text: from typing import List, Dict, Optional
  import torch
  import torch.optim._functional as F
  
  from torch import Tensor
  
  # Define a TorchScript compatible Functional RMSprop 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 _FunctionalRMSprop(object):
      def __init__(
          self,
          params: List[Tensor],
          lr: float = 1e-2,
          alpha: float = 0.99,
          eps: float = 1e-8,
          weight_decay: float = 0.0,
          momentum: float = 0.0,
          centered: bool = False
      ):
          self.defaults = {
              "lr": lr,
              "alpha": alpha,
              "eps": eps,
              "weight_decay": weight_decay,
              "momentum": momentum,
          }
          self.centered = centered
  
          if len(params) == 0:
              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}
  
          self.state = torch.jit.annotate(Dict[torch.Tensor, Dict[str, torch.Tensor]], {})
  
      def step(self, gradients: List[Optional[Tensor]]):
          params = self.param_group['params']
          params_with_grad = []
          grads = []
          square_avgs = []
          grad_avgs = []
          momentum_buffer_list = []
          lr = self.defaults['lr']
          alpha = self.defaults['alpha']
          eps = self.defaults['eps']
          momentum = self.defaults['momentum']
          weight_decay = self.defaults['weight_decay']
  
          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(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)
                      state['square_avg'] = torch.zeros_like(param, memory_format=torch.preserve_format)
                      if momentum > 0:
                          state['momentum_buffer'] = torch.zeros_like(param, memory_format=torch.preserve_format)
                      if self.centered:
                          state['grad_avg'] = torch.zeros_like(param, memory_format=torch.preserve_format)
  
                  state = self.state[param]
                  square_avgs.append(state['square_avg'])
                  if momentum > 0:
                      momentum_buffer_list.append(state['momentum_buffer'])
                  if self.centered:
                      grad_avgs.append(state['grad_avg'])
  
                  state['step'] += 1
  
          with torch.no_grad():
              F.rmsprop(params_with_grad,
                        grads,
                        square_avgs,
                        grad_avgs,
                        momentum_buffer_list,
                        lr=lr,
                        alpha=alpha,
                        eps=eps,
                        weight_decay=weight_decay,
                        momentum=momentum,
                        centered=self.centered)
  