text: import math
  import torch
  import torch.optim as optim
  from collections import defaultdict
  
  
  class SharedRMSprop(optim.Optimizer):
      """Implements RMSprop algorithm with shared states.
      """
  
      def __init__(self,
                   params,
                   lr=7e-4,
                   alpha=0.99,
                   eps=0.1,
                   weight_decay=0,
                   momentum=0,
                   centered=False):
          defaults = defaultdict(
              lr=lr,
              alpha=alpha,
              eps=eps,
              weight_decay=weight_decay,
              momentum=momentum,
              centered=centered)
          super(SharedRMSprop, self).__init__(params, defaults)
  
          for group in self.param_groups:
              for p in group['params']:
                  state = self.state[p]
                  state['step'] = torch.zeros(1)
                  state['grad_avg'] = p.data.new().resize_as_(p.data).zero_()
                  state['square_avg'] = p.data.new().resize_as_(p.data).zero_()
                  state['momentum_buffer'] = p.data.new().resize_as_(
                      p.data).zero_()
  
      def share_memory(self):
          for group in self.param_groups:
              for p in group['params']:
                  state = self.state[p]
                  state['square_avg'].share_memory_()
                  state['step'].share_memory_()
                  state['grad_avg'].share_memory_()
                  state['momentum_buffer'].share_memory_()
  
      def step(self, closure=None):
          """Performs a single optimization step.
          Arguments:
              closure (callable, optional): A closure that reevaluates the model
                  and returns the loss.
          """
          loss = None
          if closure is not None:
              loss = closure()
  
          for group in self.param_groups:
              for p in group['params']:
                  if p.grad is None:
                      continue
                  grad = p.grad.data
                  if grad.is_sparse:
                      raise RuntimeError(
                          'RMSprop does not support sparse gradients')
                  state = self.state[p]
  
                  square_avg = state['square_avg']
                  alpha = group['alpha']
  
                  state['step'] += 1
  
                  if group['weight_decay'] != 0:
                      grad = grad.add(group['weight_decay'], p.data)
  
                  square_avg.mul_(alpha).addcmul_(1 - alpha, grad, grad)
  
                  if group['centered']:
                      grad_avg = state['grad_avg']
                      grad_avg.mul_(alpha).add_(1 - alpha, grad)
                      avg = square_avg.addcmul(-1, grad_avg,
                                               grad_avg).sqrt().add_(
                                                   group['eps'])
                  else:
                      avg = square_avg.sqrt().add_(group['eps'])
  
                  if group['momentum'] > 0:
                      buf = state['momentum_buffer']
                      buf.mul_(group['momentum']).addcdiv_(grad, avg)
                      p.data.add_(-group['lr'], buf)
                  else:
                      p.data.addcdiv_(-group['lr'], grad, avg)
  
          return loss
  
  
  class SharedAdam(optim.Optimizer):
      """Implements Adam algorithm with shared states.
      """
  
      def __init__(self,
                   params,
                   lr=1e-3,
                   betas=(0.9, 0.999),
                   eps=1e-3,
                   weight_decay=0,
                   amsgrad=False):
          defaults = defaultdict(
              lr=lr,
              betas=betas,
              eps=eps,
              weight_decay=weight_decay,
              amsgrad=amsgrad)
          super(SharedAdam, self).__init__(params, defaults)
  
          for group in self.param_groups:
              for p in group['params']:
                  state = self.state[p]
                  state['step'] = torch.zeros(1)
                  state['exp_avg'] = p.data.new().resize_as_(p.data).zero_()
                  state['exp_avg_sq'] = p.data.new().resize_as_(p.data).zero_()
                  state['max_exp_avg_sq'] = p.data.new().resize_as_(
                      p.data).zero_()
  
      def share_memory(self):
          for group in self.param_groups:
              for p in group['params']:
                  state = self.state[p]
                  state['step'].share_memory_()
                  state['exp_avg'].share_memory_()
                  state['exp_avg_sq'].share_memory_()
                  state['max_exp_avg_sq'].share_memory_()
  
      def step(self, closure=None):
          """Performs a single optimization step.
          Arguments:
              closure (callable, optional): A closure that reevaluates the model
                  and returns the loss.
          """
          loss = None
          if closure is not None:
              loss = closure()
  
          for group in self.param_groups:
              for p in group['params']:
                  if p.grad is None:
                      continue
                  grad = p.grad.data
                  if grad.is_sparse:
                      raise RuntimeError(
                          'Adam does not support sparse gradients, please consider SparseAdam instead'
                      )
                  amsgrad = group['amsgrad']
  
                  state = self.state[p]
  
                  exp_avg, exp_avg_sq = state['exp_avg'], state['exp_avg_sq']
                  if amsgrad:
                      max_exp_avg_sq = state['max_exp_avg_sq']
                  beta1, beta2 = group['betas']
  
                  state['step'] += 1
  
                  if group['weight_decay'] != 0:
                      grad = grad.add(group['weight_decay'], p.data)
  
                  # Decay the first and second moment running average coefficient
                  exp_avg.mul_(beta1).add_(1 - beta1, grad)
                  exp_avg_sq.mul_(beta2).addcmul_(1 - beta2, grad, grad)
  
                  if amsgrad:
                      # Maintains the maximum of all 2nd moment running avg. till
                      # now
                      torch.max(max_exp_avg_sq, exp_avg_sq, out=max_exp_avg_sq)
                      # Use the max. for normalizing running avg. of gradient
                      denom = max_exp_avg_sq.sqrt().add_(group['eps'])
                  else:
                      denom = exp_avg_sq.sqrt().add_(group['eps'])
  
                  bias_correction1 = 1 - beta1**state['step'].item()
                  bias_correction2 = 1 - beta2**state['step'].item()
                  step_size = group['lr'] * \
                      math.sqrt(bias_correction2) / bias_correction1
  
                  p.data.addcdiv_(-step_size, exp_avg, denom)
  
          return loss
  