text: import math
  import torch
  import numpy as np
  import typing
  
  __all__ = ['AdaX', 'AdaXW']
  
  #NOTE: from https://github.com/switchablenorms/AdaX
  
  class AdaX(torch.optim.Optimizer):
      r"""Implements AdaX algorithm.
  
      - **Paper**: [AdaX: Adaptive Gradient Descent with Exponential Long Term Memory](https://arxiv.org/pdf/2004.09740.pdf)
      - **Implementation**: [GitHub @ switchablenorms](https://github.com/switchablenorms/AdaX)
      
      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, 1e-4))
          eps (float, optional): term added to the denominator to improve
              numerical stability (default: 1e-12)
          weight_decay (float, optional): L2 penalty (default: 5e-4)
  
      .. _Adam: A Method for Stochastic Optimization:
          https://arxiv.org/abs/1412.6980
      .. _On the Convergence of Adam and Beyond:
          https://openreview.net/forum?id=ryQu7f-RZ
      """
  
      def __init__(self, 
          params: typing.Iterator[torch.nn.Parameter],
          lr: float=1.5e-3,
          betas: typing.Tuple[float, float]=(0.9, 1e-4),
          eps: float=1e-12,
          weight_decay: float=5e-4
      ):
          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)
          super(AdaX, self).__init__(params, defaults)
  
      def __setstate__(self, state):
          super(AdaX, self).__setstate__(state)
  
      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('AdaX does not support sparse gradients, please consider SparseAdam instead')
  
                  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)
  
  
                  exp_avg, exp_avg_sq = state['exp_avg'], state['exp_avg_sq']
  
                  beta1, beta2 = group['betas']
                  state['step'] += 1
  
                  if group['weight_decay'] != 0:
                      grad.add_(group['weight_decay'], p.data)
                  t = state['step']
                  # Decay the first and second moment running average coefficient
                  exp_avg.mul_(beta1).add_(1 - beta1, grad)
                  exp_avg_sq.mul_(1 + beta2).addcmul_(beta2, grad, grad)
  
                  denom = exp_avg_sq.sqrt().add_(group['eps'])
  
                  bias_correction2 = ((1 + beta2) ** state['step'] - 1)
  
                  step_size = group['lr'] * math.sqrt(bias_correction2)
                  # step_size = group['lr']
                  p.data.addcdiv_(-step_size, exp_avg, denom)
  
  
          return loss
  
  class AdaXW(torch.optim.Optimizer):
      r"""Implements Adam algorithm.
  
      - **Paper**: [AdaX: Adaptive Gradient Descent with Exponential Long Term Memory](https://arxiv.org/pdf/2004.09740.pdf)
      - **Implementation**: [GitHub @ switchablenorms](https://github.com/switchablenorms/AdaX)
      
      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, 1e-4))
          eps (float, optional): term added to the denominator to improve
              numerical stability (default: 1e-8)
          weight_decay (float, optional): weight decay (default: 5e-2
      .. _Adam\: A Method for Stochastic Optimization:
          https://arxiv.org/abs/1412.6980
      .. _On the Convergence of Adam and Beyond:
          https://openreview.net/forum?id=ryQu7f-RZ
      """
  
      def __init__(self,
          params: typing.Iterator[torch.nn.Parameter],
          lr: float=0.005,
          betas: typing.Tuple[float, float]=(0.9, 1e-4),
          eps: float=1e-12,
          weight_decay: float=5e-2
      ):
          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)
          super(AdaXW, self).__init__(params, defaults)
  
      def __setstate__(self, state):
          super(AdaXW, self).__setstate__(state)
  
      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:
  
              beta1, beta2 = group['betas']
  
              for p in group['params']:
                  if p.grad is None:
                      continue
                  grad = p.grad.data
                  if grad.is_sparse:
                      raise RuntimeError('AdaX does not support sparse gradients, please consider SparseAdam instead')
  
  
                  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)
  
  
                  exp_avg, exp_avg_sq = state['exp_avg'], state['exp_avg_sq']
  
  
                  state['step'] += 1
  
                  exp_avg.mul_(beta1).add_(1 - beta1, grad)
                  exp_avg_sq.mul_(1 + beta2).addcmul_(beta2, grad, grad)
  
                  denom = exp_avg_sq.sqrt().add_(group['eps'])
  
                  bias_correction2 = ((1 + beta2) ** state['step'] - 1)
  
                  step_size = group['lr'] * math.sqrt(bias_correction2)
  
                  p.data.add_(-torch.mul(p.data, group['lr'] * group['weight_decay'])).addcdiv_(-step_size, exp_avg,
                                                                                                denom)
  
          return loss
  