text: import torch
  from . import _functional as F
  from .optimizer 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`_.
  
      Args:
          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]))
          if not 0.0 <= weight_decay:
              raise ValueError("Invalid weight_decay value: {}".format(weight_decay))
          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)
  
      @torch.no_grad()
      def step(self, closure=None):
          """Performs a single optimization step.
  
          Args:
              closure (callable, optional): A closure that reevaluates the model
                  and returns the loss.
          """
          loss = None
          if closure is not None:
              with torch.enable_grad():
                  loss = closure()
  
          for group in self.param_groups:
              params_with_grad = []
              grads = []
              exp_avgs = []
              exp_avg_sqs = []
              state_sums = []
              max_exp_avg_sqs = []
              state_steps = []
              amsgrad = group['amsgrad']
              beta1, beta2 = group['betas']
  
              for p in group['params']:
                  if p.grad is None:
                      continue
                  params_with_grad.append(p)
                  if p.grad.is_sparse:
                      raise RuntimeError('AdamW does not support sparse gradients')
                  grads.append(p.grad)
  
                  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, memory_format=torch.preserve_format)
                      # Exponential moving average of squared gradient values
                      state['exp_avg_sq'] = torch.zeros_like(p, memory_format=torch.preserve_format)
                      if amsgrad:
                          # Maintains max of all exp. moving avg. of sq. grad. values
                          state['max_exp_avg_sq'] = torch.zeros_like(p, memory_format=torch.preserve_format)
  
                  exp_avgs.append(state['exp_avg'])
                  exp_avg_sqs.append(state['exp_avg_sq'])
  
                  if amsgrad:
                      max_exp_avg_sqs.append(state['max_exp_avg_sq'])
  
                  # update the steps for each param group update
                  state['step'] += 1
                  # record the step after step update
                  state_steps.append(state['step'])
  
              F.adamw(params_with_grad,
                      grads,
                      exp_avgs,
                      exp_avg_sqs,
                      max_exp_avg_sqs,
                      state_steps,
                      amsgrad=amsgrad,
                      beta1=beta1,
                      beta2=beta2,
                      lr=group['lr'],
                      weight_decay=group['weight_decay'],
                      eps=group['eps'])
  
          return loss
  