optimizer.py

# Copyright (c) 2018 PaddlePaddle Authors. All Rights Reserved.
#
# 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
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# 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.

from __future__ import print_function

from collections import defaultdict
from .wrapped_decorator import signature_safe_contextmanager

from paddle.fluid.framework import Program, Variable, name_scope, default_main_program, default_startup_program
from paddle.fluid.distribute_lookup_table import find_distributed_lookup_table

from . import framework
from . import layers
from . import unique_name
from .backward import append_backward
from .clip import append_gradient_clip_ops, error_clip_callback
from .framework import program_guard
from .initializer import Constant
from .layer_helper import LayerHelper
from .layers import ops
from .regularizer import append_regularization_ops
from .dygraph import base as imperative_base
from .dygraph.learning_rate_scheduler import LearningRateDecay
from paddle.fluid import core
from paddle.fluid.layers import tensor
from functools import reduce
import copy

__all__ = [
    'SGD', 'Momentum', 'Adagrad', 'Adam', 'Adamax', 'DecayedAdagrad', 'Ftrl',
    'SGDOptimizer', 'MomentumOptimizer', 'AdagradOptimizer', 'AdamOptimizer',
    'AdamaxOptimizer', 'DecayedAdagradOptimizer', 'RMSPropOptimizer',
    'FtrlOptimizer', 'Adadelta', 'ModelAverage', 'LarsMomentum',
    'LarsMomentumOptimizer', 'DGCMomentumOptimizer'
]


class Optimizer(object):
    """Optimizer Base class.

    Define the common interface of an optimizer.
    User should not use this class directly,
    but need to use one of it's implementation.
    """

    def __init__(self, learning_rate, regularization=None, name=None):
        if framework.in_dygraph_mode():
            if not isinstance(learning_rate, float) and \
                    not isinstance(learning_rate, LearningRateDecay):
                raise TypeError(
                    "learning rate should be float or LearningRateDecay, got %s here"
                    % type(learning_rate))
        else:
            if not isinstance(learning_rate, float) and \
                    not isinstance(learning_rate, framework.Variable):
                raise TypeError(
                    "learning rate should be float or Variable, got %s here" %
                    type(learning_rate))

        self._name = name
        self.regularization = regularization
        self._learning_rate = learning_rate
        # the learning rate type should be inferenced from loss
        self._dtype = None
        # each program should have a independent learning rate
        # program -> Variable(learning_rate)
        self._learning_rate_map = dict()
        if isinstance(self._learning_rate, framework.Variable):
            self._learning_rate_map[framework.default_main_program(
            )] = self._learning_rate
        # Dictionary of accumulators. Some optimizer subclasses need to
        # allocate and manage extra variables associated with the parameters
        # to train. These variables are called accumulators.
        # {accum_name : { paramter_name : accumulator_for_parameter, ...}, ...}
        self._accumulators = defaultdict(lambda: dict())
        self.helper = None
        self._opti_name_list = []

    def get_opti_var_name_list(self):
        return self._opti_name_list

    def _create_global_learning_rate(self):
        if imperative_base.enabled():
            # create learning rate Variable
            if isinstance(self._learning_rate, float):
                lr = self._global_learning_rate()

                if isinstance(lr, framework.Variable):
                    return
                else:
                    self._learning_rate_map[framework.default_main_program(
                    )] = layers.create_global_var(
                        name=unique_name.generate("learning_rate"),
                        shape=[1],
                        value=float(self._learning_rate),
                        dtype='float32' if self._dtype is None else self._dtype,
                        persistable=True)
            # get learning rate Variable from LearningRateDecay
            elif isinstance(self._learning_rate, LearningRateDecay):
                self._learning_rate_map[framework.default_main_program(
                )] = self._learning_rate()
            else:
                raise TypeError(
                    "optimizer's learning rate must be float or LearningRateDecay"
                )
        else:
            lr = self._global_learning_rate()

            if isinstance(lr, framework.Variable):
                return
            else:
                if not isinstance(self._learning_rate, float):
                    raise TypeError(
                        "learning rate variable is create outside optimizer,"
                        "can not create new learning rate variable for new program"
                    )

            # create learning rate in the current main program
            self._learning_rate_map[framework.default_main_program(
            )] = layers.create_global_var(
                name=unique_name.generate("learning_rate"),
                shape=[1],
                value=float(self._learning_rate),
                dtype='float32' if self._dtype is None else self._dtype,
                persistable=True)

    def _global_learning_rate(self, program=None):
        """
        get global decayed learning rate
        :return:
        """
        if program is None:
            program = framework.default_main_program()
        return self._learning_rate_map.get(program, None)

    def _append_optimize_op(self, block, param_and_grad):
        """ append optimize operator to block and return all the added optimize_op
        """
        raise NotImplementedError()

    def _create_param_lr(self, param_and_grad):
        # create learning rate variable for every parameter
        param = param_and_grad[0]
        param_lr = param.optimize_attr['learning_rate']
        if type(param_lr) == Variable:
            return param_lr
        else:
            if param_lr == 1.0:
                return self._global_learning_rate()
            else:
                with default_main_program()._lr_schedule_guard(
                        is_with_opt=True), framework.name_scope(
                            'scale_with_param_lr'):
                    return self._global_learning_rate() * param_lr

    def _create_accumulators(self, block, parameters):
        """Create all accumulators needed by the parameters

        Args:
            block: the block in which the loss variable is present
            parameters: list of parameter variables for the optimizer
        """
        pass

    def _finish_update(self, block, parameters_and_grads):
        """Finish any custom updates needed
           before completing an optimization step

        Args:
            block: the block in which the loss variable is present
            parameters: list of parameter variables for the optimizer

        Returns:
            None
        """
        pass

    def _add_accumulator(self,
                         name,
                         param,
                         dtype=None,
                         fill_value=0.0,
                         shape=None):
        """Utility function to add an accumulator for a parameter

        Args:
            block: the block in which the loss variable is present
            name: name of the accumulator
            param: parameter variable for which accumulator is to be added
            dtype: data type of the accumulator variable
            fill_value: value to initialize the accumulator variable
        """
        if self._name is not None:
            name = self._name + "_" + name
        if (name in self._accumulators and
                param.name in self._accumulators[name]):
            if framework.in_dygraph_mode():
                return self._accumulators[name][param.name]
            raise Exception("Accumulator {} already exists for parameter {}".
                            format(name, param.name))
        if shape == None:
            shape = param.shape
        assert isinstance(self.helper, LayerHelper)

        var_name = param.name + "_" + name
        var_name = unique_name.generate(var_name)
        self._opti_name_list.append(var_name)

        var = self.helper.create_global_variable(
            name=var_name,
            persistable=True,
            dtype=dtype or param.dtype,
            type=param.type,
            shape=shape)
        self.helper.set_variable_initializer(
            var, initializer=Constant(value=float(fill_value)))
        self._accumulators[name][param.name] = var
        return var

    def _get_accumulator(self, name, param):
        """Utility function to fetch an accumulator for a parameter

        Args:
            name: name of the accumulator
            param: parameter variable for which accumulator is to be fetched

        Returns:
            accumulator variable for the parameter
        """
        if self._name is not None:
            name = self._name + "_" + name
        if (name not in self._accumulators or
                param.name not in self._accumulators[name]):
            raise Exception("Accumulator {} does not exist for parameter {}".
                            format(name, param.name))
        return self._accumulators[name][param.name]

    def _create_optimization_pass(self, parameters_and_grads):
        """Add optimization operators to update gradients to variables.

        Args:
          parameters_and_grads(list(tuple(Variable, Variable))):
            a list of (variable, gradient) pair to update.

        Returns:
          return_op_list: a list of operators that will complete one step of
            optimization. This will include parameter update ops, global step
            update ops and any other custom ops required by subclasses to manage
            their internal state.
        """
        # This is a default implementation of create_optimization_pass that
        # can be shared by most optimizers. This implementation assumes that
        # the subclass will implement the _append_optimize_op method and the
        #  _initialize_tensors method. The subclass can extend the
        # _create_accumulators method if it needs to create accumulators
        # for parameters and extend _finish_update method to add custom ops.

        # Allways called under program_guard use global block as loss block
        global_block = framework.default_main_program().global_block()
        start = len(global_block.ops)
        self.helper = LayerHelper(self.__class__.__name__)
        self._create_accumulators(global_block,
                                  [p[0] for p in parameters_and_grads])
        self._create_global_learning_rate()

        optimize_ops = []
        if framework.in_dygraph_mode():
            for param_and_grad in parameters_and_grads:
                if param_and_grad[1] is None:
                    continue
                with param_and_grad[0].block.program._optimized_guard(
                        param_and_grad):
                    if param_and_grad[0].trainable is True:
                        optimize_op = self._append_optimize_op(global_block,
                                                               param_and_grad)
                        optimize_ops.append(optimize_op)
        else:
            for param_and_grad in parameters_and_grads:
                if param_and_grad[1] is None:
                    continue
                with param_and_grad[0].block.program._optimized_guard(
                        param_and_grad), name_scope("optimizer"):
                    if param_and_grad[0].trainable is True:
                        optimize_op = self._append_optimize_op(global_block,
                                                               param_and_grad)
                        optimize_ops.append(optimize_op)

        # Get custom finish ops for subclasses
        # FIXME: Need to fix this once we figure out how to handle dependencies
        self._finish_update(global_block, parameters_and_grads)

        end = len(global_block.ops)
        return global_block._slice_ops(start, end)

    def _process_distribute_lookuptable(self, param_grads):
        """
        Because distribute lookup table only support SGD optimizer for now, not support
        other optimizer and regularization, so we should find the table parameter out,
        and avoid to add regularization and other op for it, and add sgd optimize op
        for it independently.
        :param param_grads(list((Var, Var))): list of (param, grad) pair.
        :param loss: the loss variable.
        :param startup_program: the startup program
        """
        program = framework.default_main_program()
        global_block = framework.default_main_program().global_block()
        table_name = find_distributed_lookup_table(program)
        table_param = None
        table_grad = None
        new_param_grads = []
        for p, g in param_grads:
            if p.name == table_name:
                if table_param is not None:
                    raise RuntimeError(
                        "multi dist table var found, only support one now!")
                table_param = p
                table_grad = g
            else:
                new_param_grads.append((p, g))
        sgd_op = None
        if table_param is not None:
            param_and_grad = [table_param, table_grad]
            with table_param.block.program._optimized_guard(param_and_grad), \
                    framework.name_scope("optimizer"):
                self._create_global_learning_rate()
                # create the optimize op
                sgd_op = global_block.append_op(
                    type='sgd',
                    inputs={
                        "Param": table_param,
                        "Grad": table_grad,
                        "LearningRate": self._create_param_lr(param_and_grad)
                    },
                    outputs={"ParamOut": param_and_grad[0]})
        return new_param_grads, (table_param, table_grad), sgd_op

    def _append_dgc_ops(self, param_and_grad):
        pass

    def backward(self,
                 loss,
                 startup_program=None,
                 parameter_list=None,
                 no_grad_set=None,
                 callbacks=None):
        """
        First part of `minimize`, do auto-diff to append backward ops for
        the current program.

        Args:
            loss (Variable): loss variable to run optimizations.
            startup_program (Program): startup_program for initializing parameters
                in `parameter_list`.
            parameter_list (list): list of Variables to update.
            no_grad_set (set|None): set of Variables should be ignored.
            callbacks (list|None): list of callables to run when appending backward
                operator for one parameter.

        Return:
            list: list of (param, grad) pair, grad is the output of backward.

        Examples:
            See examples in `apply_gradients`.
        """
        self._dtype = loss.dtype
        if framework.in_dygraph_mode():
            if parameter_list is not None:
                parameters = parameter_list
            else:
                parameters = framework._dygraph_tracer().all_parameters()

            params_grads = []
            for param in parameters:
                if not param.trainable:
                    continue
                if param._ivar._grad_ivar() is not None:
                    # create gradient variable
                    grad_var = Variable(
                        block=loss.block,
                        name=param._ivar._grad_name(),
                        stop_gradient=True,
                        ivar=param._ivar._grad_ivar())
                    params_grads.append((param, grad_var))
        else:
            if callbacks is None:
                callbacks = [error_clip_callback]
            else:
                assert (isinstance(callbacks, list))
            program = loss.block.program
            with program_guard(program, startup_program):
                params_grads = append_backward(loss, parameter_list,
                                               no_grad_set, callbacks)
                # Note: since we can't use all_reduce_op now,
                #  dgc_op should be the last op of one grad.
                self._append_dgc_ops(params_grads)
        return params_grads

    def apply_gradients(self, params_grads):
        """
        Second part of `minimize`, appending optimization operators for
        given `params_grads` pairs.

        Args:
            params_grads (list): list of (param, grad) pair to do optimization.

        Returns:
            list: A list of operators appended to the current program.

        Examples:
            .. code-block:: python

                loss = network()
                optimizer = fluid.optimizer.SGD(learning_rate=0.1)
                params_grads = optimizer.backward(loss)
                # you may append operations for params_grads here
                # ...
                optimizer.apply_gradients(params_grads)
        """
        params_grads = sorted(params_grads, key=lambda x: x[0].name)

        params_grads, table_param_and_grad, table_optimize_op = \
            self._process_distribute_lookuptable(params_grads)

        params_grads = append_gradient_clip_ops(params_grads)

        # Add regularization if any
        params_grads = append_regularization_ops(params_grads,
                                                 self.regularization)

        optimize_ops = self._create_optimization_pass(params_grads)
        if table_optimize_op is not None:
            optimize_ops.append(table_optimize_op)
            params_grads.append(table_param_and_grad)

        return optimize_ops

    def apply_optimize(self, loss, startup_program, params_grads):
        """
        Second part of `minimize`, appending optimization operators for
        given `params_grads` pairs.

        Args:
            loss (Variable): loss variable to run optimizations.
            startup_program (Program): startup_program for initializing parameters
                in `parameter_list`.
            params_grads (list): list of (param, grad) pair to do optimization.

        Returns:
            list: A list of operators appended to the current program.
        """
        if framework.in_dygraph_mode():
            with program_guard(framework.default_main_program(),
                               framework.default_startup_program()):
                optimize_ops = self._create_optimization_pass(params_grads)
        else:
            program = loss.block.program
            with program_guard(program, startup_program):
                optimize_ops = self.apply_gradients(params_grads)
        return optimize_ops

    def minimize(self,
                 loss,
                 startup_program=None,
                 parameter_list=None,
                 no_grad_set=None):
        """
        Add operations to minimize `loss` by updating `parameter_list`.

        This method combines interface `backward()` and
        `apply_gradients()` into one.

        Args:
            loss (Variable): loss variable to run optimizations.
            startup_program (Program): startup_program for initializing parameters
                in `parameter_list`.
            parameter_list (list): list of Variables to update.
            no_grad_set (set|None): set of Variables should be ignored.

        Returns:
            tuple: (optimize_ops, params_grads) which are, list of operators appended;
            and list of (param, grad) Variables pair for optimization.
        """
        params_grads = self.backward(
            loss,
            startup_program=startup_program,
            parameter_list=parameter_list,
            no_grad_set=no_grad_set)
        optimize_ops = self.apply_optimize(
            loss, startup_program=startup_program, params_grads=params_grads)

        return optimize_ops, params_grads


class SGDOptimizer(Optimizer):
    """
    Optimizer of the stochastic gradient descent algorithm.

    .. math::

        param\_out = param - learning\_rate * grad

    Args:
        learning_rate (float|Variable): the learning rate used to update parameters. \
        Can be a float value or a Variable with one float value as data element.
        regularization: A Regularizer, such as
                        fluid.regularizer.L2DecayRegularizer.
        name: A optional name prefix.

    Examples:
        .. code-block:: python

            sgd_optimizer = fluid.optimizer.SGD(learning_rate=0.2)
            sgd_optimizer.minimize(cost)
    """

    def __init__(self, learning_rate, regularization=None, name=None):
        assert learning_rate is not None
        super(SGDOptimizer, self).__init__(
            learning_rate=learning_rate,
            regularization=regularization,
            name=name)
        self.type = "sgd"

    def _append_optimize_op(self, block, param_and_grad):
        assert isinstance(block, framework.Block)

        # create the optimize op
        sgd_op = block.append_op(
            type=self.type,
            inputs={
                "Param": param_and_grad[0],
                "Grad": param_and_grad[1],
                "LearningRate": self._create_param_lr(param_and_grad)
            },
            outputs={"ParamOut": param_and_grad[0]},
            stop_gradient=True)

        return sgd_op


class MomentumOptimizer(Optimizer):
    """

    Simple Momentum optimizer with velocity state

    This optimizer has a flag for Nestrov Momentum.

    The update equations are as follows:

    .. math::

        & velocity = mu * velocity + gradient

        & if (use\_nesterov):

        &\quad   param = param - (gradient + mu * velocity) * learning\_rate

        & else:

        &\quad   param = param - learning\_rate * velocity

    Args:
        learning_rate (float|Variable): the learning rate used to update parameters. \
        Can be a float value or a Variable with one float value as data element.
        momentum (float): momentum factor
        use_nesterov (bool): enables Nesterov momentum
        regularization: A Regularizer, such as
                        fluid.regularizer.L2DecayRegularizer.
        name: A optional name prefix.

    Examples:
        .. code-block:: python

            optimizer = fluid.optimizer.Momentum(learning_rate=0.2, momentum=0.1)
            optimizer.minimize(cost)
    """
    _velocity_acc_str = "velocity"

    def __init__(self,
                 learning_rate,
                 momentum,
                 use_nesterov=False,
                 regularization=None,
                 name=None):
        assert learning_rate is not None
        assert momentum is not None
        super(MomentumOptimizer, self).__init__(
            learning_rate=learning_rate,
            regularization=regularization,
            name=name)
        self.type = "momentum"
        self._momentum = momentum
        self._use_nesterov = bool(use_nesterov)

    def _create_accumulators(self, block, parameters):
        assert isinstance(block, framework.Block)

        for p in parameters:
            self._add_accumulator(self._velocity_acc_str, p)

    def _append_optimize_op(self, block, param_and_grad):
        assert isinstance(block, framework.Block)

        velocity_acc = self._get_accumulator(self._velocity_acc_str,
                                             param_and_grad[0])
        # create the momentum optimize op
        momentum_op = block.append_op(
            type=self.type,
            inputs={
                "Param": param_and_grad[0],
                "Grad": param_and_grad[1],
                "Velocity": velocity_acc,
                "LearningRate": self._create_param_lr(param_and_grad)
            },
            outputs={
                "ParamOut": param_and_grad[0],
                "VelocityOut": velocity_acc
            },
            attrs={"mu": self._momentum,
                   "use_nesterov": self._use_nesterov},
            stop_gradient=True)

        return momentum_op


class DGCMomentumOptimizer(MomentumOptimizer):
    """

    Original paper is https://arxiv.org/abs/1712.01887

    DGC reduces the communication bandwidth by sending only the important gradients (sparse update):\
        only gradients larger than a threshold are transmitted.

    To avoid losing information, DGC accumulates the rest of the gradients locally.

    Eventually, these gradients become large enough to be transmitted.

    Thus, DGC sends the large gradients immediately but eventually send all of the gradients over time.

    To ensure no loss of accuracy, DGC employs momentum correction and local gradient clipping on top of the gradient sparsification to maintain model performance.

    DGC also uses momentum factor masking and warmup training to overcome the staleness problem caused by reduced communication.

    This optimizer will do two things:

        1. Compress the gradient by get TopK import value from tensor \
            and use it for allreduce to reduce network bandwidth.

        2. Call momentum to optimize on the cost.

    Args:
        learning_rate (float|Variable): the learning rate used to update parameters. \
            Can be a float value or a Variable with one float value as data element.
        momentum (float): Momentum factor.
        rampup_begin_step (int): The beginning step from which gradient compression is implemented.
        rampup_step (int): How long it use the sparsity periods. Default is 1.
            for example: If the sparsity is [0.75, 0.9375, 0.984375, 0.996, 0.999], and the rampup_step is 5, \
                it will use 0.75 at 0 step, and 0.9375 at 1 step, and so on. And when reach sparsity array ends, \
                it will use 0.999 then and after.
        sparsity (list[float]): Get top important element from gradient tensor, the ratio is (1 - current sparsity).
        use_nesterov (bool): Enables Nesterov momentum. True means use nesterov.
        local_grad_clip_norm (float): Clip norm value if needed.
        num_trainers: The number of training nodes.
        regularization: A Regularizer, such as fluid.regularizer.L2DecayRegularizer.
        name: An optional name prefix.

    Examples:
        .. code-block:: python

            optimizer = fluid.optimizer.DGCMomentumOptimizer(
                        learning_rate=0.0001,
                        momentum=0.9,
                        rampup_step=1000,
                        rampup_begin_step=1252,
                        sparsity=[0.999, 0.999])

    """

    def __init__(self,
                 learning_rate,
                 momentum,
                 rampup_begin_step,
                 rampup_step=1,
                 sparsity=[0.999],
                 use_nesterov=False,
                 local_grad_clip_norm=None,
                 num_trainers=None,
                 regularization=None,
                 name=None):
        self._sparsity = sparsity
        self._rampup_step = rampup_step
        self._rampup_step_var = None

        self._rampup_begin_step = rampup_begin_step
        self._rampup_begin_step_var = None

        self._global_step_var = None
        self._local_grad_clip_norm = None
        self._clip_norm = None

        if local_grad_clip_norm is not None:
            assert isinstance(num_trainers, int)
            assert isinstance(local_grad_clip_norm, float)
            assert num_trainers > 0

            self._local_grad_clip_norm = local_grad_clip_norm
            self._num_trainers = num_trainers
            self._clip_norm = local_grad_clip_norm / (num_trainers *
                                                      num_trainers)

        super(DGCMomentumOptimizer, self).__init__(
            learning_rate, momentum, use_nesterov, regularization, name)

        core.init_dgc()

    def _add_auto_increment_var(self, counter_name, begin, step=1):
        helper = LayerHelper('global_step_counter')
        counter, is_new_var = helper.create_or_get_global_variable(
            name=counter_name, dtype='float32', shape=[1], persistable=True)
        if is_new_var:
            helper.set_variable_initializer(
                counter,
                initializer=Constant(
                    value=float(begin - 1), force_cpu=True))
            helper.main_program.global_block()._prepend_op(
                type='increment',
                inputs={'X': [counter]},
                outputs={'Out': [counter]},
                attrs={'step': float(step)},
                stop_gradient=True)
            counter.stop_gradient = True

        return counter

    def _append_dgc_ops(self, param_and_grads):
        start_program = default_startup_program()
        main_program = default_main_program()
        main_program._enable_dgc = True

        # step counter
        self._global_step_var = self._add_auto_increment_var(
            counter_name=core.dgc.kDGCCounterName(), begin=0)

        # rampup begin step var for all_reduce_op_handle
        self._rampup_begin_step_var = tensor.create_global_var(
            shape=[1],
            dtype=core.VarDesc.VarType.FP32,
            persistable=True,
            name=core.dgc.kDGCRampUpBeginStepName(),
            value=self._rampup_begin_step * 1.0,
            force_cpu=True)

        for param_var, grad_var in param_and_grads:
            var_numel = abs(reduce(lambda x, y: x * y, param_var.shape))
            if var_numel < 16384 or \
                param_var.type == core.VarDesc.VarType.SELECTED_ROWS  or \
                grad_var.type == core.VarDesc.VarType.SELECTED_ROWS  or  \
                    param_var.dtype != core.VarDesc.VarType.FP32 :
                continue

            u_var = tensor.create_global_var(
                shape=param_var.shape,
                dtype=param_var.dtype,
                persistable=True,
                name=param_var.name + core.dgc.kDGCUName(),
                value=0.0)
            v_var = tensor.create_global_var(
                shape=param_var.shape,
                dtype=param_var.dtype,
                persistable=True,
                name=param_var.name + core.dgc.kDGCVName(),
                value=0.0)

            k_var = tensor.create_global_var(
                shape=[1],
                dtype=param_var.dtype,
                persistable=True,
                name=param_var.name + core.dgc.kDGCKName(),
                value=0.0,
                force_cpu=True)

            encoded_var = tensor.create_global_var(
                shape=[1],
                dtype=param_var.dtype,
                persistable=True,
                name=param_var.name + core.dgc.kDGCEncodedName(),
                value=0.0,
                force_cpu=False)

            # del back oprolevarname
            op_maker = core.op_proto_and_checker_maker
            backward = core.op_proto_and_checker_maker.OpRole.Backward
            for op in main_program.global_block().ops:
                if not self._is_the_backward_op(op):
                    continue

                var_attr = op.all_attrs()[op_maker.kOpRoleVarAttrName()]
                if param_var.name not in var_attr:
                    continue

                var_attr.remove(param_var.name)
                var_attr.remove(grad_var.name)
                if len(var_attr) > 1:
                    op._set_attr(op_maker.kOpRoleVarAttrName(), var_attr)
                else:
                    op._remove_attr(op_maker.kOpRoleVarAttrName())

            clip_var = grad_var
            if self._local_grad_clip_norm is not None:
                clip_var = self._append_clip_norm(grad_var, self._clip_norm)
            self._dgc_op(param_var, clip_var, grad_var, u_var, v_var, k_var,
                         encoded_var)

    def _is_the_backward_op(self, op):
        op_maker = core.op_proto_and_checker_maker
        backward = core.op_proto_and_checker_maker.OpRole.Backward
        if op_maker.kOpRoleVarAttrName() in op.attr_names and \
                int(op.all_attrs()[op_maker.kOpRoleAttrName()]) == int(backward):
            return True
        return False

    def _clip_by_norm(self, x, max_norm, name=None):
        args = {'x': x, 'max_norm': max_norm, 'name': name}

        helper = LayerHelper("dgc_clip_by_norm_op", **args)

        if name is None:
            name = unique_name.generate(".".join([helper.name, 'tmp']))

        out = helper.create_variable(
            type=x.type, name=name, dtype=x.dtype, persistable=False)

        helper.append_op(
            type="dgc_clip_by_norm",
            inputs={"X": x,
                    "current_step": self._global_step_var},
            attrs={
                "max_norm": max_norm,
                "rampup_begin_step": float(self._rampup_begin_step)
            },
            outputs={"Out": out})
        return out

    def _append_clip_norm(self, grad_var, clip_norm):
        with grad_var.block.program._backward_role_guard():
            return self._clip_by_norm(
                x=grad_var, max_norm=clip_norm, name=grad_var.name)

    def _dgc_op(self, param_var, clip_var, grad_var, u_var, v_var, k_var,
                encoded_var):
        block = framework.default_main_program().global_block()
        op_maker = core.op_proto_and_checker_maker
        dgc_op = block.append_op(
            type="dgc",
            inputs={
                "U": u_var,
                "V": v_var,
                "Grad": clip_var,
                "current_step": self._global_step_var
            },
            outputs={
                "U_out": u_var,
                "V_out": v_var,
                "EncodeGrad": encoded_var,
                "k": k_var,
                "Grad_out": grad_var
            },
            attrs={
                "m": self._momentum,
                "sparsity": self._sparsity,
                "use_nesterov": self._use_nesterov,
                "rampup_begin_step": float(self._rampup_begin_step),
                "rampup_step": float(self._rampup_step)
            },
            stop_gradient=True)

        backward = op_maker.OpRole.Backward
        dgc_op._set_attr(op_maker.kOpRoleAttrName(), backward)
        dgc_op._set_attr(op_maker.kOpRoleVarAttrName(),
                         [param_var.name, grad_var.name])


class LarsMomentumOptimizer(Optimizer):
    """
    Momentum optimizer with LARS support

    The update equations are as follows:

    .. math::

        & local\_learning\_rate = learning\_rate * lars\_coeff * \\
          \\frac{||param||}{||gradient|| + lars\_weight\_decay * ||param||}

        & velocity = mu * velocity + local\_learning\_rate * (gradient + lars\_weight\_decay * param)

        & param = param - velocity

    Args:
        learning_rate (float|Variable): the learning rate used to update parameters. \
        Can be a float value or a Variable with one float value as data element.
        momentum (float): momentum factor
        lars_coeff (float): defines how much we trust the layer to change its weights.
        lars_weight_decay (float): weight decay coefficient for decaying using LARS.
        regularization: A Regularizer, such as
                        fluid.regularizer.L2DecayRegularizer.
        name: A optional name prefix.


    Examples:
        .. code-block:: python

            optimizer = fluid.optimizer.LarsMomentum(learning_rate=0.2, momentum=0.1, lars_weight_decay=0.001)
            optimizer.minimize(cost)
    """
    _velocity_acc_str = "velocity"

    def __init__(self,
                 learning_rate,
                 momentum,
                 lars_coeff=0.001,
                 lars_weight_decay=0.0005,
                 regularization=None,
                 name=None):
        assert learning_rate is not None
        assert momentum is not None
        super(LarsMomentumOptimizer, self).__init__(
            learning_rate=learning_rate,
            regularization=regularization,
            name=name)
        self.type = "lars_momentum"
        self._momentum = momentum
        self._lars_coeff = float(lars_coeff)
        self._lars_weight_decay = float(lars_weight_decay)

    def _create_accumulators(self, block, parameters):
        assert isinstance(block, framework.Block)

        for p in parameters:
            self._add_accumulator(self._velocity_acc_str, p)

    def _append_optimize_op(self, block, param_and_grad):
        assert isinstance(block, framework.Block)

        velocity_acc = self._get_accumulator(self._velocity_acc_str,
                                             param_and_grad[0])
        # create the momentum optimize op
        momentum_op = block.append_op(
            type=self.type,
            inputs={
                "Param": param_and_grad[0],
                "Grad": param_and_grad[1],
                "Velocity": velocity_acc,
                "LearningRate": self._create_param_lr(param_and_grad)
            },
            outputs={
                "ParamOut": param_and_grad[0],
                "VelocityOut": velocity_acc
            },
            attrs={
                "mu": self._momentum,
                "lars_coeff": self._lars_coeff,
                "lars_weight_decay": self._lars_weight_decay
            },
            stop_gradient=True)

        return momentum_op


class AdagradOptimizer(Optimizer):
    """
    **Adaptive Gradient Algorithm (Adagrad)**

    The update is done as follows:

    .. math::

        moment\_out &= moment + grad * grad

        param\_out &= param - \\frac{learning\_rate * grad}{\sqrt{moment\_out} + \epsilon}

    The original paper(http://www.jmlr.org/papers/volume12/duchi11a/duchi11a.pdf)
    does not have the epsilon attribute. It is added here in our implementation
    as also proposed here: http://cs231n.github.io/neural-networks-3/#ada
    for numerical stability to avoid the division by zero error.

    Args:
        learning_rate (float|Variable): the learning rate used to update parameters. \
        Can be a float value or a Variable with one float value as data element.
        epsilon (float): a small float value for numerical stability.
        regularization: A Regularizer, such as
                        fluid.regularizer.L2DecayRegularizer.
        name: A optional name prefix.
        initial_accumulator_value (float): Initial value for moment accumulator.

    Examples:
        .. code-block:: python

            optimizer = fluid.optimizer.Adagrad(learning_rate=0.2)
            optimizer.minimize(cost)
    """
    _moment_acc_str = "moment"

    def __init__(self,
                 learning_rate,
                 epsilon=1.0e-6,
                 regularization=None,
                 name=None,
                 initial_accumulator_value=0.0):
        assert learning_rate is not None
        assert epsilon is not None
        super(AdagradOptimizer, self).__init__(
            learning_rate=learning_rate,
            regularization=regularization,
            name=name)
        self.type = "adagrad"
        self._epsilon = epsilon
        self.initial_accumulator_value = initial_accumulator_value

    def _create_accumulators(self, block, parameters):
        assert isinstance(block, framework.Block)

        for p in parameters:
            self._add_accumulator(self._moment_acc_str, p)

    def _append_optimize_op(self, block, param_and_grad):
        assert isinstance(block, framework.Block)

        moment_acc = self._get_accumulator(self._moment_acc_str,
                                           param_and_grad[0])
        startup_block = framework.default_startup_program().global_block()
        startup_block.append_op(
            type='fill_constant',
            inputs={},
            outputs={'Out': [moment_acc]},
            attrs={
                'dtype': moment_acc.dtype,
                'value': self.initial_accumulator_value,
                'shape': moment_acc.shape,
            })

        # Create the adagrad optimizer op
        adagrad_op = block.append_op(
            type=self.type,
            inputs={
                "Param": param_and_grad[0],
                "Grad": param_and_grad[1],
                "Moment": moment_acc,
                "LearningRate": self._create_param_lr(param_and_grad)
            },
            outputs={"ParamOut": param_and_grad[0],
                     "MomentOut": moment_acc},
            attrs={"epsilon": self._epsilon},
            stop_gradient=True)

        return adagrad_op


class AdamOptimizer(Optimizer):
    """
    This implements the Adam optimizer from Section 2 of the Adam
    paper : https://arxiv.org/abs/1412.6980.
    Adam is a first-order gradient-based optimization method based on
    adaptive estimates of lower-order moments.

    Adam updates:

    .. math::

        t & = t + 1

        moment\_1\_out & = {\\beta}_1 * moment\_1 + (1 - {\\beta}_1) * grad

        moment\_2\_out & = {\\beta}_2 * moment\_2 + (1 - {\\beta}_2) * grad * grad

        learning\_rate & = learning\_rate * \\
                          \\frac{\sqrt{1 - {\\beta}_2^t}}{1 - {\\beta}_1^t}

        param\_out & = param - learning\_rate * \\frac{moment\_1}{\sqrt{moment\_2} + \epsilon}

    Args:
        learning_rate (float|Variable): the learning rate used to update parameters. \
        Can be a float value or a Variable with one float value as data element.
        beta1 (float): The exponential decay rate for the 1st moment estimates.
        beta2 (float): The exponential decay rate for the 2nd moment estimates.
        epsilon (float): a small float value for numerical stability.
        regularization: A Regularizer, such as fluid.regularizer.L2DecayRegularizer.
        name: A optional name prefix.
        lazy_mode(bool: false): The official Adam algorithm has two moving-average accumulators
        the accumulators are updated at every step. Every element of the two moving-average is updated
        in both dense mode and sparse mode. If the size of parameter is very large, then the update
        may be very slow. The lazy mode only update the element that has gradient is the current
        mini-batch, so it will be much more faster. But this mode has different semantics with the
        original Adam algorithm and may lead to different result.

    Examples:
        .. code-block:: python

            optimizer = fluid.optimizer.Adam(learning_rate=0.2)
            optimizer.minimize(cost)

    """
    _moment1_acc_str = "moment1"
    _moment2_acc_str = "moment2"
    _beta1_pow_acc_str = "beta1_pow_acc"
    _beta2_pow_acc_str = "beta2_pow_acc"

    def __init__(self,
                 learning_rate=0.001,
                 beta1=0.9,
                 beta2=0.999,
                 epsilon=1e-8,
                 regularization=None,
                 name=None,
                 lazy_mode=False):
        assert learning_rate is not None
        assert beta1 is not None
        assert beta2 is not None
        assert epsilon is not None
        super(AdamOptimizer, self).__init__(
            learning_rate=learning_rate,
            regularization=regularization,
            name=name)
        self.type = "adam"
        self._beta1 = beta1
        self._beta2 = beta2
        self._epsilon = epsilon
        self._lazy_mode = lazy_mode

    def _create_accumulators(self, block, parameters):
        assert isinstance(block, framework.Block)

        # Create accumulator tensors for first and second moments
        for p in parameters:
            self._add_accumulator(self._moment1_acc_str, p)
            self._add_accumulator(self._moment2_acc_str, p)
            self._add_accumulator(
                name=self._beta1_pow_acc_str,
                param=p,
                dtype='float32',
                fill_value=self._beta1,
                shape=[1])
            self._add_accumulator(
                name=self._beta2_pow_acc_str,
                param=p,
                dtype='float32',
                fill_value=self._beta2,
                shape=[1])

    def _append_optimize_op(self, block, param_and_grad):
        assert isinstance(block, framework.Block)

        moment1 = self._get_accumulator(self._moment1_acc_str,
                                        param_and_grad[0])
        moment2 = self._get_accumulator(self._moment2_acc_str,
                                        param_and_grad[0])
        beta1_pow_acc = self._get_accumulator(self._beta1_pow_acc_str,
                                              param_and_grad[0])
        beta2_pow_acc = self._get_accumulator(self._beta2_pow_acc_str,
                                              param_and_grad[0])

        # create the adam optimize op
        adam_op = block.append_op(
            type=self.type,
            inputs={
                "Param": param_and_grad[0],
                "Grad": param_and_grad[1],
                "LearningRate": self._create_param_lr(param_and_grad),
                "Moment1": moment1,
                "Moment2": moment2,
                "Beta1Pow": beta1_pow_acc,
                "Beta2Pow": beta2_pow_acc
            },
            outputs={
                "ParamOut": param_and_grad[0],
                "Moment1Out": moment1,
                "Moment2Out": moment2
            },
            attrs={
                "beta1": self._beta1,
                "beta2": self._beta2,
                "epsilon": self._epsilon,
                "lazy_mode": self._lazy_mode,
                "min_row_size_to_use_multithread": 1000
            },
            stop_gradient=True)

        return adam_op

    def _finish_update(self, block, param_and_grads):
        """Update Beta1 and Beta2 Power accumulators
        """
        assert isinstance(block, framework.Block)
        main_block = block.program.global_block()
        for param, grad in param_and_grads:
            if grad is None:
                continue
            with param.block.program._optimized_guard(
                [param, grad]), name_scope("optimizer"):
                beta1_pow_acc = self._get_accumulator(self._beta1_pow_acc_str,
                                                      param)
                beta2_pow_acc = self._get_accumulator(self._beta2_pow_acc_str,
                                                      param)
                main_block.append_op(
                    type="scale",
                    inputs={"X": beta1_pow_acc},
                    outputs={"Out": beta1_pow_acc},
                    attrs={"scale": self._beta1},
                    stop_gradient=True)

                main_block.append_op(
                    type="scale",
                    inputs={"X": beta2_pow_acc},
                    outputs={"Out": beta2_pow_acc},
                    attrs={"scale": self._beta2},
                    stop_gradient=True)


class AdamaxOptimizer(Optimizer):
    """
    We implement the Adamax optimizer from Section 7 of the Adam
    paper: https://arxiv.org/abs/1412.6980. Adamax is a variant of the
    Adam algorithm based on the infinity norm.

    Adamax updates:

    .. math::

        t & = t + 1

        moment\_out & = {\\beta}_1 * moment + (1 - {\\beta}_1) * grad

        inf\_norm\_out & = max({\\beta}_2 * inf\_norm + \epsilon, |grad|)

        learning\_rate & = \\frac{learning\_rate}{1 - {\\beta}_1^t}

        param\_out & = param - learning\_rate * \\frac{moment\_out}{inf\_norm\_out}


    The original paper does not have an epsilon attribute.
    However, it is added here for numerical stability to prevent the
    division by 0 error.

    Args:
        learning_rate (float|Variable): the learning rate used to update parameters. \
        Can be a float value or a Variable with one float value as data element.
        beta1 (float): The exponential decay rate for the 1st moment estimates.
        beta2 (float): The exponential decay rate for the 2nd moment estimates.
        epsilon (float): a small float value for numerical stability.
        regularization: A Regularizer, such as
                        fluid.regularizer.L2DecayRegularizer.
        name: A optional name prefix.

    Examples:
        .. code-block:: python

            optimizer = fluid.optimizer.Adamax(learning_rate=0.2)
            optimizer.minimize(cost)

    Notes:
       Currently, AdamaxOptimizer doesn't support sparse parameter optimization.
    """
    _moment_acc_str = "moment"
    _inf_norm_acc_str = "inf_norm"
    _beta1_pow_acc_str = "beta1_pow_acc"

    def __init__(self,
                 learning_rate=0.001,
                 beta1=0.9,
                 beta2=0.999,
                 epsilon=1e-8,
                 regularization=None,
                 name=None):
        assert learning_rate is not None
        assert beta1 is not None
        assert beta2 is not None
        assert epsilon is not None
        super(AdamaxOptimizer, self).__init__(
            learning_rate=learning_rate,
            regularization=regularization,
            name=name)
        self.type = "adamax"
        self._beta1 = beta1
        self._beta2 = beta2
        self._epsilon = epsilon

    def _create_accumulators(self, block, parameters):
        # Create accumulator tensors for first moment and infinity norm
        for p in parameters:
            self._add_accumulator(self._moment_acc_str, p)
            self._add_accumulator(self._inf_norm_acc_str, p)
            self._add_accumulator(
                name=self._beta1_pow_acc_str,
                param=p,
                dtype='float32',
                fill_value=self._beta1,
                shape=[1])

    def _append_optimize_op(self, block, param_and_grad):
        assert isinstance(block, framework.Block)

        moment = self._get_accumulator(self._moment_acc_str, param_and_grad[0])
        inf_norm = self._get_accumulator(self._inf_norm_acc_str,
                                         param_and_grad[0])
        beta1_pow_acc = self._get_accumulator(self._beta1_pow_acc_str,
                                              param_and_grad[0])
        # create the adamax optimize op
        adamax_op = block.append_op(
            type=self.type,
            inputs={
                "Param": param_and_grad[0],
                "Grad": param_and_grad[1],
                "LearningRate": self._create_param_lr(param_and_grad),
                "Moment": moment,
                "InfNorm": inf_norm,
                "Beta1Pow": beta1_pow_acc
            },
            outputs={
                "ParamOut": param_and_grad[0],
                "MomentOut": moment,
                "InfNormOut": inf_norm
            },
            attrs={
                "beta1": self._beta1,
                "beta2": self._beta2,
                "epsilon": self._epsilon
            },
            stop_gradient=True)

        return adamax_op

    def _finish_update(self, block, parameters_and_grads):
        """Update Beta1 Power accumulator
        """
        assert isinstance(block, framework.Block)
        main_block = block.program.global_block()
        for param, grad in parameters_and_grads:
            if grad is None:
                continue
            with param.block.program._optimized_guard(
                [param, grad]), name_scope('adamx'):
                beta1_pow_acc = self._get_accumulator(self._beta1_pow_acc_str,
                                                      param)
                main_block.append_op(
                    type="scale",
                    inputs={"X": beta1_pow_acc},
                    outputs={"Out": beta1_pow_acc},
                    attrs={"scale": self._beta1},
                    stop_gradient=True)


class DecayedAdagradOptimizer(Optimizer):
    """
    **Decayed Adagrad Optimizer**

    The original paper(http://www.jmlr.org/papers/volume12/duchi11a/duchi11a.pdf)

    The update is done as follows:

    .. math::

        moment\_out & = decay * moment + (1 - decay) * grad * grad

        param\_out & = param - \\frac{learning\_rate * grad}{\sqrt{moment\_out} + \epsilon}

    The original paper(http://www.jmlr.org/papers/volume12/duchi11a/duchi11a.pdf)
    does not have an epsilon attribute. It is added here for numerical
    stability to avoid the division by zero error.

    Args:
        learning_rate (float|Variable): the learning rate used to update parameters. \
        Can be a float value or a Variable with one float value as data element.
        decay (float): decay rate.
        epsilon (float): a small float value for numerical stability.
        regularization: A Regularizer, such as
                        fluid.regularizer.L2DecayRegularizer.
        name: A optional name prefix.

    Examples:
        .. code-block:: python

            optimizer = fluid.optimizer.DecayedAdagrad(learning_rate=0.2)
            optimizer.minimize(cost)

    Notes:
       Currently, DecayedAdagradOptimizer doesn't support sparse parameter optimization.
    """
    _moment_acc_str = "moment"

    def __init__(self,
                 learning_rate,
                 decay=0.95,
                 epsilon=1.0e-6,
                 regularization=None,
                 name=None):
        assert learning_rate is not None
        assert decay is not None
        assert epsilon is not None

        super(DecayedAdagradOptimizer, self).__init__(
            learning_rate=learning_rate,
            regularization=regularization,
            name=name)
        self.type = "decayed_adagrad"
        self._decay = decay
        self._epsilon = epsilon

    def _create_accumulators(self, block, parameters):
        assert isinstance(block, framework.Block)

        for p in parameters:
            self._add_accumulator(self._moment_acc_str, p)

    def _append_optimize_op(self, block, param_and_grad):
        assert isinstance(block, framework.Block)

        moment_acc = self._get_accumulator(self._moment_acc_str,
                                           param_and_grad[0])

        # Create the decayed adagrad optimizer op
        decayed_adagrad_op = block.append_op(
            type=self.type,
            inputs={
                "Param": param_and_grad[0],
                "Grad": param_and_grad[1],
                "Moment": moment_acc,
                "LearningRate": self._create_param_lr(param_and_grad)
            },
            outputs={"ParamOut": param_and_grad[0],
                     "MomentOut": moment_acc},
            attrs={"epsilon": self._epsilon},
            stop_gradient=True)

        return decayed_adagrad_op


class AdadeltaOptimizer(Optimizer):
    """
    **Adadelta Optimizer**

    Simple Adadelta optimizer with average squared grad state and
    average squared update state.
    The details of adadelta please refer to this
    `ADADELTA: AN ADAPTIVE LEARNING RATE METHOD
    <http://www.matthewzeiler.com/pubs/googleTR2012/googleTR2012.pdf>`_.

    ..  math::

        E(g_t^2) &= \\rho * E(g_{t-1}^2) + (1-\\rho) * g^2 \\\\
        learning\\_rate &= sqrt( ( E(dx_{t-1}^2) + \\epsilon ) / ( \\
                          E(g_t^2) + \\epsilon ) ) \\\\
        E(dx_t^2) &= \\rho * E(dx_{t-1}^2) + (1-\\rho) * (-g*learning\\_rate)^2

    Args:
        learning_rate(float): global learning rate
        rho(float): rho in equation
        epsilon(float): epsilon in equation
        regularization: A Regularizer, such as
                        fluid.regularizer.L2DecayRegularizer.
        name: A optional name prefix.

    Examples:
        .. code-block:: python

            optimizer = fluid.optimizer.Adadelta(
                learning_rate=0.0003, epsilon=1.0e-6, rho=0.95)
            _, params_grads = optimizer.minimize(cost)

    Notes:
       Currently, AdadeltaOptimizer doesn't support sparse parameter optimization.
    """

    _avg_squared_grad_acc_str = "_avg_squared_grad"
    _avg_squared_update_acc_str = "_avg_squared_update"

    def __init__(self,
                 learning_rate,
                 epsilon=1.0e-6,
                 rho=0.95,
                 regularization=None,
                 name=None):
        if learning_rate is None:
            raise ValueError("learning_rate is not set.")
        if epsilon is None:
            raise ValueError("epsilon is not set.")
        if rho is None:
            raise ValueError("rho is not set.")
        super(AdadeltaOptimizer, self).__init__(
            learning_rate=learning_rate,
            regularization=regularization,
            name=name)
        self.type = "adadelta"
        self._epsilon = epsilon
        self._rho = rho

    def _create_accumulators(self, block, parameters):
        if not isinstance(block, framework.Block):
            raise TypeError("block is not instance of framework.Block.")

        for p in parameters:
            self._add_accumulator(self._avg_squared_grad_acc_str, p)
            self._add_accumulator(self._avg_squared_update_acc_str, p)

    def _append_optimize_op(self, block, param_and_grad):
        if not isinstance(block, framework.Block):
            raise TypeError("block is not instance of framework.Block.")

        avg_squared_grad_acc = self._get_accumulator(
            self._avg_squared_grad_acc_str, param_and_grad[0])
        avg_squared_update_acc = self._get_accumulator(
            self._avg_squared_update_acc_str, param_and_grad[0])

        # Create the adadelta optimizer op
        adadelta_op = block.append_op(
            type=self.type,
            inputs={
                "Param": param_and_grad[0],
                "Grad": param_and_grad[1],
                "AvgSquaredGrad": avg_squared_grad_acc,
                "AvgSquaredUpdate": avg_squared_update_acc
            },
            outputs={
                "ParamOut": param_and_grad[0],
                "AvgSquaredGradOut": avg_squared_grad_acc,
                "AvgSquaredUpdateOut": avg_squared_update_acc
            },
            attrs={"epsilon": self._epsilon,
                   "rho": self._rho},
            stop_gradient=True)

        return adadelta_op


class RMSPropOptimizer(Optimizer):
    """
    Root Mean Squared Propagation (RMSProp) is an unpublished, adaptive learning
    rate method. The original slides proposed RMSProp: Slide 29 of
    http://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf .

    The original equation is as follows:

    ..  math::

        r(w, t) & = \\rho r(w, t-1) + (1 - \\rho)(\\nabla Q_{i}(w))^2

        w & = w - \\frac{\\eta} {\\sqrt{r(w,t) + \\epsilon}} \\nabla Q_{i}(w)

    The first equation calculates moving average of the squared gradient for
    each weight. Then dividing the gradient by :math:`sqrt{v(w,t)}`.

    In some cases, adding a momentum term :math: `\\beta` is beneficial.
    In our implementation, Nesterov momentum is used:

    ..  math::

        r(w, t) & = \\rho r(w, t-1) + (1 - \\rho)(\\nabla Q_{i}(w))^2

        v(w, t) & = \\beta v(w, t-1) + \\frac{\\eta} {\\sqrt{r(w,t) +
            \\epsilon}} \\nabla Q_{i}(w)

        w & = w - v(w, t)

    if centered is True:

    ..  math::

        r(w, t) & = \\rho r(w, t-1) + (1 - \\rho)(\\nabla Q_{i}(w))^2

        g(w, t) & = \\rho g(w, t-1) + (1 - \\rho)\\nabla Q_{i}(w)

        v(w, t) & = \\beta v(w, t-1) + \\frac{\\eta} {\\sqrt{r(w,t) - (g(w, t))^2 +
            \\epsilon}} \\nabla Q_{i}(w)

        w & = w - v(w, t)

    where, :math:`\\rho` is a hyperparameter and typical values are 0.9, 0.95
    and so on. :math: `beta` is the momentum term. :math: `\\epsilon` is a
    smoothing term to avoid division by zero, usually set somewhere in range
    from 1e-4 to 1e-8.


    Args:
        learning_rate(float): global learning rate.
        rho(float): rho is :math: `\\rho` in equation, set 0.95 by default.
        epsilon(float): :math: `\\epsilon` in equation is smoothing term to
            avoid division by zero, set 1e-6 by default.
        momentum(float): :math:`\\beta` in equation is the momentum term,
            set 0.0 by default.
        centered(bool): If True, gradients are normalized by the estimated variance of
            the gradient; if False, by the uncentered second moment. Setting this to
            True may help with training, but is slightly more expensive in terms of
            computation and memory. Defaults to False.
        regularization: A Regularizer, such as
                        fluid.regularizer.L2DecayRegularizer.
        name: A optional name prefix.

    Raises:
        ValueError: If learning_rate, rho, epsilon, momentum are None.

    Examples:
          .. code-block:: python

              optimizer = fluid.optimizer.RMSProp(0.0001)
              _, params_grads = optimizer.minimize(cost)
    """

    _momentum_acc_str = "momentum"
    _mean_square_acc_str = "mean_square"
    _mean_grad_acc_str = "mean_grad"

    def __init__(self,
                 learning_rate,
                 rho=0.95,
                 epsilon=1.0e-6,
                 momentum=0.0,
                 centered=False,
                 regularization=None,
                 name=None):
        super(RMSPropOptimizer, self).__init__(
            learning_rate=learning_rate,
            regularization=regularization,
            name=name)
        if learning_rate is None:
            raise ValueError("learning_rate is not set.")
        if rho is None:
            raise ValueError("rho is not set.")
        if epsilon is None:
            raise ValueError("epsilon is not set.")
        if momentum is None:
            raise ValueError("momentum is not set.")

        self.type = "rmsprop"
        self._rho = rho
        self._epsilon = epsilon
        self._momentum = momentum
        self._centered = centered

    def _create_accumulators(self, block, parameters):
        if not isinstance(block, framework.Block):
            raise TypeError("block is not instance of framework.Block.")

        for p in parameters:
            self._add_accumulator(self._momentum_acc_str, p)
            self._add_accumulator(self._mean_square_acc_str, p)
            self._add_accumulator(self._mean_grad_acc_str, p)

    def _append_optimize_op(self, block, param_and_grad):
        if not isinstance(block, framework.Block):
            raise TypeError("block is not instance of framework.Block.")

        momentum_acc = self._get_accumulator(self._momentum_acc_str,
                                             param_and_grad[0])
        mean_square_acc = self._get_accumulator(self._mean_square_acc_str,
                                                param_and_grad[0])
        mean_grad_acc = self._get_accumulator(self._mean_grad_acc_str,
                                              param_and_grad[0])
        rmsprop_op = block.append_op(
            type=self.type,
            inputs={
                "Param": param_and_grad[0],
                "Grad": param_and_grad[1],
                "Moment": momentum_acc,
                "MeanSquare": mean_square_acc,
                "MeanGrad": mean_grad_acc,
                "LearningRate": self._create_param_lr(param_and_grad),
            },
            outputs={
                "ParamOut": param_and_grad[0],
                "MomentOut": momentum_acc,
                "MeanSquareOut": mean_square_acc,
                "MeanGradOut": mean_grad_acc
            },
            attrs={
                "epsilon": self._epsilon,
                "decay": self._rho,
                "momentum": self._momentum,
                "centered": self._centered
            },
            stop_gradient=True)

        return rmsprop_op


class FtrlOptimizer(Optimizer):
    """
    FTRL (Follow The Regularized Leader) Optimizer.

    The paper that proposed Follow The Regularized Leader (FTRL):
    (https://www.eecs.tufts.edu/~dsculley/papers/ad-click-prediction.pdf)

    ..  math::

        &new\_accum = squared\_accum + grad^2

        &if (lr\_power == -0.5):

        &\quad  linear\_accum += grad - \\frac{\\sqrt{new\_accum} - \\sqrt{squared\_accum}}{learning\_rate * param}

        &else:

        &\quad   linear\_accum += grad - \\frac{new\_accum^{-lr\_power} - accum^{-lr\_power}}{learning\_rate * param}


        &x = l1 * sign(linear\_accum) - linear\_accum

        &if (lr\_power == -0.5):

        &\quad   y = \\frac{\\sqrt{new\_accum}}{learning\_rate} + (2 * l2)

        &\quad   pre\_shrink = \\frac{x}{y}

        &\quad   param = (abs(linear\_accum) > l1).select(pre\_shrink, 0.0)

        &else:

        &\quad   y = \\frac{new\_accum^{-lr\_power}}{learning\_rate} + (2 * l2)

        &\quad   pre\_shrink = \\frac{x}{y}

        &\quad   param = (abs(linear\_accum) > l1).select(pre\_shrink, 0.0)

        &squared\_accum += grad^2

    Args:
        learning_rate (float|Variable): global learning rate.
        l1 (float): L1 regularization strength.
        l2 (float): L2 regularization strength.
        lr_power (float): Learning Rate Power.
        regularization: A Regularizer, such as
                        fluid.regularizer.L2DecayRegularizer.
        name: A optional name prefix.

    Raises:
        ValueError: If learning_rate, rho, epsilon, momentum are None.

    Examples:
          .. code-block:: python

              optimizer = fluid.optimizer.Ftrl(0.0001)
              _, params_grads = optimizer.minimize(cost)

    Notes:
       Currently, FtrlOptimizer doesn't support sparse parameter optimization.
    """

    _squared_acc_str = "squared"
    _linear_acc_str = "linear"

    def __init__(self,
                 learning_rate,
                 l1=0.0,
                 l2=0.0,
                 lr_power=-0.5,
                 regularization=None,
                 name=None):
        super(FtrlOptimizer, self).__init__(
            learning_rate=learning_rate,
            regularization=regularization,
            name=name)
        if learning_rate is None:
            raise ValueError("learning_rate is not set.")

        self.type = "ftrl"
        self._l1 = l1
        self._l2 = l2
        self._lr_power = lr_power

    def _create_accumulators(self, block, parameters):
        if not isinstance(block, framework.Block):
            raise TypeError("block is not instance of framework.Block.")

        for p in parameters:
            self._add_accumulator(self._squared_acc_str, p)
            self._add_accumulator(self._linear_acc_str, p)

    def _append_optimize_op(self, block, param_and_grad):
        if not isinstance(block, framework.Block):
            raise TypeError("block is not instance of framework.Block.")

        squared_acc = self._get_accumulator(self._squared_acc_str,
                                            param_and_grad[0])
        linear_acc = self._get_accumulator(self._linear_acc_str,
                                           param_and_grad[0])
        ftrl_op = block.append_op(
            type=self.type,
            inputs={
                "Param": param_and_grad[0],
                "Grad": param_and_grad[1],
                "SquaredAccumulator": squared_acc,
                "LinearAccumulator": linear_acc,
                "LearningRate": self._create_param_lr(param_and_grad),
            },
            outputs={
                "ParamOut": param_and_grad[0],
                "SquaredAccumOut": squared_acc,
                "LinearAccumOut": linear_acc
            },
            attrs={"l1": self._l1,
                   "l2": self._l1,
                   "lr_power": self._lr_power},
            stop_gradient=True)

        return ftrl_op


# We short the class name, since users will use the optimizer with the package
# name. The sample code:
#
# import paddle.fluid as fluid
#
# sgd = fluid.optimizer.SGD(...)
#
# It is no need to add an `Optimizer` as the class suffix
SGD = SGDOptimizer
Momentum = MomentumOptimizer
Adagrad = AdagradOptimizer
Adam = AdamOptimizer
Adamax = AdamaxOptimizer
DecayedAdagrad = DecayedAdagradOptimizer
Adadelta = AdadeltaOptimizer
RMSProp = RMSPropOptimizer
Ftrl = FtrlOptimizer
LarsMomentum = LarsMomentumOptimizer


class ModelAverage(Optimizer):
    """Accumulate the average of parameters whtin sliding window. The average
    result will be saved in temporary variables which can be applied to
    parameter variables of current model by calling 'apply()' method. And the
    'restore()' method is used to restored the parameter values of current model.

    The size of average window is determined by average_window_rate,
    min_average_window, max_average_window and current update times.

    Args:
        average_window_rate: The rate of average window.
        min_average_window: The minimum size of average window.
        max_average_window: The maximum size of average window.
        regularization: A Regularizer, such as
                        fluid.regularizer.L2DecayRegularizer.
        name: A optional name prefix.
    Examples:

      .. code-block:: python

        optimizer = fluid.optimizer.Momentum()
        optimizer.minimize(cost)
        model_average = fluid.optimizer.ModelAverage(0.15,
                                                min_average_window=10000,
                                                max_average_window=20000)
        for pass_id in range(args.pass_num):
            for data in train_reader():
                exe.run(fluid.default_main_program()...)

            with model_average.apply(exe):
                for data in test_reader():
                    exe.run(inference_program...)
    """

    def __init__(self,
                 average_window_rate,
                 min_average_window=10000,
                 max_average_window=10000,
                 regularization=None,
                 name=None):
        super(ModelAverage, self).__init__(
            0.0, regularization=regularization, name=name)
        self.average_window = average_window_rate
        self.min_average_window = min_average_window
        self.max_average_window = max_average_window

        self.params_grads = []
        for param in framework.default_main_program().global_block(
        ).all_parameters():
            if param.do_model_average != False:
                grad = param.block.create_var(
                    name=unique_name.generate(".".join([param.name, 'tmp'])),
                    dtype=param.dtype,
                    persistable=False,
                    stop_gradient=True)
                self.params_grads.append((param, grad))

        for param, grad in self.params_grads:
            if grad is None:
                continue
            with param.block.program._optimized_guard(
                [param, grad]), name_scope('move_average'):
                self._append_average_accumulate_op(param)

        self.apply_program = Program()
        block = self.apply_program.global_block()
        with program_guard(main_program=self.apply_program):
            for param_grad in self.params_grads:
                self._add_average_apply_op(block, param_grad)

        self.restore_program = Program()
        block = self.restore_program.global_block()
        with program_guard(main_program=self.restore_program):
            for param_grad in self.params_grads:
                self._add_average_restore_op(block, param_grad)

    def _add_average_apply_op(self, block, param_grad):
        param = block._clone_variable(param_grad[0])
        grad = block._clone_variable(param_grad[1])
        sum_1 = block._clone_variable(self._get_accumulator('sum_1', param))
        sum_2 = block._clone_variable(self._get_accumulator('sum_2', param))
        sum_3 = block._clone_variable(self._get_accumulator('sum_3', param))
        num_accumulates = block._clone_variable(
            self._get_accumulator('num_accumulates', param))
        old_num_accumulates = block._clone_variable(
            self._get_accumulator('old_num_accumulates', param))
        num_updates = block._clone_variable(
            self._get_accumulator('num_updates', param))
        # backup param value to grad
        layers.assign(input=param, output=grad)
        # param = (sum_1 + sum_2 + sum_3) / (num_accumulates + old_num_accumulates)
        tmp = layers.sum(x=[num_accumulates, old_num_accumulates])
        sum = layers.sum(x=[sum_1, sum_2, sum_3])
        tmp = layers.cast(
            x=tmp, dtype='float32' if self._dtype == None else self._dtype)
        sum = layers.cast(
            x=sum, dtype='float32' if self._dtype == None else self._dtype)
        ops._elementwise_div(x=sum, y=tmp, out=param)

    def _add_average_restore_op(self, block, param_grad):
        param = block._clone_variable(param_grad[0])
        grad = block._clone_variable(param_grad[1])
        layers.assign(input=grad, output=param)

    def _append_average_accumulate_op(self, param):
        self.helper = LayerHelper("average_accumulate")
        sum_1 = self._add_accumulator('sum_1', param)
        sum_2 = self._add_accumulator('sum_2', param)
        sum_3 = self._add_accumulator('sum_3', param)
        num_accumulates = self._add_accumulator(
            'num_accumulates', param, dtype='int64', shape=[1])
        old_num_accumulates = self._add_accumulator(
            'old_num_accumulates', param, dtype='int64', shape=[1])
        num_updates = self._add_accumulator(
            'num_updates', param, dtype='int64', shape=[1])

        self.helper.append_op(
            type='average_accumulates',
            inputs={
                "param": param,
                "in_sum_1": sum_1,
                "in_sum_2": sum_2,
                "in_sum_3": sum_3,
                "in_num_accumulates": num_accumulates,
                "in_old_num_accumulates": old_num_accumulates,
                "in_num_updates": num_updates
            },
            outputs={
                "out_sum_1": sum_1,
                "out_sum_2": sum_2,
                "out_sum_3": sum_3,
                "out_num_accumulates": num_accumulates,
                "out_old_num_accumulates": old_num_accumulates,
                "out_num_updates": num_updates,
            },
            attrs={
                "average_window": self.average_window,
                "min_average_window": self.min_average_window,
                "max_average_window": self.max_average_window,
            },
            stop_gradient=True)

    @signature_safe_contextmanager
    def apply(self, executor, need_restore=True):
        """Apply average values to parameters of current model.
        """
        executor.run(self.apply_program)
        try:
            yield
        finally:
            if need_restore:
                self.restore(executor)

    def restore(self, executor):
        """Restore parameter values of current model.
        """
        executor.run(self.restore_program)