text: # Copyright (c) 2020, NVIDIA CORPORATION.  All rights reserved.
  #
  # Licensed under the Apache License, Version 2.0 (the "License");
  # you may not use this file except in compliance with the License.
  # You may obtain a copy of the License at
  #
  #     http://www.apache.org/licenses/LICENSE-2.0
  #
  # Unless required by applicable law or agreed to in writing, software
  # distributed under the License is distributed on an "AS IS" BASIS,
  # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  # See the License for the specific language governing permissions and
  # limitations under the License.
  
  import torch
  from torch.optim.optimizer import Optimizer
  
  __all__ = ['Novograd']
  
  
  def _check_valid_opt_params(lr, eps, betas):
      if lr < 0:
          raise ValueError(f"Invalid learning rate: {lr}")
      if eps < 0:
          raise ValueError(f"Invalid epsilon value: {eps}")
      if not (0.0 <= betas[0] < 1.0 and 0.0 <= betas[1] < 1.0):
          raise ValueError(f"Betas have to be between 0 and 1: {betas}")
  
  
  class Novograd(Optimizer):
      """Implements Novograd algorithm.
      It has been proposed  in "Stochastic Gradient Methods with Layer-wise
      Adaptive Moments for Training of Deep Networks"
      (https://arxiv.org/abs/1905.11286)
      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 (L2 penalty) (default: 0)
          amsgrad (boolean, optional): whether to use the AMSGrad variant of this
              algorithm from the paper "On the Convergence of Adam and Beyond"
      """
  
      def __init__(
          self,
          params,
          lr=1e-3,
          betas=(0.95, 0.98),
          eps=1e-8,
          weight_decay=0,
          grad_averaging=False,
          amsgrad=False,
          luc=False,
          luc_trust=1e-3,
          luc_eps=1e-8,
      ):
          _check_valid_opt_params(lr, eps, betas)
          defaults = dict(
              lr=lr, betas=betas, eps=eps, weight_decay=weight_decay, grad_averaging=grad_averaging, amsgrad=amsgrad,
          )
          self.luc = luc
          self.luc_trust = luc_trust
          self.luc_eps = luc_eps
          super(Novograd, self).__init__(params, defaults)
  
      def __setstate__(self, state):
          super(Novograd, 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
                  grad = p.grad.data
                  if grad.is_sparse:
                      raise RuntimeError("Sparse gradients are not supported.")
                  amsgrad = group["amsgrad"]
                  state = self.state[p]
  
                  # State initialization
                  if not state:
                      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([]).to(state["exp_avg"].device)
                      if amsgrad:
                          # Maintains max of all exp moving avg of squared grad
                          state["max_exp_avg_sq"] = torch.zeros([]).to(state["exp_avg"].device)
  
                  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
  
                  norm = grad.norm().pow(2)
  
                  if exp_avg_sq == 0:
                      exp_avg_sq.copy_(norm)
                  else:
                      exp_avg_sq.mul_(beta2).add_(norm, alpha=1.0 - beta2)
  
                  if amsgrad:
                      # Maintains max 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"])
  
                  grad.div_(denom)
                  if group["weight_decay"] != 0:
                      grad.add_(p.data, alpha=group["weight_decay"])
                  if group["grad_averaging"]:
                      grad.mul_(1 - beta1)
                  exp_avg.mul_(beta1).add_(grad)
  
                  if self.luc:
                      # Clip update so that updates are less than eta*weights
                      data_norm = torch.norm(p.data)
                      grad_norm = torch.norm(exp_avg.data)
                      luc_factor = self.luc_trust * data_norm / (grad_norm + self.luc_eps)
                      luc_factor = min(luc_factor, group["lr"])
                      p.data.add_(exp_avg, alpha=-luc_factor)
                  else:
                      p.data.add_(exp_avg, alpha=-group["lr"])
  
          return loss
  
  