# Copyright (c) 2016 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.

import paddle.trainer_config_helpers.config_parser_utils as config_parser_utils
import paddle.trainer_config_helpers.optimizers as v1_optimizers
from paddle.proto.OptimizerConfig_pb2 import OptimizerConfig

__all__ = [
    'Momentum', 'Adam', 'Adamax', 'AdaGrad', 'DecayedAdaGrad', 'AdaDelta',
    'RMSProp', 'ModelAverage', 'L2Regularization'
]


class Optimizer(object):
    def __init__(self, **kwargs):
        import py_paddle.swig_paddle as swig_api
        if 'batch_size' in kwargs:
            del kwargs['batch_size']  # not important for python library.

        def __impl__():
            v1_optimizers.settings(batch_size=1, **kwargs)

        self.__opt_conf_proto__ = config_parser_utils.parse_optimizer_config(
            __impl__)
        self.__opt_conf__ = swig_api.OptimizationConfig.createFromProto(
            self.__opt_conf_proto__)

    def enable_types(self):
        """
        get enable_types for each optimizer.
        enable_types = [value, gradient, momentum, etc]
        For each optimizer(SGD, Adam), GradientMachine should enable different
        buffers.
        """
        import py_paddle.swig_paddle as swig_api
        tmp = swig_api.ParameterOptimizer.create(self.__opt_conf__)
        assert isinstance(tmp, swig_api.ParameterOptimizer)
        return tmp.getParameterTypes()

    def __create_local_updater__(self):
        import py_paddle.swig_paddle as swig_api
        return swig_api.ParameterUpdater.createLocalUpdater(self.__opt_conf__)

    def __create_remote_updater__(self, pass_num, use_sparse_updater):
        import py_paddle.swig_paddle as swig_api
        return swig_api.ParameterUpdater.createRemoteUpdater(
            self.__opt_conf__, pass_num, use_sparse_updater)

    def __create_new_remote_updater__(self, pserver_spec, use_etcd):
        import py_paddle.swig_paddle as swig_api
        return swig_api.ParameterUpdater.createNewRemoteUpdater(
            self.__opt_conf__, pserver_spec, use_etcd)

    def create_updater(self, is_local, num_passes, use_sparse_updater,
                       pserver_spec, use_etcd):
        """
        create proper parameter_updater by configuration.
        :param is_local: create local or remote parameter updater
        :param num_passes: remote parameter updater will use this to config
        parameter server.
        :param use_sparse_updater: when use remote updater, if some parameter is
        sparse, updater should do some extra thing:

        ..  code-block:: python

            if use_sparse_remote_updater:
                        gradient_machine.prefetch(in_args)
                        parameter_updater.getParametersRemote()

        :param pserver_spec: pserver location, eg: localhost:3000, if use etcd,
        pserver_spec should be the etcd endpoints, eg: http://localhost:2379
        :return: parameter_updater
        """
        if is_local:
            parameter_updater = self.__create_local_updater__()
        else:
            if pserver_spec is None:
                parameter_updater = self.__create_remote_updater__(
                    num_passes, use_sparse_updater)
            else:
                parameter_updater = self.__create_new_remote_updater__(
                    pserver_spec, use_etcd)
        return parameter_updater


class Momentum(Optimizer):
    """
    Momentum Optimizer.

    When sparse=False, the momentum update formula is as follows:

    ..  math::

        v_{t} &= k * v_{t-1} - \\gamma_t (g_{t} + \\lambda w_{t-1}) \\\\
        w_{t} &= w_{t-1} + v_{t} \\\\

    where, :math:`k` is momentum, :math:`\\lambda` is decay rate,
    :math:`\\gamma_t` is learning rate at the t'th iteration.
    :math:`w_{t}` is the weight as the t'th iteration.
    And the :math:`v_{t}` is the history momentum variable.

    When sparse=True, the update scheme:

    ..  math::

        \\alpha_t &= \\alpha_{t-1} / k \\\\
        \\beta_t &= \\beta_{t-1} / (1 + \\lambda \\gamma_t) \\\\
        u_t &= u_{t-1} - \\alpha_t \\gamma_t g_t \\\\
        v_t &= v_{t-1} + \\tau_{t-1} \\alpha_t \\gamma_t g_t \\\\
        \\tau_t &= \\tau_{t-1} + \\beta_t / \\alpha_t
    
    where :math:`k` is momentum, :math:`\\lambda` is decay rate, 
    :math:`\\gamma_t` is learning rate at the t'th iteration.

    :param momentum: the momentum factor.
    :type momentum: float
    :param sparse: with sparse support or not, False by default.
    :type sparse: bool
    """

    def __init__(self, momentum=None, sparse=False, **kwargs):
        learning_method = v1_optimizers.MomentumOptimizer(
            momentum=momentum, sparse=sparse)
        super(Momentum, self).__init__(
            learning_method=learning_method, **kwargs)


class Adam(Optimizer):
    """
    Adam optimizer.
    The details of please refer `Adam: A Method for Stochastic Optimization
    <https://arxiv.org/abs/1412.6980>`_

    ..  math::

        m(w, t) & = \\beta_1 m(w, t-1) + (1 - \\beta_1) \\nabla Q_i(w) \\\\
        v(w, t) & = \\beta_2 v(w, t-1) + (1 - \\beta_2)(\\nabla Q_i(w)) ^2 \\\\
        w & = w - \\frac{\\eta m(w, t)}{\\sqrt{v(w,t) + \\epsilon}}

    :param beta1: the :math:`\\beta_1` in equation.
    :type beta1: float
    :param beta2: the :math:`\\beta_2` in equation.
    :type beta2: float
    :param epsilon: the :math:`\\epsilon` in equation. It is used to prevent
                        divided by zero.
    :type epsilon: float
    """

    def __init__(self, beta1=0.9, beta2=0.999, epsilon=1e-8, **kwargs):
        learning_method = v1_optimizers.AdamOptimizer(
            beta1=beta1, beta2=beta2, epsilon=epsilon)
        super(Adam, self).__init__(learning_method=learning_method, **kwargs)


class Adamax(Optimizer):
    """
    Adamax optimizer.

    The details of please refer this `Adam: A Method for Stochastic Optimization
    <https://arxiv.org/abs/1412.6980>`_

    ..  math::

        m_t & = \\beta_1 * m_{t-1} + (1-\\beta_1)* \\nabla Q_i(w) \\\\
        u_t & = max(\\beta_2*u_{t-1}, abs(\\nabla Q_i(w))) \\\\
        w_t & = w_{t-1} - (\\eta/(1-\\beta_1^t))*m_t/u_t

    :param beta1: the :math:`\\beta_1` in the equation.
    :type beta1: float
    :param beta2: the :math:`\\beta_2` in the equation.
    :type beta2: float
    """

    def __init__(self, beta1=0.9, beta2=0.999, **kwargs):
        learning_method = v1_optimizers.AdamaxOptimizer(
            beta1=beta1, beta2=beta2)
        super(Adamax, self).__init__(learning_method=learning_method, **kwargs)


class AdaGrad(Optimizer):
    """
    Adagrad(for ADAptive GRAdient algorithm) optimizer.

    For details please refer this `Adaptive Subgradient Methods for
    Online Learning and Stochastic Optimization
    <http://www.magicbroom.info/Papers/DuchiHaSi10.pdf>`_.

    ..  math::

        G &= \\sum_{\\tau=1}^{t} g_{\\tau} g_{\\tau}^T \\\\
        w & = w - \\eta diag(G)^{-\\frac{1}{2}} \\circ g
    """

    def __init__(self, **kwargs):
        learning_method = v1_optimizers.AdaGradOptimizer()
        super(AdaGrad, self).__init__(learning_method=learning_method, **kwargs)


class DecayedAdaGrad(Optimizer):
    """
    AdaGrad method with decayed sum gradients. The equations of this method
    show as follow.

    ..  math::

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

    :param rho: The :math:`\\rho` parameter in that equation
    :type rho: float
    :param epsilon: The :math:`\\epsilon` parameter in that equation.
    :type epsilon: float
    """

    def __init__(self, rho=0.95, epsilon=1e-06, **kwargs):
        learning_method = v1_optimizers.DecayedAdaGradOptimizer(
            rho=rho, epsilon=epsilon)
        super(DecayedAdaGrad, self).__init__(
            learning_method=learning_method, **kwargs)


class AdaDelta(Optimizer):
    """
    AdaDelta method. 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

    :param rho: :math:`\\rho` in equation
    :type rho: float
    :param epsilon: :math:`\\rho` in equation
    :type epsilon: float
    """

    def __init__(self, rho=0.95, epsilon=1e-06, **kwargs):
        learning_method = v1_optimizers.AdaDeltaOptimizer(
            rho=rho, epsilon=epsilon)
        super(AdaDelta, self).__init__(
            learning_method=learning_method, **kwargs)


class RMSProp(Optimizer):
    """
    RMSProp(for Root Mean Square Propagation) optimizer. For details please
    refer this `slide <http://www.cs.toronto.edu/~tijmen/csc321/slides/
    lecture_slides_lec6.pdf>`_.

    The equations of this method as follows:

    ..  math::

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

    :param rho: the :math:`\\rho` in the equation. The forgetting factor.
    :type rho: float
    :param epsilon: the :math:`\\epsilon` in the equation.
    :type epsilon: float
    """

    def __init__(self, rho=0.95, epsilon=1e-6, **kwargs):
        learning_method = v1_optimizers.RMSPropOptimizer(
            rho=rho, epsilon=epsilon)
        super(RMSProp, self).__init__(learning_method=learning_method, **kwargs)


ModelAverage = v1_optimizers.ModelAverage
L2Regularization = v1_optimizers.L2Regularization

if __name__ == '__main__':
    import py_paddle.swig_paddle as swig_api
    swig_api.initPaddle('--use_gpu=false')
    for opt in [
            Momentum(), Adam(), Adamax(), AdaGrad(), DecayedAdaGrad(),
            AdaDelta(), RMSProp(), Adam(
                model_average=ModelAverage(average_window=0.5),
                regularization=L2Regularization(rate=0.5),
                gradient_clipping_threshold=25)
    ]:
        print opt, opt.enable_types()

# Copyright (c) 2016 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.

import paddle.trainer_config_helpers.config_parser_utils as config_parser_utils
import paddle.trainer_config_helpers.optimizers as v1_optimizers
from paddle.proto.OptimizerConfig_pb2 import OptimizerConfig

__all__ = [
    'Momentum', 'Adam', 'Adamax', 'AdaGrad', 'DecayedAdaGrad', 'AdaDelta',
    'RMSProp', 'ModelAverage', 'L2Regularization'
]


class Optimizer(object):
    def __init__(self, **kwargs):
        import py_paddle.swig_paddle as swig_api
        if 'batch_size' in kwargs:
            del kwargs['batch_size']  # not important for python library.

        def __impl__():
            v1_optimizers.settings(batch_size=1, **kwargs)

        self.__opt_conf_proto__ = config_parser_utils.parse_optimizer_config(
            __impl__)
        self.__opt_conf__ = swig_api.OptimizationConfig.createFromProto(
            self.__opt_conf_proto__)

    def enable_types(self):
        """
        get enable_types for each optimizer.
        enable_types = [value, gradient, momentum, etc]
        For each optimizer(SGD, Adam), GradientMachine should enable different
        buffers.
        """
        import py_paddle.swig_paddle as swig_api
        tmp = swig_api.ParameterOptimizer.create(self.__opt_conf__)
        assert isinstance(tmp, swig_api.ParameterOptimizer)
        return tmp.getParameterTypes()

    def __create_local_updater__(self):
        import py_paddle.swig_paddle as swig_api
        return swig_api.ParameterUpdater.createLocalUpdater(self.__opt_conf__)

    def __create_remote_updater__(self, pass_num, use_sparse_updater):
        import py_paddle.swig_paddle as swig_api
        return swig_api.ParameterUpdater.createRemoteUpdater(
            self.__opt_conf__, pass_num, use_sparse_updater)

    def __create_new_remote_updater__(self, pserver_spec, use_etcd):
        import py_paddle.swig_paddle as swig_api
        return swig_api.ParameterUpdater.createNewRemoteUpdater(
            self.__opt_conf__, pserver_spec, use_etcd)

    def create_updater(self, is_local, num_passes, use_sparse_updater,
                       pserver_spec, use_etcd):
        """
        create proper parameter_updater by configuration.
        :param is_local: create local or remote parameter updater
        :param num_passes: remote parameter updater will use this to config
        parameter server.
        :param use_sparse_updater: when use remote updater, if some parameter is
        sparse, updater should do some extra thing:

        ..  code-block:: python

            if use_sparse_remote_updater:
                        gradient_machine.prefetch(in_args)
                        parameter_updater.getParametersRemote()

        :param pserver_spec: pserver location, eg: localhost:3000, if use etcd,
        pserver_spec should be the etcd endpoints, eg: http://localhost:2379
        :return: parameter_updater
        """
        if is_local:
            parameter_updater = self.__create_local_updater__()
        else:
            if pserver_spec is None:
                parameter_updater = self.__create_remote_updater__(
                    num_passes, use_sparse_updater)
            else:
                parameter_updater = self.__create_new_remote_updater__(
                    pserver_spec, use_etcd)
        return parameter_updater


class Momentum(Optimizer):
    """
    Momentum Optimizer.

    When sparse=False, the momentum update formula is as follows:

    ..  math::

        v_{t} &= k * v_{t-1} - \\gamma_t (g_{t} + \\lambda w_{t-1}) \\\\
        w_{t} &= w_{t-1} + v_{t} \\\\

    where, :math:`k` is momentum, :math:`\\lambda` is decay rate,
    :math:`\\gamma_t` is learning rate at the t'th iteration.
    :math:`w_{t}` is the weight as the t'th iteration.
    And the :math:`v_{t}` is the history momentum variable.

    When sparse=True, the update scheme:

    ..  math::

        \\alpha_t &= \\alpha_{t-1} / k \\\\
        \\beta_t &= \\beta_{t-1} / (1 + \\lambda \\gamma_t) \\\\
        u_t &= u_{t-1} - \\alpha_t \\gamma_t g_t \\\\
        v_t &= v_{t-1} + \\tau_{t-1} \\alpha_t \\gamma_t g_t \\\\
        \\tau_t &= \\tau_{t-1} + \\beta_t / \\alpha_t
    
    where :math:`k` is momentum, :math:`\\lambda` is decay rate, 
    :math:`\\gamma_t` is learning rate at the t'th iteration.

    :param momentum: the momentum factor.
    :type momentum: float
    :param sparse: with sparse support or not, False by default.
    :type sparse: bool
    """

    def __init__(self, momentum=None, sparse=False, **kwargs):
        learning_method = v1_optimizers.MomentumOptimizer(
            momentum=momentum, sparse=sparse)
        super(Momentum, self).__init__(
            learning_method=learning_method, **kwargs)


class Adam(Optimizer):
    """
    Adam optimizer.
    The details of please refer `Adam: A Method for Stochastic Optimization
    <https://arxiv.org/abs/1412.6980>`_

    ..  math::

        m(w, t) & = \\beta_1 m(w, t-1) + (1 - \\beta_1) \\nabla Q_i(w) \\\\
        v(w, t) & = \\beta_2 v(w, t-1) + (1 - \\beta_2)(\\nabla Q_i(w)) ^2 \\\\
        w & = w - \\frac{\\eta m(w, t)}{\\sqrt{v(w,t) + \\epsilon}}

    :param beta1: the :math:`\\beta_1` in equation.
    :type beta1: float
    :param beta2: the :math:`\\beta_2` in equation.
    :type beta2: float
    :param epsilon: the :math:`\\epsilon` in equation. It is used to prevent
                        divided by zero.
    :type epsilon: float
    """

    def __init__(self, beta1=0.9, beta2=0.999, epsilon=1e-8, **kwargs):
        learning_method = v1_optimizers.AdamOptimizer(
            beta1=beta1, beta2=beta2, epsilon=epsilon)
        super(Adam, self).__init__(learning_method=learning_method, **kwargs)


class Adamax(Optimizer):
    """
    Adamax optimizer.

    The details of please refer this `Adam: A Method for Stochastic Optimization
    <https://arxiv.org/abs/1412.6980>`_

    ..  math::

        m_t & = \\beta_1 * m_{t-1} + (1-\\beta_1)* \\nabla Q_i(w) \\\\
        u_t & = max(\\beta_2*u_{t-1}, abs(\\nabla Q_i(w))) \\\\
        w_t & = w_{t-1} - (\\eta/(1-\\beta_1^t))*m_t/u_t

    :param beta1: the :math:`\\beta_1` in the equation.
    :type beta1: float
    :param beta2: the :math:`\\beta_2` in the equation.
    :type beta2: float
    """

    def __init__(self, beta1=0.9, beta2=0.999, **kwargs):
        learning_method = v1_optimizers.AdamaxOptimizer(
            beta1=beta1, beta2=beta2)
        super(Adamax, self).__init__(learning_method=learning_method, **kwargs)


class AdaGrad(Optimizer):
    """
    Adagrad(for ADAptive GRAdient algorithm) optimizer.

    For details please refer this `Adaptive Subgradient Methods for
    Online Learning and Stochastic Optimization
    <http://www.magicbroom.info/Papers/DuchiHaSi10.pdf>`_.

    ..  math::

        G &= \\sum_{\\tau=1}^{t} g_{\\tau} g_{\\tau}^T \\\\
        w & = w - \\eta diag(G)^{-\\frac{1}{2}} \\circ g
    """

    def __init__(self, **kwargs):
        learning_method = v1_optimizers.AdaGradOptimizer()
        super(AdaGrad, self).__init__(learning_method=learning_method, **kwargs)


class DecayedAdaGrad(Optimizer):
    """
    AdaGrad method with decayed sum gradients. The equations of this method
    show as follow.

    ..  math::

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

    :param rho: The :math:`\\rho` parameter in that equation
    :type rho: float
    :param epsilon: The :math:`\\epsilon` parameter in that equation.
    :type epsilon: float
    """

    def __init__(self, rho=0.95, epsilon=1e-06, **kwargs):
        learning_method = v1_optimizers.DecayedAdaGradOptimizer(
            rho=rho, epsilon=epsilon)
        super(DecayedAdaGrad, self).__init__(
            learning_method=learning_method, **kwargs)


class AdaDelta(Optimizer):
    """
    AdaDelta method. 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

    :param rho: :math:`\\rho` in equation
    :type rho: float
    :param epsilon: :math:`\\rho` in equation
    :type epsilon: float
    """

    def __init__(self, rho=0.95, epsilon=1e-06, **kwargs):
        learning_method = v1_optimizers.AdaDeltaOptimizer(
            rho=rho, epsilon=epsilon)
        super(AdaDelta, self).__init__(
            learning_method=learning_method, **kwargs)


class RMSProp(Optimizer):
    """
    RMSProp(for Root Mean Square Propagation) optimizer. For details please
    refer this `slide <http://www.cs.toronto.edu/~tijmen/csc321/slides/
    lecture_slides_lec6.pdf>`_.

    The equations of this method as follows:

    ..  math::

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

    :param rho: the :math:`\\rho` in the equation. The forgetting factor.
    :type rho: float
    :param epsilon: the :math:`\\epsilon` in the equation.
    :type epsilon: float
    """

    def __init__(self, rho=0.95, epsilon=1e-6, **kwargs):
        learning_method = v1_optimizers.RMSPropOptimizer(
            rho=rho, epsilon=epsilon)
        super(RMSProp, self).__init__(learning_method=learning_method, **kwargs)


ModelAverage = v1_optimizers.ModelAverage
L2Regularization = v1_optimizers.L2Regularization

if __name__ == '__main__':
    import py_paddle.swig_paddle as swig_api
    swig_api.initPaddle('--use_gpu=false')
    for opt in [
            Momentum(), Adam(), Adamax(), AdaGrad(), DecayedAdaGrad(),
            AdaDelta(), RMSProp(), Adam(
                model_average=ModelAverage(average_window=0.5),
                regularization=L2Regularization(rate=0.5),
                gradient_clipping_threshold=25)
    ]:
        print opt, opt.enable_types()

# Copyright (c) 2016 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. # not important for python library. get enable_types for each optimizer.         enable_types = [value, gradient, momentum, etc]         For each optimizer(SGD, Adam), GradientMachine should enable different         buffers. create proper parameter_updater by configuration.         :param is_local: create local or remote parameter updater         :param num_passes: remote parameter updater will use this to config         parameter server.         :param use_sparse_updater: when use remote updater, if some parameter is         sparse, updater should do some extra thing:          ..  code-block:: python              if use_sparse_remote_updater:                         gradient_machine.prefetch(in_args)                         parameter_updater.getParametersRemote()          :param pserver_spec: pserver location, eg: localhost:3000, if use etcd,         pserver_spec should be the etcd endpoints, eg: http://localhost:2379         :return: parameter_updater Momentum Optimizer.      When sparse=False, the momentum update formula is as follows:      ..  math::          v_{t} &= k * v_{t-1} - \\gamma_t (g_{t} + \\lambda w_{t-1}) \\\\         w_{t} &= w_{t-1} + v_{t} \\\\      where, :math:`k` is momentum, :math:`\\lambda` is decay rate,     :math:`\\gamma_t` is learning rate at the t'th iteration.     :math:`w_{t}` is the weight as the t'th iteration.     And the :math:`v_{t}` is the history momentum variable.      When sparse=True, the update scheme:      ..  math::          \\alpha_t &= \\alpha_{t-1} / k \\\\         \\beta_t &= \\beta_{t-1} / (1 + \\lambda \\gamma_t) \\\\         u_t &= u_{t-1} - \\alpha_t \\gamma_t g_t \\\\         v_t &= v_{t-1} + \\tau_{t-1} \\alpha_t \\gamma_t g_t \\\\         \\tau_t &= \\tau_{t-1} + \\beta_t / \\alpha_t          where :math:`k` is momentum, :math:`\\lambda` is decay rate,      :math:`\\gamma_t` is learning rate at the t'th iteration.      :param momentum: the momentum factor.     :type momentum: float     :param sparse: with sparse support or not, False by default.     :type sparse: bool Adam optimizer.     The details of please refer `Adam: A Method for Stochastic Optimization     <https://arxiv.org/abs/1412.6980>`_      ..  math::          m(w, t) & = \\beta_1 m(w, t-1) + (1 - \\beta_1) \\nabla Q_i(w) \\\\         v(w, t) & = \\beta_2 v(w, t-1) + (1 - \\beta_2)(\\nabla Q_i(w)) ^2 \\\\         w & = w - \\frac{\\eta m(w, t)}{\\sqrt{v(w,t) + \\epsilon}}      :param beta1: the :math:`\\beta_1` in equation.     :type beta1: float     :param beta2: the :math:`\\beta_2` in equation.     :type beta2: float     :param epsilon: the :math:`\\epsilon` in equation. It is used to prevent                         divided by zero.     :type epsilon: float Adamax optimizer.      The details of please refer this `Adam: A Method for Stochastic Optimization     <https://arxiv.org/abs/1412.6980>`_      ..  math::          m_t & = \\beta_1 * m_{t-1} + (1-\\beta_1)* \\nabla Q_i(w) \\\\         u_t & = max(\\beta_2*u_{t-1}, abs(\\nabla Q_i(w))) \\\\         w_t & = w_{t-1} - (\\eta/(1-\\beta_1^t))*m_t/u_t      :param beta1: the :math:`\\beta_1` in the equation.     :type beta1: float     :param beta2: the :math:`\\beta_2` in the equation.     :type beta2: float Adagrad(for ADAptive GRAdient algorithm) optimizer.      For details please refer this `Adaptive Subgradient Methods for     Online Learning and Stochastic Optimization     <http://www.magicbroom.info/Papers/DuchiHaSi10.pdf>`_.      ..  math::          G &= \\sum_{\\tau=1}^{t} g_{\\tau} g_{\\tau}^T \\\\         w & = w - \\eta diag(G)^{-\\frac{1}{2}} \\circ g AdaGrad method with decayed sum gradients. The equations of this method     show as follow.      ..  math::          E(g_t^2) &= \\rho * E(g_{t-1}^2) + (1-\\rho) * g^2 \\\\         learning\\_rate &= 1/sqrt( ( E(g_t^2) + \\epsilon )      :param rho: The :math:`\\rho` parameter in that equation     :type rho: float     :param epsilon: The :math:`\\epsilon` parameter in that equation.     :type epsilon: float AdaDelta method. 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      :param rho: :math:`\\rho` in equation     :type rho: float     :param epsilon: :math:`\\rho` in equation     :type epsilon: float RMSProp(for Root Mean Square Propagation) optimizer. For details please     refer this `slide <http://www.cs.toronto.edu/~tijmen/csc321/slides/     lecture_slides_lec6.pdf>`_.      The equations of this method as follows:      ..  math::          v(w, t) & = \\rho v(w, t-1) + (1 - \\rho)(\\nabla Q_{i}(w))^2 \\\\         w & = w - \\frac{\\eta} {\\sqrt{v(w,t) + \\epsilon}} \\nabla Q_{i}(w)      :param rho: the :math:`\\rho` in the equation. The forgetting factor.     :type rho: float     :param epsilon: the :math:`\\epsilon` in the equation.     :type epsilon: float