.
+
+In the old version of PaddlePaddle, a C++ class `RecurrentGradientMachine` implements the general sequence decoder based on beam search, 
+due to the complexity, the implementation relays on a lot of special data structures, 
+quite trivial and hard to be customized by users.
+
+There are a lot of heuristic tricks in the sequence generation tasks, 
+so the flexibility of sequence decoder is very important to users.
+
+During PaddlePaddle's refactoring work,
+some new concept is proposed such as [LoDTensor](https://github.com/PaddlePaddle/Paddle/blob/develop/paddle/framework/lod_tensor.md) and [TensorArray](https://github.com/PaddlePaddle/Paddle/blob/develop/doc/design/tensor_array.md) that can better support sequence usage,
+and they can help to make the implementation of beam search based sequence decoder **more transparent and modular** .
+
+For example, the RNN sates, candidates IDs and probabilities of beam search can be represented as `LoDTensors`;
+the selected candidate's IDs in each time step can be stored in a `TensorArray`, and `Packed` to the sentences translated.
+
+## Changing LoD's absolute offset to relative offsets
+The current `LoDTensor` is designed to store levels of variable-length sequences,
+it stores several arrays of integers each represents a level.
+
+The integers in each level represents the begin and end (not inclusive) offset of a sequence **in the underlying tensor**, 
+let's call this format the **absolute-offset LoD** for clear.
+
+The relative-offset LoD can fast retrieve any sequence but fails to represent empty sequences, for example, a two-level LoD is as follows
+```python
+[[0, 3, 9]
+ [0, 2, 3, 3, 3, 9]]
+```
+The first level tells that there are two sequences:
+- the first's offset is `[0, 3)`
+- the second's offset is `[3, 9)`
+
+while on the second level, there are several empty sequences that both begin and end at `3`.
+It is impossible to tell how many empty second-level sequences exist in the first-level sequences.
+
+There are many scenarios that relay on empty sequence representation,
+such as machine translation or image to text, one instance has no translations or the empty candidate set for a prefix.
+
+So let's introduce another format of LoD, 
+it stores **the offsets of the lower level sequences** and is called **relative-offset** LoD.
+
+For example, to represent the same sequences of the above data
+
+```python
+[[0, 3, 6]
+ [0, 2, 3, 3, 3, 9]]
+```
+
+the first level represents that there are two sequences, 
+their offsets in the second-level LoD is `[0, 3)` and `[3, 5)`.
+
+The second level is the same with the relative offset example because the lower level is a tensor.
+It is easy to find out the second sequence in the first-level LoD has two empty sequences.
+
+The following demos are based on relative-offset LoD.
+
+## Usage in a simple machine translation model
+Let's start from a simple machine translation model that is simplified from [machine translation chapter](https://github.com/PaddlePaddle/book/tree/develop/08.machine_translation) to draw a simple blueprint of what a sequence decoder can do and how to use it.
+
+The model has an encoder that learns the semantic vector from a sequence,
+and a decoder which uses the sequence decoder to generate new sentences.
+
+**Encoder**
+```python
+import paddle as pd
+
+dict_size = 8000
+source_dict_size = dict_size
+target_dict_size = dict_size
+word_vector_dim = 128
+encoder_dim = 128
+decoder_dim = 128
+beam_size = 5
+max_length = 120
+
+# encoder
+src_word_id = pd.data(
+    name='source_language_word',
+    type=pd.data.integer_value_sequence(source_dict_dim))
+src_embedding = pd.embedding(size=source_dict_size, size=word_vector_dim)
+
+src_word_vec = pd.lookup(src_embedding, src_word_id)
+
+encoder_out_seq = pd.gru(input=src_word_vec, size=encoder_dim)
+
+encoder_ctx = pd.last_seq(encoder_out_seq)
+# encoder_ctx_proj is the learned semantic vector
+encoder_ctx_proj = pd.fc(
+    encoder_ctx, size=decoder_dim, act=pd.activation.Tanh(), bias=None)
+```
+
+**Decoder**
+
+```python
+def generate():
+    decoder = pd.while_loop()
+    with decoder.step():
+        decoder_mem = decoder.memory(init=encoder_ctx)  # mark the memory
+        generated_ids = decoder.memory() # TODO init to batch_size <s>s
+        generated_scores = decoder.memory() # TODO init to batch_size 1s or 0s
+
+        target_word = pd.lookup(trg_embedding, gendrated_ids)
+        # expand encoder_ctx's batch to fit target_word's lod
+        # for example
+        # decoder_mem.lod is
+        # [[0 1 3],
+        #  [0 1 3 6]]
+        # its tensor content is [a1 a2 a3 a4 a5]
+        # which means there are 2 sentences to translate
+        #   - the first sentence has 1 translation prefixes, the offsets are [0, 1)
+        #   - the second sentence has 2 translation prefixes, the offsets are [1, 3) and [3, 6)
+        # the target_word.lod is 
+        # [[0, 1, 6]
+        #  [0, 2, 4, 7, 9 12]]
+        # which means 2 sentences to translate, each has 1 and 5 prefixes
+        # the first prefix has 2 candidates
+        # the following has 2, 3, 2, 3 candidates
+        # the encoder_ctx_expanded's content will be
+        # [a1 a1 a2 a2 a3 a3 a3 a4 a4 a5 a5 a5]
+        encoder_ctx_expanded = pd.lod_expand(encoder_ctx, target_word)
+        decoder_input = pd.fc(
+            act=pd.activation.Linear(),
+            input=[target_word, encoder_ctx],
+            size=3 * decoder_dim)
+        gru_out, cur_mem = pd.gru_step(
+            decoder_input, mem=decoder_mem, size=decoder_dim)
+        scores = pd.fc(
+            gru_out,
+            size=trg_dic_size,
+            bias=None,
+            act=pd.activation.Softmax())
+        # K is an config
+        topk_scores, topk_ids = pd.top_k(scores, K)
+        topk_generated_scores = pd.add_scalar(topk_scores, generated_scores)
+
+        selected_ids, selected_generation_scores = decoder.beam_search(
+            topk_ids, topk_generated_scores)
+
+        # update the states
+        decoder_mem.update(cur_mem)  # tells how to update state
+        generated_ids.update(selected_ids)
+        generated_scores.update(selected_generation_scores)
+
+        decoder.output(selected_ids)
+        decoder.output(selected_generation_scores)
+
+translation_ids, translation_scores = decoder()
+```
+The `decoder.beam_search` is a operator that given the candidates and the scores of translations including the candidates,
+return the result of the beam search algorithm.
+
+In this way, users can customize anything on the inputs or outputs of beam search, for example, two ways to prune some translation prefixes
+
+1. meke the correspondind elements in `topk_generated_scores` zero or some small values, beam_search will discard this candidate.
+2. remove some specific candidate in `selected_ids`
+3. get the final `translation_ids`, remove the translation sequence in it.
+
+The implementation of sequence decoder can reuse the C++ class [RNNAlgorithm](https://github.com/Superjom/Paddle/blob/68cac3c0f8451fe62a4cdf156747d6dc0ee000b3/paddle/operators/dynamic_recurrent_op.h#L30),
+so the python syntax is quite similar to a [RNN](https://github.com/Superjom/Paddle/blob/68cac3c0f8451fe62a4cdf156747d6dc0ee000b3/doc/design/block.md#blocks-with-for-and-rnnop).
+
+Both of them are two-level `LoDTensors`
+
+- the first level represents `batch_size` of (source) sentences;
+- the second level represents the candidate ID sets for translation prefix.
+
+for example, 3 source sentences to translate, and has 2, 3, 1 candidates.
+
+Unlike an RNN, in sequence decoder, the previous state and the current state have different LoD and shape,
+a `lod_expand` operator is used to expand the LoD of the previous state to fit the current state.
+
+For example, the previous state
+
+* LoD is `[0, 1, 3][0, 2, 5, 6]`
+* content of tensor is `a1 a2 b1 b2 b3 c1`
+
+the current state stored in `encoder_ctx_expanded`
+
+* LoD is `[0, 2, 7][0 3 5 8 9 11 11]`
+* the content is 
+  - a1 a1 a1 (a1 has 3 candidates, so the state should be copied 3 times for each candidates)
+  - a2 a2
+  - b1 b1 b1
+  - b2
+  - b3 b3
+  - None (c1 has 0 candidates, so c1 is dropped)
+
+Benefit from the relative offset LoD, empty candidate set can be represented naturally.
+
+the status in each time step can be stored in `TensorArray`, and `Pack`ed to a final LoDTensor, the corresponding syntax is 
+
+```python
+decoder.output(selected_ids)
+decoder.output(selected_generation_scores)
+```
+
+the `selected_ids` is the candidate ids for the prefixes, 
+it will be `Packed` by `TensorArray` to a two-level `LoDTensor`,
+the first level represents the source sequences,
+the second level represents generated sequences.
+
+Pack the `selected_scores` will get a `LoDTensor` that stores scores of each candidate of translations.
+
+Pack the `selected_generation_scores` will get a `LoDTensor`, and each tail is the probability of the translation.
+
+## LoD and shape changes during decoding
+<p align="center">
+  <img src="./images/LOD-and-shape-changes-during-decoding.jpg"/>
+</p>
+
+According the image above, the only phrase to change LoD is beam search.
+
+## Beam search design
+The beam search algorthm will be implemented as one method of the sequence decoder, it has 3 inputs
+
+1. `topk_ids`, top K candidate ids for each prefix.
+2. `topk_scores`, the corresponding scores for `topk_ids`
+3. `generated_scores`, the score of the prefixes.
+
+All of the are LoDTensors, so that the sequence affilication is clear.
+Beam search will keep a beam for each prefix and select a smaller candidate set for each prefix.
+
+It will return three variables
+
+1. `selected_ids`, the final candidate beam search function selected for the next step.
+2. `selected_scores`, the scores for the candidates.
+3. `generated_scores`, the updated scores for each prefixes (with the new candidates appended).
+
+## Introducing the LoD-based `Pack` and `Unpack` methods in `TensorArray`
+The `selected_ids`, `selected_scores` and `generated_scores` are LoDTensors,
+and they exist in each time step,
+so it is natural to store them in arrays.
+
+Currently, PaddlePaddle has a module called `TensorArray` which can store an array of tensors,
+the results of beam search are better to store in a `TensorArray`.
+
+The `Pack` and `UnPack` in `TensorArray` are used to package tensors in the array to a `LoDTensor` or split the `LoDTensor` to an array of tensors. 
+It needs some extensions to support pack or unpack an array of `LoDTensors`.