backward.py 30.0 KB
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#   Copyright (c) 2018 PaddlePaddle Authors. All Rights Reserved.
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#
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# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
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#
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#     http://www.apache.org/licenses/LICENSE-2.0
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#
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# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

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from paddle.fluid import framework as framework
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from . import core
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import collections
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import copy
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import six
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from .. import compat as cpt
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from . import unique_name
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__all__ = ['append_backward']
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def _rename_arg_(op_descs, old_name, new_name, begin_idx=None, end_idx=None):
    """
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    Traverse all ops in op_descs[begin_idx : end_idx],
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    if any op has inputs/outputs named "old_name", rename it as 'new_name'
    """
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    if begin_idx is None:
        begin_idx = 0
    if end_idx is None:
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        end_idx = len(op_descs)
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    for i in range(begin_idx, end_idx):
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        op_desc = op_descs[i]
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        if isinstance(op_desc, tuple):
            op_desc = op_desc[0]
        op_desc.rename_input(old_name, new_name)
        op_desc.rename_output(old_name, new_name)
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def _create_op_desc_(op_type, inputs, outputs, attrs):
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    """
    Create a C++ OpDesc object with specified inputs, outputs and attributes.
    """
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    op_desc = core.OpDesc()
    op_desc.set_type(op_type)
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    for para, args in six.iteritems(inputs):
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        op_desc.set_input(
            para,
            list(
                map(lambda arg: arg.decode() if isinstance(arg, six.binary_type) else arg,
                    args)))
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    for para, args in six.iteritems(outputs):
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        op_desc.set_output(
            para,
            list(
                map(lambda arg: arg.decode() if isinstance(arg, six.binary_type) else arg,
                    args)))
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    op_role_attr_name = core.op_proto_and_checker_maker.kOpRoleAttrName()

    if op_role_attr_name not in attrs:
        attrs[
            op_role_attr_name] = core.op_proto_and_checker_maker.OpRole.Backward
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    for name, val in six.iteritems(attrs):
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        if isinstance(val, framework.Block):
            op_desc.set_block_attr(name, val.desc)
        else:
            op_desc.set_attr(name, val)
    return op_desc


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def _infer_var_data_type_(grad_var_name, block):
    """
    Infer the data type of given grad variable
    """
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    grad_var = block.desc.find_var(cpt.to_bytes(grad_var_name))
    fwd_name = _strip_grad_suffix_(grad_var_name)
    if block.desc.has_var_recursive(cpt.to_bytes(fwd_name)):
        fwd_var = block.desc.find_var_recursive(cpt.to_bytes(fwd_name))
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        grad_var.set_dtype(fwd_var.dtype())
    else:
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        grad_var.set_dtype(core.VarDesc.VarType.FP32)
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def _all_in_set_(cands, s):
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    """
    Test if all elements of 'cands' are in set 's'
    """
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    if len(cands) == 0:
        return False
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    for c in cands:
        if not c in s:
            return False
    return True


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def _some_in_set_(cands, s):
    """
    Test if some elements of 'cands' are in set 's'
    """
    if len(cands) == 0:
        return False
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    literal_set = cpt.to_text(s)
    literal_cands = cpt.to_text(cands)
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    for c in literal_cands:
        if c in literal_set:
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            return True
    return False


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def _strip_grad_suffix_(name):
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    """
    Strip the grad suffix from the given varibale name
    e.g. x@GRAD ==> x
         y@GRAD@RENAME@1 ==> y
    """
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    name = cpt.to_text(name)
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    pos = name.find(core.grad_var_suffix())
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    return name[:pos] if pos != -1 else name
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def _append_grad_suffix_(name):
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    """
    Append grad suffix to the given variable name
    e.g. x ==> x@GRAD
    """
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    return cpt.to_text(name) + core.grad_var_suffix()
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def _addup_repetitive_outputs_(op_descs):
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    """
    In backward part, an variable may be the output of more than one ops.
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    And one op may yield its multiple outputs to the same variable.
    In these cases, the variable should be the accumulation of all the outputs.
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    `sum_op`s are added to implement the accumulate.
    """
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    pending_sum_ops = []
    var_rename_count = collections.defaultdict(int)
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    renamed_vars = collections.defaultdict(list)
    for idx, op_desc in enumerate(op_descs):
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        for var_name in op_desc.input_arg_names():
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            if len(renamed_vars[var_name]) > 1:
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                pending_sum_ops.append((_create_op_desc_(
                    "sum", {"X": renamed_vars[var_name]}, {"Out": [var_name]},
                    {"use_mkldnn": False}), idx))
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                renamed_vars[var_name] = [var_name]
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        for param_idx, param_name in enumerate(op_desc.output_names()):
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            arg_names = op_desc.output(param_name)
            for arg_idx, var_name in enumerate(arg_names):
                if var_name == core.empty_var_name(
                ) or var_name in op_desc.input_arg_names():
                    # empty variable or inplace op
                    continue
                if len(renamed_vars[var_name]) == 0:
                    # it's the first time we get the variable
                    renamed_vars[var_name] = [var_name]
                else:
                    if len(renamed_vars[var_name]) == 1:
                        new_name = var_name + "@RENAME@" + \
                            str(var_rename_count[var_name])
                        var_rename_count[var_name] += 1
                        # rename original var_name
                        renamed_vars[var_name][0] = new_name
                        _rename_arg_(op_descs, var_name, new_name, 0, idx)
                        _rename_arg_(pending_sum_ops, var_name, new_name)

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                        for p in op_desc.output_names()[:param_idx]:
                            p_arg_names = op_desc.output(p)
                            if var_name in p_arg_names:
                                op_desc.set_output(p, [
                                    new_name if x == var_name else x
                                    for x in p_arg_names
                                ])

                        arg_names = [
                            new_name if x == var_name else x
                            for x in arg_names[:arg_idx]
                        ] + arg_names[arg_idx:]

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                    new_name = var_name + "@RENAME@" + \
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                        str(var_rename_count[var_name])
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                    var_rename_count[var_name] += 1
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                    arg_names[arg_idx] = new_name
                    op_desc.set_output(param_name, arg_names)
                    renamed_vars[var_name].append(new_name)
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    for var_name, inputs in six.iteritems(renamed_vars):
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        if len(inputs) > 1:
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            pending_sum_ops.append(
                (_create_op_desc_("sum", {"X": inputs}, {"Out": [var_name]},
                                  {"use_mkldnn": False}), len(op_descs)))
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    # sum_op descs are sorted according to their insert position
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    for p in reversed(pending_sum_ops):
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        op_descs.insert(p[1], p[0])

    return op_descs


def _remove_no_grad_branch_(op_descs, no_grad_set):
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    """
    Remove unnecessary grad ops
    A grad op can be removed in two cases:
        1. all outputs of the grad op are in 'no_grad_set'
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        2. all grad inputs of the grad op are in 'no_grad_set'
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    """
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    def _op_can_be_removed_(op_desc, no_grad_set):
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        out_arg_names = op_desc.output_arg_names()
        if len(out_arg_names) == 0 or _all_in_set_(out_arg_names, no_grad_set):
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            return True
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        if _all_in_set_([
                name for name in op_desc.input_arg_names()
                if name.find(core.grad_var_suffix()) != -1
        ], no_grad_set):
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            no_grad_set.update(out_arg_names)
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            return True
        return False

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    # Remove ops whose outputs are all in no_grad_dict
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    op_descs = [
        op_desc for op_desc in op_descs
        if not _op_can_be_removed_(op_desc, no_grad_set)
    ]
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    # Insert fill_zeros_like_op
    to_insert = []
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    for idx, op_desc in enumerate(op_descs):
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        for arg in op_desc.input_arg_names():
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            if core.grad_var_suffix() in arg and arg in no_grad_set:
                to_insert.append((_create_op_desc_("fill_zeros_like", {
                    "X": [_strip_grad_suffix_(arg)]
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                }, {"Out": [arg]}, {}), idx))
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    list([op_descs.insert(p[1], p[0]) for p in reversed(to_insert)])
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    return op_descs


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from .proto import framework_pb2
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def serialize_op_decs(op_desc):
    protostr = op_desc.serialize_to_string()
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    proto = framework_pb2.OpDesc.FromString(six.binary_type(protostr))
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    return proto.__str__()


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def _callback_lookup_(op):
    """
    Only used in _append_backward_ops_
    Build and returns a callback function for certain op. For example

    parallel_do:           AllReduce

    :param op:
    :return: callback function
    """
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    if op.type == 'parallel_do' and op.attr('use_nccl'):
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        all_vars = op.block.vars
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        param_names = set(op.input('parameters'))
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        param_names = [
            name for name in param_names
            if all_vars[name].stop_gradient is False
        ]
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        param_grad_names = [n + "@GRAD" for n in param_names]

        class ParallelDoCallBack(object):
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            def __init__(self, param_grad_names, parallel_scopes_name):
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                self.has_inserted_nccl_init = False
                self.param_grad_names = param_grad_names
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                self.parallel_scopes_name = parallel_scopes_name
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            def __call__(self, block, context):
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                if not self.has_inserted_nccl_init:
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                    op_desc = _create_op_desc_(
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                        "ncclInit",
                        {"parallel_scopes": self.parallel_scopes_name},
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                        {"Communicator": ['nccl_com__do_not_change_']}, {})
                    block.program.global_block().desc.append_op().copy_from(
                        op_desc)
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                    self.has_inserted_nccl_init = True

                current_op_desc = context["__current_op_desc__"]
                for o_param in current_op_desc.output_names():
                    for o_argu in current_op_desc.output(o_param):
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                        if o_argu in self.param_grad_names:
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                            allreduce_out_name = o_argu + "__nccl_all_reduce__"
                            op_desc = _create_op_desc_(
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                                "ncclReduce",
                                {
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                                    "X": [o_argu],
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                                    "Communicator":
                                    ['nccl_com__do_not_change_']
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                                },
                                {"Out": [allreduce_out_name]},
                                {"reduction": "ncclSum",
                                 "root": 0}, )
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                            block.desc.append_op().copy_from(op_desc)

                            op_desc = _create_op_desc_(
                                "assign", {"X": [allreduce_out_name]},
                                {"Out": [o_argu]}, {})
                            block.desc.append_op().copy_from(op_desc)
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        return ParallelDoCallBack(param_grad_names,
                                  op.output("parallel_scopes"))
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    else:
        return None


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def _append_backward_ops_(block,
                          ops,
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                          target_block,
                          no_grad_dict,
                          grad_to_var,
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                          callbacks=None):
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    """
    Create all grad ops, and insert them into given block

    Args:
        block(Block): the block where forward ops are
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        ops(Op): the forward operators whose backward ops need to be added
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        target_block(Block): the block which is going to hold new generated grad ops
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        no_grad_dict(dict):
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            key(int)  block index
            val(set) a set of varibale names. These varibales have no gradient
        grad_to_var(dict)(output argument):
            key(str): grad variable name
            val(str): corresponding forward variable name
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        callback(callable object): a callable object used to decorate new generated grad ops
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    """
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    if callbacks is not None:
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        assert (isinstance(callbacks, list))
        for cb in callbacks:
            if not hasattr(cb, '__call__'):
                raise ValueError("'callback' must be a callable object.")
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    # grad_op_descs holds created grad_op, and will be appended to target_block
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    grad_op_descs = []
    program = block.program
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    for op in reversed(ops):
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        grad_sub_block_list = []
        # If the op has its own sub-block, deal with the sub-block first
        if op.has_attr("sub_block"):
            sub_block = program.block(op.block_attr("sub_block"))
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            grad_sub_block = program.create_block()
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            grad_sub_block._set_forward_block_idx(sub_block.idx)
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            cb = _callback_lookup_(op)
            if cb is not None:
                if callbacks is None:
                    new_callbacks = [cb]
                else:
                    new_callbacks = callbacks + [_callback_lookup_(op)]
                _append_backward_ops_(sub_block, sub_block.ops, grad_sub_block,
                                      no_grad_dict, grad_to_var, new_callbacks)
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            else:
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                _append_backward_ops_(sub_block, sub_block.ops, grad_sub_block,
                                      no_grad_dict, grad_to_var, callbacks)
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            program.rollback()
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            grad_sub_block_list.append(grad_sub_block.desc)

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        # Getting op's corresponding grad_op
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        grad_op_desc, op_grad_to_var = core.get_grad_op_desc(
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            op.desc, cpt.to_text(no_grad_dict[block.idx]), grad_sub_block_list)
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        grad_op_descs.extend(grad_op_desc)
        grad_to_var.update(op_grad_to_var)

    grad_op_descs = _addup_repetitive_outputs_(grad_op_descs)

    grad_op_descs = _remove_no_grad_branch_(grad_op_descs,
                                            no_grad_dict[block.idx])
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    # append op_desc in grad_op_descs to target_block
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    op_role_attr_name = core.op_proto_and_checker_maker.kOpRoleAttrName()
    backward = core.op_proto_and_checker_maker.OpRole.Backward
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    for op_desc in grad_op_descs:
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        new_op_desc = target_block.desc.append_op()
        new_op_desc.copy_from(op_desc)
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        new_op_desc.set_attr(op_role_attr_name, backward)
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        grad_to_var["__current_op_desc__"] = new_op_desc
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        if callbacks is not None:
            assert (isinstance(callbacks, list))
            for cb in callbacks:
                cb(block=target_block, context=grad_to_var)
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def _append_backward_vars_(block, start_op_idx, grad_to_var, grad_info_map):
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    """
    Create new variables required by backward pass.

    Args:
        block(Block): the block where new variables will be created
        start_op_idx(int): Only variables required by ops in block.ops[start_op_idx : ] will be created
        grad_to_var(dict):
            key(str): grad variable name
            val(str): corresponding forward variable name
            In most cases, this dict is generated by _append_backward_ops_()
        grad_info_map(dict)(output argument):
            key(str): forward variable name
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            val(tuple): a tuple of (str, Block), str is the corresponding grad name, Block is the block containing grad variable
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    """
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    for op_idx in range(start_op_idx, block.desc.op_size()):
        op_desc = block.desc.op(op_idx)
        if op_desc.has_attr("sub_block"):
            sub_block = block.program.block(op_desc.block_attr("sub_block"))
            _append_backward_vars_(sub_block, 0, grad_to_var, grad_info_map)
        new_vars = set()
        # create new gradient variables
        for grad_var_name in op_desc.output_arg_names():
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            if block.desc.has_var_recursive(cpt.to_bytes(
                    grad_var_name)) or grad_var_name == core.empty_var_name():
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                continue
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            block.desc.var(cpt.to_bytes(grad_var_name))
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            new_vars.add(grad_var_name)
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            if grad_var_name not in grad_to_var:
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                continue
            grad_info_map[grad_to_var[grad_var_name]] = (grad_var_name, block)
        # infer_shape and infer_type
        op_desc.infer_var_type(block.desc)
        op_desc.infer_shape(block.desc)
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        # ncclInit dones't need to set data_type
        if op_desc.type() == 'ncclInit':
            continue
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        for arg in op_desc.output_arg_names():
            if arg in new_vars:
                _infer_var_data_type_(arg, block)
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def _rename_grad_(block, start_op_idx, grad_to_var, target_grad_map):
    var_map = copy.copy(target_grad_map)
    for op_idx in range(start_op_idx, block.desc.op_size()):
        op_desc = block.desc.op(op_idx)
        for name in op_desc.input_arg_names():
            if name in var_map:
                op_desc.rename_input(name, var_map[name])

        for name in op_desc.output_arg_names():
            if block.desc.find_var(name.encode("ascii")):
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                new_name = unique_name.generate(name)
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                op_desc.rename_output(name, new_name)
                var_map[name] = new_name

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    for g, ng in six.iteritems(var_map):
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        if g in grad_to_var:
            grad_to_var[ng] = grad_to_var[g]
            grad_to_var.pop(g)


def _get_stop_gradients_(program):
    no_grad_dict = dict()
    assert isinstance(program, framework.Program)
    for block in program.blocks:
        assert isinstance(block, framework.Block)
        block_no_grad_set = set()
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        for var in list(block.vars.values()):
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            assert isinstance(var, framework.Variable)
            if var.stop_gradient:
                block_no_grad_set.add(_append_grad_suffix_(var.name))
        no_grad_dict[block.idx] = block_no_grad_set
    return no_grad_dict


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def append_backward(loss, parameter_list=None, no_grad_set=None,
                    callbacks=None):
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    """
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    Append backward part to main_program.

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    A complete neural network training is made up of forward and backward
    propagation. However, when we configure a network, we only need to
    specify its forwrd part. The backward part is generated automatically
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    according to the forward part by this function.

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    In most cases, users do not need to invoke this function manually. It
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    will be automatically invoked by the optimizer's `minimize` function.
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    Args:
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        loss(Variable): The loss variable of the network.
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        parameter_list(list[string]|None): Names of parameters that need
                                           to be updated by optimizers.
                                           If it is None, all parameters
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                                           will be updated.
                                           Default: None
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        no_grad_set(set|None): Variables in the Block 0 whose gradients
                               should be ignored. All variables with
                               `step_gradient=True` from all blocks will
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                               be automatically added into this set.
                               Default: None
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        callbacks(list[callable object]|None): The callbacks are used for
                                               doing some custom jobs during
                                               backward part building. All
                                               callable objects in it will
                                               be invoked once each time a
                                               new gradient operator is added
                                               into the program. The callable
                                               object must has two input
                                               parameters: 'block' and 'context'.
                                               The 'block' is the block which
                                               the new gradient operator will
                                               be added to. The 'context' is a
                                               map, whose keys are gradient
                                               variable names and values are
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                                               corresponding original variables.
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                                               In addition to this, the 'context'
                                               has another special key-value pair:
                                               the key is string '__current_op_desc__'
                                               and the value is the op_desc of the
                                               gradient operator who has just
                                               triggered the callable object.
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    Returns:
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        list[(Variable,Variable)]: Pairs of parameter and its
        corresponding gradients. The key is the parameter and the
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        value is gradient variable.

    Raises:
        AssertionError: If `loss` is not an instance of Variable.

    Examples:
        .. code-block:: python

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            # network configuration code
            # ...
            avg_loss = fluid.layers.mean(loss)
            param_grad_list = fluid.backward.append_backward(loss=avg_loss)
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    """
    assert isinstance(loss, framework.Variable)
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    if loss.op is None:
        # the loss is from a cloned program. Find loss op manually.
        for op in reversed(loss.block.ops):
            assert isinstance(op, framework.Operator)
            if len(op.output_arg_names) == 1 and op.output_arg_names[
                    0] == loss.name:
                loss.op = op
                break
        if loss.op is None:
            raise ValueError("loss.op is None. Should not happend")

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    loss.op.set_attr(core.op_proto_and_checker_maker.kOpRoleAttrName(),
                     int(core.op_proto_and_checker_maker.OpRole.Forward) |
                     int(core.op_proto_and_checker_maker.OpRole.Loss))

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    if callbacks is not None:
        isinstance(callbacks, list)
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    program = loss.block.program
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    if no_grad_set is None:
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        no_grad_set = set()
    no_grad_set = copy.copy(no_grad_set)
    no_grad_dict = _get_stop_gradients_(program)
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    no_grad_dict[0].update(list(map(_append_grad_suffix_, no_grad_set)))
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    grad_info_map = dict()
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    root_block = program.block(0)
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    fwd_op_num = root_block.desc.op_size()
    current_block_idx = program.current_block_idx
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    grad_to_var = dict()

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    op_desc = _create_op_desc_(
        "fill_constant", {}, {"Out": [_append_grad_suffix_(loss.name)]}, {
            "shape": [1],
            "value": 1.0,
            "dtype": loss.dtype,
            "force_cpu": False,
            core.op_proto_and_checker_maker.kOpRoleAttrName():
            int(core.op_proto_and_checker_maker.OpRole.Backward) |
            int(core.op_proto_and_checker_maker.OpRole.Loss),
        })
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    root_block.desc.append_op().copy_from(op_desc)

    block_no_grad_set = set(map(_strip_grad_suffix_, no_grad_dict[0]))
    op_path = _find_op_path_(root_block, [loss], [], block_no_grad_set)
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    no_grad_dict[0].update(list(map(_append_grad_suffix_, block_no_grad_set)))
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    _append_backward_ops_(root_block, op_path, root_block, no_grad_dict,
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                          grad_to_var, callbacks)
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    # Because calc_gradient may be called multiple times,
    # we need rename the internal gradient variables so that they have
    # different names.
    _rename_grad_(root_block, fwd_op_num, grad_to_var, {})

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    _append_backward_vars_(root_block, fwd_op_num, grad_to_var, grad_info_map)
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    program.current_block_idx = current_block_idx
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    program._sync_with_cpp()
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    if parameter_list is not None:
        parameters = parameter_list
    else:
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        params = program.global_block().all_parameters()
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        program.global_block().iter_parameters()
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        parameters = [param.name for param in params]
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    params_and_grads = []
    for param in parameters:
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        if cpt.to_text(param) not in grad_info_map:
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            continue
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        grad_info = grad_info_map[param]
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        grad_block = grad_info[1]
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        if not grad_block.has_var(grad_info[0]):
            raise ValueError("grad block[{0}] did not have grad var {1}".format(
                grad_info[1], grad_info[0]))
        # Get the param var from the global block
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        param_var = program.global_block().var(param)
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        grad_var = grad_block.var(grad_info[0])
        if loss.block.has_var(grad_info[0]):
            params_and_grads.append((param_var, grad_var))
        else:
            params_and_grads.append((param_var, None))
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    op_role_var_attr_name = core.op_proto_and_checker_maker.kOpRoleVarAttrName()
    for p, g in params_and_grads:
        if g is None:
            continue
        for op in reversed(program.global_block().ops):
            assert isinstance(op, framework.Operator)
            if g.name in op.output_arg_names:
                g.op = op
                break

        if g.op is None:
            raise ValueError("Unexpected branch")
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        attr_val = [p.name, g.name]
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        if g.op.has_attr(op_role_var_attr_name):
            attr_val.extend(g.op.attr(op_role_var_attr_name))
        g.op.set_attr(op_role_var_attr_name, attr_val)
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    return params_and_grads
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def _as_list(x):
    if x is None:
        return []
    return list(x) if isinstance(x, collections.Sequence) else [x]


def _find_op_path_(block, outputs, inputs, no_grad_set):
    """
    no_grad_set will also be changed
    """
    input_names = set([inp.name for inp in inputs])
    output_names = set([out.name for out in outputs])

    relevant_op_flags = [True] * len(block.ops)

    # All the inputs of the block are used if inputs is empty,
    if inputs:
        for i, op in enumerate(block.ops):
            if _some_in_set_(op.desc.input_arg_names(), input_names):
                for name in op.desc.output_arg_names():
                    if name not in no_grad_set:
                        input_names.add(name)
            else:
                relevant_op_flags[i] = False

    for i, op in reversed(list(enumerate(block.ops))):
        if _some_in_set_(op.desc.output_arg_names(), output_names):
            for name in op.desc.input_arg_names():
                if name not in no_grad_set:
                    output_names.add(name)
        else:
            relevant_op_flags[i] = False

    op_path = [
        block.ops[i] for i in range(len(block.ops)) if relevant_op_flags[i]
    ]

    if inputs:
        for op in op_path:
            for name in op.desc.input_arg_names():
                if name not in input_names:
                    no_grad_set.add(name)

    return op_path


def calc_gradient(targets, inputs, target_gradients=None, no_grad_set=None):
    """
    Backpropagate the graidents of targets to inputs.

    Args:
        targets(Variable|list[Variable]): The target variables
        inputs(Variable|list[Variable]): The input variables
        no_grad_set(set[string]): The names of variables that have no gradients
            in Block 0. All variables with `stop_gradient=True` from all blocks
            will be automatically added.

    Return:
        (list[Variable]): list of gradients for inputs
        If an input does not affect targets, the corresponding gradient variable
        will be None
    """
    targets = _as_list(targets)
    inputs = _as_list(inputs)
    target_gradients = _as_list(target_gradients)

    block = targets[0].block
    prog = block.program
    block_idx = block.idx

    if not target_gradients:
        target_gradients = [None] * len(targets)

    if len(targets) != len(target_gradients):
        raise ValueError(
            "Should have the same number of target_gradients as targets")

    if no_grad_set is None:
        no_grad_set = set()
    no_grad_set = copy.copy(no_grad_set)
    no_grad_dict = _get_stop_gradients_(prog)
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    no_grad_dict[0].update(list(map(_append_grad_suffix_, no_grad_set)))
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    fwd_op_num = block.desc.op_size()

    target_grad_map = {}
    for i, grad in enumerate(target_gradients):
        target = targets[i]
        if grad is None:
            grad_name = _append_grad_suffix_(target.name)
            op_desc = _create_op_desc_("fill_constant_batch_size_like",
                                       {"Input": [target.name]},
                                       {"Out": [grad_name]}, {
                                           "shape": target.shape,
                                           "value": 1.0,
                                           "dtype": target.dtype,
                                           'input_dim_idx': 0,
                                           'output_dim_idx': 0
                                       })
            block.desc.append_op().copy_from(op_desc)
        else:
            if target.block.idx != block_idx or target.block.program != prog:
                raise ValueError("all targets must be in the same block")
            if target.shape != grad.shape:
                raise ValueError(
                    "The shapes of target and grad are different: %s %s" % (
                        target.name, grad.name))
            target_grad_map[_append_grad_suffix_(target.name)] = grad.name

    for input in inputs:
        if input.block.program != prog:
            raise "input must be in the same program as targets"

    block_no_grad_set = set(map(_strip_grad_suffix_, no_grad_dict[0]))
    op_path = _find_op_path_(block, targets, inputs, block_no_grad_set)
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    no_grad_dict[0].update(list(map(_append_grad_suffix_, block_no_grad_set)))
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    grad_to_var = dict()
    grad_info_map = dict()
    _append_backward_ops_(block, op_path, block, no_grad_dict, grad_to_var)

    # Because calc_gradient may be called multiple times,
    # we need rename the internal gradient variables so that they have
    # different names.
    _rename_grad_(block, fwd_op_num, grad_to_var, target_grad_map)

    _append_backward_vars_(block, fwd_op_num, grad_to_var, grad_info_map)
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    prog._sync_with_cpp()
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    grad_vars = []
    for input_var in inputs:
        if input_var.name not in grad_info_map:
            grad_vars.append(None)
        else:
            grad_info = grad_info_map[input_var.name]
            grad_block = grad_info[1]
            grad_var = grad_block.var(grad_info[0])
            grad_vars.append(grad_var)

    if len(grad_vars) == 1:
        return grad_vars[0]
    else:
        return grad_vars