text: from typing import List, Dict, Optional, Tuple
  import torch
  import torch.optim._functional as F
  
  from torch import Tensor
  
  # Define a TorchScript compatible Functional AdamW Optimizer
  # where we use these optimizer in a functional way.
  # Instead of using the `param.grad` when updating parameters,
  # we explicitly allow the distributed optimizer pass gradients to
  # the `step` function. In this way, we could separate the gradients
  # and parameters and allow multithreaded trainer to update the
  # parameters without data traces on accumulating to the same .grad.
  # NOTE: This should be only used by distributed optimizer internals
  # and not meant to expose to the user.
  @torch.jit.script
  class _FunctionalAdamW(object):
      def __init__(
          self,
          params: List[Tensor],
          lr: float = 1e-3,
          betas: Tuple[float, float] = (0.9, 0.999),
          eps: float = 1e-8,
          weight_decay: float = 1e-2,
          amsgrad: bool = False,
          _allow_empty_param_list: bool = 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))
  
          self.defaults = {
              "lr": lr,
              "eps": eps,
              "beta1": betas[0],
              "beta2": betas[1],
              "weight_decay": weight_decay,
          }
          self.amsgrad = amsgrad
          self.state = torch.jit.annotate(Dict[torch.Tensor, Dict[str, torch.Tensor]], {})
  
          if len(params) == 0 and not _allow_empty_param_list:
              raise ValueError("optimizer got an empty parameter list")
  
          # NOTE: we only have one param_group and don't allow user to add additional
          # param group as it's not a common use case.
          self.param_group = {"params": params}
  
      def step(self, gradients: List[Optional[Tensor]]):
          params = self.param_group['params']
          params_with_grad = []
          grads = []
          exp_avgs = []
          exp_avg_sqs = []
          max_exp_avg_sqs = []
          state_steps: List[int] = []
  
          if len(params) != len(gradients):
              raise ValueError(
                  "the gradients passed in does not equal to the size of the parameters!"
                  + f"Params length: {len(params)}. "
                  + f"Gradients length: {len(gradients)}"
              )
  
          for param, gradient in zip(self.param_group['params'], gradients):
              if gradient is not None:
                  params_with_grad.append(param)
                  grads.append(gradient)
                  # Lazy state initialization
                  if param not in self.state:
                      self.state[param] = {}
                      state = self.state[param]
                      state['step'] = torch.tensor(0.0)
                      # Exponential moving average of gradient values
                      state['exp_avg'] = torch.zeros_like(param, memory_format=torch.preserve_format)
                      # Exponential moving average of squared gradient values
                      state['exp_avg_sq'] = torch.zeros_like(param, memory_format=torch.preserve_format)
                      if self.amsgrad:
                          # Maintains max of all exp. moving avg. of sq. grad. values
                          state['max_exp_avg_sq'] = torch.zeros_like(param, memory_format=torch.preserve_format)
  
                  state = self.state[param]
  
                  exp_avgs.append(state['exp_avg'])
                  exp_avg_sqs.append(state['exp_avg_sq'])
  
                  if self.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'].item())
  
          with torch.no_grad():
              F.adamw(params_with_grad,
                      grads,
                      exp_avgs,
                      exp_avg_sqs,
                      max_exp_avg_sqs,
                      state_steps,
                      amsgrad=self.amsgrad,
                      beta1=self.defaults['beta1'],
                      beta2=self.defaults['beta2'],
                      lr=self.defaults['lr'],
                      weight_decay=self.defaults['weight_decay'],
                      eps=self.defaults['eps'])
  