text: """Weight updating functions."""
  import math
  import pickle
  import logging
  import warnings
  import numpy
  from .ndarray import (NDArray, zeros, clip, sqrt, sign, array, maximum, abs as NDabs)
  from .ndarray import (sgd_update, sgd_mom_update, adam_update, rmsprop_update, rmspropalex_update,
                        mp_sgd_update, mp_sgd_mom_update)
  from .random import normal
  
  
  class Optimizer(object):
      """The base class inherited by all optimizers.
  
      Parameters
      ----------
      rescale_grad : float, optional
          Multiply the gradient with `rescale_grad` before updating. Often
          choose to be ``1.0/batch_size``.
  
      param_idx2name : dict from int to string, optional
          A dictionary that maps int index to string name.
  
      clip_gradient : float, optional
          Clip the gradient by projecting onto the box ``[-clip_gradient, clip_gradient]``.
  
      learning_rate : float, optional
          The initial learning rate.
  
      lr_scheduler : LRScheduler, optional
          The learning rate scheduler.
  
      wd : float, optional
          The weight decay (or L2 regularization) coefficient. Modifies objective
          by adding a penalty for having large weights.
  
      sym: Symbol, optional
          The Symbol this optimizer is applying to.
  
      begin_num_update : int, optional
          The initial number of updates.
      """
      def __init__(self, rescale_grad=1., param_idx2name=None, wd=0.,
                   clip_gradient=None, learning_rate=0.01,
                   lr_scheduler=None, sym=None, begin_num_update=0):
          self.rescale_grad = rescale_grad
          self.lr = learning_rate
          self.lr_scheduler = lr_scheduler
          if lr_scheduler is not None:
              self.lr_scheduler.base_lr = learning_rate
  
          self.wd = wd
          self.lr_mult = {}
          self.wd_mult = {}
          self.begin_num_update = begin_num_update
          self.num_update = begin_num_update
          self._index_update_count = {}
          self.clip_gradient = clip_gradient
  
          if param_idx2name is None:
              param_idx2name = {}
          assert isinstance(param_idx2name, dict), \
              'param_idx2name should be a dict of param indexes to names.'
          self.idx2name = param_idx2name.copy()
          self.sym = sym
  
          self.set_lr_mult({})
          self.set_wd_mult({})
  
      opt_registry = {}
  
      @staticmethod
      def register(klass):
          """Registers a new optimizer.
  
          Once an optimizer is registered, we can create an instance of this
          optimizer with `create_optimizer` later.
  
          Examples
          --------
  
          >>> @mx.optimizer.Optimizer.register
          ... class MyOptimizer(mx.optimizer.Optimizer):
          ...     pass
          >>> optim = mx.optimizer.Optimizer.create_optimizer('MyOptimizer')
          >>> print(type(optim))
          <class '__main__.MyOptimizer'>
          """
          assert(isinstance(klass, type))
          name = klass.__name__.lower()
          if name in Optimizer.opt_registry:
              logging.warning('WARNING: New optimizer %s.%s is overriding '
                              'existing optimizer %s.%s',
                              klass.__module__, klass.__name__,
                              Optimizer.opt_registry[name].__module__,
                              Optimizer.opt_registry[name].__name__)
          Optimizer.opt_registry[name] = klass
          return klass
  
      @staticmethod
      def create_optimizer(name, **kwargs):
          """Instantiates an optimizer with a given name and kwargs.
  
          .. note:: We can use the alias `create` for ``Optimizer.create_optimizer``.
  
          Parameters
          ----------
          name: str
              Name of the optimizer. Should be the name
              of a subclass of Optimizer. Case insensitive.
  
          kwargs: dict
              Parameters for the optimizer.
  
          Returns
          -------
          Optimizer
              An instantiated optimizer.
  
          Examples
          --------
          >>> sgd = mx.optimizer.Optimizer.create_optimizer('sgd')
          >>> type(sgd)
          <class 'mxnet.optimizer.SGD'>
          >>> adam = mx.optimizer.create('adam', learning_rate=.1)
          >>> type(adam)
          <class 'mxnet.optimizer.Adam'>
          """
          if name.lower() in Optimizer.opt_registry:
              return Optimizer.opt_registry[name.lower()](**kwargs)
          else:
              raise ValueError('Cannot find optimizer %s' % name)
  
  
      def create_state(self, index, weight):
          """Creates auxiliary state for a given weight.
  
          Some optimizers require additional states, e.g. as momentum, in addition
          to gradients in order to update weights. This function creates state
          for a given weight which will be used in `update`. This function is
          called only once for each weight.
  
          Parameters
          ----------
          index : int
              An unique index to identify the weight.
          weight : NDArray
              The weight.
  
          Returns
          -------
          state : any obj
              The state associated with the weight.
          """
  
      def update(self, index, weight, grad, state):
          """Updates the given parameter using the corresponding gradient and state.
  
          Parameters
          ----------
          index : int
              The unique index of the parameter into the individual learning
              rates and weight decays. Learning rates and weight decay
              may be set via `set_lr_mult()` and `set_wd_mult()`, respectively.
          weight : NDArray
              The parameter to be updated.
          grad : NDArray
              The gradient of the objective with respect to this parameter.
          state : any obj
              The state returned by `create_state()`.
          """
          raise NotImplementedError()
  
      def set_lr_scale(self, args_lrscale): # pylint: disable=unused-argument
          """[DEPRECATED] Sets lr scale. Use set_lr_mult instead."""
          raise DeprecationWarning
  
      def set_lr_mult(self, args_lr_mult):
          """Sets an individual learning rate multiplier for each parameter.
  
          If you specify a learning rate multiplier for a parameter, then
          the learning rate for the parameter will be set as the product of
          the global learning rate `self.lr` and its multiplier.
  
          .. note:: The default learning rate multiplier of a `Variable`
              can be set with `lr_mult` argument in the constructor.
  
          Parameters
          ----------
          args_lr_mult : dict of str/int to float
              For each of its key-value entries, the learning rate multipler for the
              parameter specified in the key will be set as the given value.
  
              You can specify the parameter with either its name or its index.
              If you use the name, you should pass `sym` in the constructor,
              and the name you specified in the key of `args_lr_mult` should match
              the name of the parameter in `sym`. If you use the index, it should
              correspond to the index of the parameter used in the `update` method.
  
              Specifying a parameter by its index is only supported for backward
              compatibility, and we recommend to use the name instead.
          """
          self.lr_mult = {}
          if self.sym is not None:
              attr = self.sym.attr_dict()
              for name in self.sym.list_arguments():
                  if name in attr and '__lr_mult__' in attr[name]:
                      self.lr_mult[name] = float(attr[name]['__lr_mult__'])
          self.lr_mult.update(args_lr_mult)
  
      def set_wd_mult(self, args_wd_mult):
          """Sets an individual weight decay multiplier for each parameter.
  
          By default, if `param_idx2name` was provided in the
          constructor, the weight decay multipler is set as 0 for all
          parameters whose name don't end with ``_weight`` or
          ``_gamma``.
  
          .. note:: The default weight decay multiplier for a `Variable`
              can be set with its `wd_mult` argument in the constructor.
  
          Parameters
          ----------
          args_wd_mult : dict of string/int to float
              For each of its key-value entries, the weight decay multipler for the
              parameter specified in the key will be set as the given value.
  
              You can specify the parameter with either its name or its index.
              If you use the name, you should pass `sym` in the constructor,
              and the name you specified in the key of `args_lr_mult` should match
              the name of the parameter in `sym`. If you use the index, it should
              correspond to the index of the parameter used in the `update` method.
  
              Specifying a parameter by its index is only supported for backward
              compatibility, and we recommend to use the name instead.
          """
          self.wd_mult = {}
          for n in self.idx2name.values():
              if not (n.endswith('_weight') or n.endswith('_gamma')):
                  self.wd_mult[n] = 0.0
          if self.sym is not None:
              attr = self.sym.attr_dict()
              for name in self.sym.list_arguments():
                  if name in attr and '__wd_mult__' in attr[name]:
                      self.wd_mult[name] = float(attr[name]['__wd_mult__'])
          self.wd_mult.update(args_wd_mult)
  
      def _update_count(self, index):
          """Updates num_update.
  
          Parameters
          ----------
          index : int
              The index to be updated.
          """
          if index not in self._index_update_count:
              self._index_update_count[index] = self.begin_num_update
          self._index_update_count[index] += 1
          self.num_update = max(self._index_update_count[index], self.num_update)
  
      def _get_lr(self, index):
          """Gets the learning rate given the index of the weight.
  
          Parameters
          ----------
          index : int
              The index corresponding to the weight.
  
          Returns
          -------
          lr : float
              Learning rate for this index.
          """
          if self.lr_scheduler is not None:
              lr = self.lr_scheduler(self.num_update)
          else:
              lr = self.lr
  
          if index in self.lr_mult:
              lr *= self.lr_mult[index]
          elif index in self.idx2name:
              lr *= self.lr_mult.get(self.idx2name[index], 1.0)
          return lr
  
      def _get_wd(self, index):
          """Gets weight decay for index.
          Returns 0 for non-weights if the name of weights are provided for `__init__`.
  
          Parameters
          ----------
          index : int
              The index for weight.
  
          Returns
          -------
          wd : float
              Weight decay for this index.
          """
          wd = self.wd
          if index in self.wd_mult:
              wd *= self.wd_mult[index]
          elif index in self.idx2name:
              wd *= self.wd_mult.get(self.idx2name[index], 1.0)
          return wd
  
  # convenience wrapper for Optimizer.Register
  register = Optimizer.register   # pylint: disable=invalid-name
  
  @register
  class SGD(Optimizer):
      """The SGD optimizer with momentum and weight decay.
  
      The optimizer updates the weight by::
  
          state = momentum * state + lr * rescale_grad * clip(grad, clip_gradient) + wd * weight
          weight = weight - state
  
      For details of the update algorithm see :class:`~mxnet.ndarray.sgd_update` and
      :class:`~mxnet.ndarray.sgd_mom_update`.
  
      This optimizer accepts the following parameters in addition to those accepted
      by :class:`.Optimizer`.
  
      Parameters
      ----------
      momentum : float, optional
         The momentum value.
      multi_precision: bool, optional
         Flag to control the internal precision of the optimizer.
         ``False`` results in using the same precision as the weights (default),
         ``True`` makes internal 32-bit copy of the weights and applies gradients
                  in 32-bit precision even if actual weights used in the model have lower precision.
                  Turning this on can improve convergence and accuracy when training with float16.
      """
      def __init__(self, momentum=0.0, multi_precision=False, **kwargs):
          super(SGD, self).__init__(**kwargs)
          self.momentum = momentum
          self.multi_precision = multi_precision
  
      def create_state(self, index, weight):
          momentum = None
          weight_master_copy = None
          if self.multi_precision and weight.dtype == numpy.float16:
              weight_master_copy = array(weight, ctx=weight.context, dtype=numpy.float32)
              if self.momentum != 0.0:
                  momentum = zeros(weight.shape, weight.context, dtype=numpy.float32)
              return (momentum, weight_master_copy)
          if weight.dtype == numpy.float16 and not self.multi_precision:
              warnings.warn("Accumulating with float16 in optimizer can lead to "
                            "poor accuracy or slow convergence. "
                            "Consider using multi_precision=True option of the "
                            "SGD optimizer")
          if self.momentum != 0.0:
              momentum = zeros(weight.shape, weight.context, dtype=weight.dtype)
          return momentum
  
      def update(self, index, weight, grad, state):
          assert(isinstance(weight, NDArray))
          assert(isinstance(grad, NDArray))
          lr = self._get_lr(index)
          wd = self._get_wd(index)
          self._update_count(index)
  
          kwargs = {'rescale_grad': self.rescale_grad}
          if self.momentum > 0:
              kwargs['momentum'] = self.momentum
          if self.clip_gradient:
              kwargs['clip_gradient'] = self.clip_gradient
          use_multi_precision = isinstance(state, (list, tuple))
  
          if not use_multi_precision:
              if state is not None:
                  sgd_mom_update(weight, grad, state, out=weight,
                                 lr=lr, wd=wd, **kwargs)
              else:
                  sgd_update(weight, grad, out=weight,
                             lr=lr, wd=wd, **kwargs)
          else:
              if state[0] is not None:
                  mp_sgd_mom_update(weight, grad, state[0], state[1], out=weight,
                                    lr=lr, wd=wd, **kwargs)
              else:
                  mp_sgd_update(weight, grad, state[1], out=weight,
                                lr=lr, wd=wd, **kwargs)
  
  @register
  class DCASGD(Optimizer):
      """The DCASGD optimizer.
  
      This class implements the optimizer described in *Asynchronous Stochastic Gradient Descent
      with Delay Compensation for Distributed Deep Learning*,
      available at https://arxiv.org/abs/1609.08326.
  
      This optimizer accepts the following parameters in addition to those accepted
      by :class:`.Optimizer`.
  
      Parameters
      ----------
      momentum : float, optional
         The momentum value.
  
      lamda : float, optional
         Scale DC value.
      """
      def __init__(self, momentum=0.0, lamda=0.04, **kwargs):
          super(DCASGD, self).__init__(**kwargs)
          self.momentum = momentum
          self.weight_previous = {}
          self.lamda = lamda
  
      def create_state(self, index, weight):
          if self.momentum == 0.0:
              return (None,
                      weight.copy())  # previous weight
          else:
              return (zeros(weight.shape, weight.context, dtype=weight.dtype), # momentum
                      weight.copy())  # previous weight
  
      def update(self, index, weight, grad, state):
          assert(isinstance(weight, NDArray))
          assert(isinstance(grad, NDArray))
          lr = self._get_lr(index)
          wd = self._get_wd(index)
          self._update_count(index)
  
          grad = grad * self.rescale_grad
          if self.clip_gradient is not None:
              grad = clip(grad, -self.clip_gradient, self.clip_gradient)
  
          mom, previous_weight = state
          if mom:
              mom[:] *= self.momentum
              mom[:] += -lr * (grad + wd * weight + self.lamda \
                               * grad * grad * (weight - previous_weight))
          else:
              assert(self.momentum == 0.0)
              mom = -lr * (grad + wd * weight + self.lamda \
                           * grad * grad * (weight - previous_weight))
          previous_weight[:] = weight
          weight[:] += mom
  
  @register
  class NAG(SGD):
      """Nesterov accelerated SGD.
  
      This optimizer updates each weight by::
  
          state = momentum * state + grad + wd * weight
          weight = weight - (lr * (grad + momentum * state))
  
      This optimizer accepts the same arguments as :class:`.SGD`.
      """
      def __init__(self, **kwargs):
          super(NAG, self).__init__(**kwargs)
  
      def update(self, index, weight, grad, state):
          assert(isinstance(weight, NDArray))
          assert(isinstance(grad, NDArray))
          lr = self._get_lr(index)
          wd = self._get_wd(index)
          self._update_count(index)
  
          grad = grad * self.rescale_grad
          if self.clip_gradient is not None:
              grad = clip(grad, -self.clip_gradient, self.clip_gradient)
  
          if state is not None:
              mom = state
              mom[:] *= self.momentum
              grad += wd * weight
              mom[:] += grad
              grad[:] += self.momentum * mom
              weight[:] += -lr * grad
          else:
              assert self.momentum == 0.0
              weight[:] += -lr * (grad + wd * weight)
  
  @register
  class SGLD(Optimizer):
      """Stochastic Gradient Riemannian Langevin Dynamics.
  
      This class implements the optimizer described in the paper *Stochastic Gradient
      Riemannian Langevin Dynamics on the Probability Simplex*, available at
      https://papers.nips.cc/paper/4883-stochastic-gradient-riemannian-langevin-dynamics-on-the-probability-simplex.pdf.
  
      """
      def __init__(self, **kwargs):
          super(SGLD, self).__init__(**kwargs)
  
      def create_state(self, index, weight):
          return None
  
      def update(self, index, weight, grad, state):
          assert(isinstance(weight, NDArray))
          assert(isinstance(grad, NDArray))
          lr = self._get_lr(index)
          wd = self._get_wd(index)
          self._update_count(index)
  
          grad = grad * self.rescale_grad
          if self.clip_gradient is not None:
              grad = clip(grad, -self.clip_gradient, self.clip_gradient)
          weight[:] += - lr/2 * (grad + wd * weight) + normal(0, math.sqrt(lr),
                                                              weight.shape, weight.context)
  
  
  @register  # pylint: disable=invalid-name
  class ccSGD(SGD):
      """[DEPRECATED] Same as `SGD`. Left here for backward compatibility."""
      def __init__(self, *args, **kwargs):
          super(ccSGD, self).__init__(*args, **kwargs)
  
  @register
  class Adam(Optimizer):
      """The Adam optimizer.
  
      This class implements the optimizer described in *Adam: A Method for
      Stochastic Optimization*, available at http://arxiv.org/abs/1412.6980.
  
      This optimizer accepts the following parameters in addition to those accepted
      by :class:`.Optimizer`.
  
      For details of the update algorithm, see :class:`ndarray.adam_update`.
  
      Parameters
      ----------
      beta1 : float, optional
          Exponential decay rate for the first moment estimates.
      beta2 : float, optional
          Exponential decay rate for the second moment estimates.
      epsilon : float, optional
          Small value to avoid division by 0.
      """
      def __init__(self, learning_rate=0.001, beta1=0.9, beta2=0.999, epsilon=1e-8,
                   **kwargs):
          super(Adam, self).__init__(learning_rate=learning_rate, **kwargs)
          self.beta1 = beta1
          self.beta2 = beta2
          self.epsilon = epsilon
  
      def create_state(self, index, weight):
          return (zeros(weight.shape, weight.context, dtype=weight.dtype),  # mean
                  zeros(weight.shape, weight.context, dtype=weight.dtype))  # variance
  
      def update(self, index, weight, grad, state):
          assert(isinstance(weight, NDArray))
          assert(isinstance(grad, NDArray))
          lr = self._get_lr(index)
          wd = self._get_wd(index)
          self._update_count(index)
  
          t = self._index_update_count[index]
          coef1 = 1. - self.beta1**t
          coef2 = 1. - self.beta2**t
          lr *= math.sqrt(coef2)/coef1
  
          kwargs = {'beta1': self.beta1, 'beta2': self.beta2, 'epsilon': self.epsilon,
                    'rescale_grad': self.rescale_grad}
          if self.clip_gradient:
              kwargs['clip_gradient'] = self.clip_gradient
  
          mean, var = state
          adam_update(weight, grad, mean, var, out=weight,
                      lr=lr, wd=wd, **kwargs)
  
  @register
  class AdaGrad(Optimizer):
      """AdaGrad optimizer.
  
      This class implements the AdaGrad optimizer described in *Adaptive Subgradient
      Methods for Online Learning and Stochastic Optimization*, and available at
      http://www.jmlr.org/papers/volume12/duchi11a/duchi11a.pdf.
  
      This optimizer accepts the following parameters in addition to those accepted
      by :class:`.Optimizer`.
  
      Parameters
      ----------
      eps: float, optional
          Small value to avoid division by 0.
      """
      def __init__(self, eps=1e-7, **kwargs):
          super(AdaGrad, self).__init__(**kwargs)
          self.float_stable_eps = eps
  
      def create_state(self, index, weight):
          return zeros(weight.shape, weight.context)  # history
  
      def update(self, index, weight, grad, state):
          assert(isinstance(weight, NDArray))
          assert(isinstance(grad, NDArray))
          lr = self._get_lr(index)
          wd = self._get_wd(index)
          self._update_count(index)
  
          grad = grad * self.rescale_grad
          if self.clip_gradient is not None:
              grad = clip(grad, -self.clip_gradient, self.clip_gradient)
          history = state
          history[:] += (grad * grad)
          weight[:] += -lr * (grad / sqrt(history + self.float_stable_eps) + wd * weight)
  
  @register
  class RMSProp(Optimizer):
      """The RMSProp optimizer.
  
      Two versions of RMSProp are implemented:
  
      If ``centered=False``, we follow
      http://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf by
      Tieleman & Hinton, 2012.
      For details of the update algorithm see :class:`~mxnet.ndarray.rmsprop_update`.
  
      If ``centered=True``, we follow http://arxiv.org/pdf/1308.0850v5.pdf (38)-(45)
      by Alex Graves, 2013.
      For details of the update algorithm see :class:`~mxnet.ndarray.rmspropalex_update`.
  
      This optimizer accepts the following parameters in addition to those accepted
      by :class:`.Optimizer`.
  
      Parameters
      ----------
      gamma1: float, optional
          A decay factor of moving average over past squared gradient.
      gamma2: float, optional
          A "momentum" factor. Only used if `centered`=``True``.
      epsilon : float, optional
          Small value to avoid division by 0.
      centered : bool, optional
          Flag to control which version of RMSProp to use.
          ``True`` will use Graves's version of `RMSProp`,
          ``False`` will use Tieleman & Hinton's version of `RMSProp`.
      clip_weights : float, optional
          Clips weights into range ``[-clip_weights, clip_weights]``.
      """
      def __init__(self, learning_rate=0.001, gamma1=0.9, gamma2=0.9,
                   epsilon=1e-8, centered=False, clip_weights=None, **kwargs):
          super(RMSProp, self).__init__(learning_rate=learning_rate, **kwargs)
          self.gamma1 = gamma1
          self.gamma2 = gamma2
          self.centered = centered
          self.epsilon = epsilon
          self.clip_weights = clip_weights
  
      def create_state(self, index, weight):
          if self.centered:
              return (
                  zeros(weight.shape, weight.context),  # n
                  zeros(weight.shape, weight.context),  # g
                  zeros(weight.shape, weight.context))  # delta
          else:
              return (zeros(weight.shape, weight.context), )  # n
  
      def update(self, index, weight, grad, state):
          assert(isinstance(weight, NDArray))
          assert(isinstance(grad, NDArray))
          lr = self._get_lr(index)
          wd = self._get_wd(index)
          self._update_count(index)
  
          kwargs = {'gamma1': self.gamma1, 'epsilon': self.epsilon,
                    'rescale_grad': self.rescale_grad}
          if self.centered:
              kwargs['gamma2'] = self.gamma2
          if self.clip_gradient:
              kwargs['clip_gradient'] = self.clip_gradient
          if self.clip_weights:
              kwargs['clip_weights'] = self.clip_weights
  
          if not self.centered:
              (n, ) = state
              rmsprop_update(
                  weight, grad, n, out=weight, lr=lr, wd=wd, **kwargs)
          else:
              n, g, delta = state
              rmspropalex_update(weight, grad, n, g, delta, out=weight,
                                 lr=lr, wd=wd, **kwargs)
  
  @register
  class AdaDelta(Optimizer):
      """The AdaDelta optimizer.
  
      This class implements AdaDelta, an optimizer described in  *ADADELTA: An adaptive
      learning rate method*, available at https://arxiv.org/abs/1212.5701.
  
      This optimizer accepts the following parameters in addition to those accepted
      by :class:`.Optimizer`.
  
      Parameters
      ----------
      rho: float
          Decay rate for both squared gradients and delta.
      epsilon : float
          Small value to avoid division by 0.
      """
      def __init__(self, rho=0.90, epsilon=1e-5, **kwargs):
          super(AdaDelta, self).__init__(**kwargs)
          self.rho = rho
          self.epsilon = epsilon
  
      def create_state(self, index, weight):
          return (zeros(weight.shape, weight.context),  # accumulated g
                  zeros(weight.shape, weight.context))  # accumulated delta
  
      def update(self, index, weight, grad, state):
          assert(isinstance(weight, NDArray))
          assert(isinstance(grad, NDArray))
          wd = self._get_wd(index)
          self._update_count(index)
  
          # preprocess grad
          grad *= self.rescale_grad
          if self.clip_gradient is not None:
              grad = clip(grad, -self.clip_gradient, self.clip_gradient)
  
          # accumulated g and delta initlization
          acc_g, acc_delta = state
  
          # update g, delta
          acc_g[:] = self.rho * acc_g + (1. - self.rho) * grad * grad
          current_delta = sqrt(acc_delta + self.epsilon) / sqrt(acc_g + self.epsilon) * grad
          acc_delta[:] = self.rho * acc_delta + (1. - self.rho) * current_delta * current_delta
  
          # update weight
          weight[:] -= current_delta + wd * weight
  
  #pylint: disable=invalid-name
  @register
  class Ftrl(Optimizer):
      """The Ftrl optimizer.
  
      Referenced from *Ad Click Prediction: a View from the Trenches*, available at
      http://dl.acm.org/citation.cfm?id=2488200.
  
      Parameters
      ----------
      lamda1 : float, optional
          L1 regularization coefficient.
      learning_rate : float, optional
          The initial learning rate.
      beta : float, optional
          Per-coordinate learning rate correlation parameter.
      eta :
          .. math::
             \\eta_{t,i} = \\frac{learningrate}{\\beta+\\sqrt{\\sum_{s=1}^tg_{s,i}^t}}
      """
  
      def __init__(self, lamda1=0.01, learning_rate=0.1, beta=1, **kwargs):
          super(Ftrl, self).__init__(**kwargs)
          self.lamda1 = lamda1
          self.beta = beta
          self.lr = learning_rate
  
      def create_state(self, index, weight):
          return (zeros(weight.shape, weight.context),  # dn
                  zeros(weight.shape, weight.context))  # n
  
      def update(self, index, weight, grad, state):
          assert(isinstance(weight, NDArray))
          assert(isinstance(grad, NDArray))
          self._update_count(index)
          wd = self._get_wd(index)
          lr = self._get_lr(index)
  
          # preprocess grad
          grad *= self.rescale_grad
          if self.clip_gradient is not None:
              grad = clip(grad, -self.clip_gradient, self.clip_gradient)
  
          # accumulated g and delta initialization
          dn, n = state
  
          #update dn, n
          dn += grad - (sqrt(n + grad * grad) - sqrt(n)) * weight / lr
          n += grad * grad
  
          # update weight
          weight[:] = (sign(dn) * self.lamda1 - dn) / \
                      ((self.beta + sqrt(n)) / lr + wd) * (NDabs(dn) > self.lamda1)
  
  @register
  class Adamax(Optimizer):
      """The AdaMax optimizer.
  
      It is a variant of Adam based on the infinity norm
      available at http://arxiv.org/abs/1412.6980 Section 7.
  
      This optimizer accepts the following parameters in addition to those accepted
      by :class:`.Optimizer`.
  
      Parameters
      ----------
      beta1 : float, optional
          Exponential decay rate for the first moment estimates.
      beta2 : float, optional
          Exponential decay rate for the second moment estimates.
      """
      def __init__(self, learning_rate=0.002, beta1=0.9, beta2=0.999, **kwargs):
          super(Adamax, self).__init__(learning_rate=learning_rate, **kwargs)
          self.beta1 = beta1
          self.beta2 = beta2
  
      def create_state(self, index, weight):
          return (zeros(weight.shape, weight.context, dtype=weight.dtype),  # mean
                  zeros(weight.shape, weight.context, dtype=weight.dtype))  # variance
  
      def update(self, index, weight, grad, state):
          assert(isinstance(weight, NDArray))
          assert(isinstance(grad, NDArray))
          lr = self._get_lr(index)
          wd = self._get_wd(index)
          self._update_count(index)
  
          t = self._index_update_count[index]
          lr /= (1. - self.beta1**t)
  
          # preprocess grad
          grad = grad * self.rescale_grad + wd * weight
          if self.clip_gradient is not None:
              grad = clip(grad, -self.clip_gradient, self.clip_gradient)
  
          # update m_t and u_t
          m_t, u_t = state
          m_t[:] = self.beta1 * m_t + (1. - self.beta1) * grad
          u_t[:] = maximum(self.beta2 * u_t, NDabs(grad))
  
          # update weight
          weight[:] -= lr * m_t / u_t
  
  @register
  class Nadam(Optimizer):
      """The Nesterov Adam optimizer.
  
      Much like Adam is essentially RMSprop with momentum,
      Nadam is Adam RMSprop with Nesterov momentum available
      at http://cs229.stanford.edu/proj2015/054_report.pdf.
  
      This optimizer accepts the following parameters in addition to those accepted
      by :class:`.Optimizer`.
  
      Parameters
      ----------
      beta1 : float, optional
          Exponential decay rate for the first moment estimates.
      beta2 : float, optional
          Exponential decay rate for the second moment estimates.
      epsilon : float, optional
          Small value to avoid division by 0.
      schedule_decay : float, optional
          Exponential decay rate for the momentum schedule
      """
      def __init__(self, learning_rate=0.001, beta1=0.9, beta2=0.999, epsilon=1e-8,
                   schedule_decay=0.004, **kwargs):
          super(Nadam, self).__init__(learning_rate=learning_rate, **kwargs)
          self.beta1 = beta1
          self.beta2 = beta2
          self.epsilon = epsilon
          self.schedule_decay = schedule_decay
          self.m_schedule = 1.
  
      def create_state(self, index, weight):
          return (zeros(weight.shape, weight.context, dtype=weight.dtype),  # mean
                  zeros(weight.shape, weight.context, dtype=weight.dtype))  # variance
  
      def update(self, index, weight, grad, state):
          assert(isinstance(weight, NDArray))
          assert(isinstance(grad, NDArray))
          lr = self._get_lr(index)
          wd = self._get_wd(index)
          self._update_count(index)
  
          t = self._index_update_count[index]
  
          # preprocess grad
          grad *= self.rescale_grad + wd * weight
          if self.clip_gradient is not None:
              grad = clip(grad, -self.clip_gradient, self.clip_gradient)
  
          # warming momentum schedule
          momentum_t = self.beta1 * (1. - 0.5 * (pow(0.96, t * self.schedule_decay)))
          momentum_t_1 = self.beta1 * (1. - 0.5 * (pow(0.96, (t + 1) * self.schedule_decay)))
          self.m_schedule = self.m_schedule * momentum_t
          m_schedule_next = self.m_schedule * momentum_t_1
  
          # update m_t and v_t
          m_t, v_t = state
          m_t[:] = self.beta1 * m_t + (1. - self.beta1) * grad
          v_t[:] = self.beta2 * v_t + (1. - self.beta2) * grad * grad
  
          grad_prime = grad / (1. - self.m_schedule)
          m_t_prime = m_t / (1. - m_schedule_next)
          v_t_prime = v_t / (1. - pow(self.beta2, t))
          m_t_bar = (1. - momentum_t) * grad_prime + momentum_t_1 * m_t_prime
  
          # update weight
          weight[:] -= lr * m_t_bar / (sqrt(v_t_prime) + self.epsilon)
  
  @register
  class Test(Optimizer):
      """The Test optimizer"""
      def __init__(self, **kwargs):
          super(Test, self).__init__(**kwargs)
  
      def create_state(self, index, weight):
          """Creates a state to duplicate weight."""
          return zeros(weight.shape, weight.context)
  
      def update(self, index, weight, grad, state):
          """Performs w += rescale_grad * grad."""
          weight[:] += grad * self.rescale_grad
          state[:] = weight
  
  # backward compatibility wrapper for Optimizer.CreateOptimizer
  create = Optimizer.create_optimizer  # pylint: disable=invalid-name
  
  class Updater(object):
      """Updater for kvstore."""
      def __init__(self, optimizer):
          self.optimizer = optimizer
          self.states = {}
          self.states_synced = {}
  
      def __call__(self, index, grad, weight):
          """Updates weight given gradient and index."""
          if index not in self.states:
              self.states[index] = self.optimizer.create_state(index, weight)
              self.states_synced[index] = True
          elif not self.states_synced[index]:
              self.states[index] = \
                  self.sync_state_context(self.states[index], weight.context)
              self.states_synced[index] = True
          self.optimizer.update(index, weight, grad, self.states[index])
  
      def sync_state_context(self, state, context):
          if isinstance(state, NDArray):
              return state.as_in_context(context)
          elif isinstance(state, (tuple, list)):
              synced_state = (self.sync_state_context(i, context) for i in state)
              if isinstance(state, tuple):
                  return tuple(synced_state)
              else:
                  return list(synced_state)
          else:
              return state
  
      def set_states(self, states):
          """Sets updater states."""
          self.states = pickle.loads(states)
          self.states_synced = dict.fromkeys(self.states.keys(), False)
  
      def get_states(self):
          """Gets updater states."""
          return pickle.dumps(self.states)
  
  def get_updater(optimizer):
      """Returns a closure of the updater needed for kvstore.
  
      Parameters
      ----------
      optimizer: Optimizer
           The optimizer.
  
      Returns
      -------
      updater: function
           The closure of the updater.
      """
      return Updater(optimizer)
  