text: import math
  import torch
  from torch.optim import Optimizer
  
  
  class AdamW(Optimizer):
      r"""Implements AdamW algorithm.
  
      The original Adam algorithm was proposed in `Adam: A Method for Stochastic Optimization`_.
      The AdamW variant was proposed in `Decoupled Weight Decay Regularization`_.
  
      Arguments:
          params (iterable): iterable of parameters to optimize or dicts defining
              parameter groups
          lr (float, optional): learning rate (default: 1e-3)
          betas (Tuple[float, float], optional): coefficients used for computing
              running averages of gradient and its square (default: (0.9, 0.999))
          eps (float, optional): term added to the denominator to improve
              numerical stability (default: 1e-8)
          weight_decay (float, optional): weight decay coefficient (default: 1e-2)
          amsgrad (boolean, optional): whether to use the AMSGrad variant of this
              algorithm from the paper `On the Convergence of Adam and Beyond`_
              (default: False)
  
      .. _Adam\: A Method for Stochastic Optimization:
          https://arxiv.org/abs/1412.6980
      .. _Decoupled Weight Decay Regularization:
          https://arxiv.org/abs/1711.05101
      .. _On the Convergence of Adam and Beyond:
          https://openreview.net/forum?id=ryQu7f-RZ
      """
  
      def __init__(self, params, lr=1e-3, betas=(0.9, 0.999), eps=1e-8,
                   weight_decay=1e-2, amsgrad=False):
          if not 0.0 <= lr:
              raise ValueError("Invalid learning rate: {}".format(lr))
          if not 0.0 <= eps:
              raise ValueError("Invalid epsilon value: {}".format(eps))
          if not 0.0 <= betas[0] < 1.0:
              raise ValueError("Invalid beta parameter at index 0: {}".format(betas[0]))
          if not 0.0 <= betas[1] < 1.0:
              raise ValueError("Invalid beta parameter at index 1: {}".format(betas[1]))
          defaults = dict(lr=lr, betas=betas, eps=eps,
                          weight_decay=weight_decay, amsgrad=amsgrad)
          super(AdamW, self).__init__(params, defaults)
  
      def __setstate__(self, state):
          super(AdamW, self).__setstate__(state)
          for group in self.param_groups:
              group.setdefault('amsgrad', False)
  
      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
  
                  # Perform stepweight decay
                  p.data.mul_(1 - group['lr'] * group['weight_decay'])
  
                  # Perform optimization step
                  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]
  
                  # State initialization
                  if len(state) == 0:
                      state['step'] = 0
                      # Exponential moving average of gradient values
                      state['exp_avg'] = torch.zeros_like(p.data)
                      # Exponential moving average of squared gradient values
                      state['exp_avg_sq'] = torch.zeros_like(p.data)
                      if amsgrad:
                          # Maintains max of all exp. moving avg. of sq. grad. values
                          state['max_exp_avg_sq'] = torch.zeros_like(p.data)
  
                  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
                  bias_correction1 = 1 - beta1 ** state['step']
                  bias_correction2 = 1 - beta2 ** state['step']
  
                  # 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() / math.sqrt(bias_correction2)).add_(group['eps'])
                  else:
                      denom = (exp_avg_sq.sqrt() / math.sqrt(bias_correction2)).add_(group['eps'])
  
                  step_size = group['lr'] / bias_correction1
  
                  p.data.addcdiv_(-step_size, exp_avg, denom)
  
          return loss
  
  