text: import math
  import torch
  from torch.optim.optimizer import Optimizer
  from torch.nn.utils import clip_grad_norm_
  
  
  def warmup_cosine(x, warmup=0.002):
      s = 1 if x <= warmup else 0
      return s*(x/warmup) + (1-s)*(0.5 * (1 + torch.cos(math.pi * x)))
  
  
  def warmup_constant(x, warmup=0.002):
      s = 1 if x <= warmup else 0
      return s*(x/warmup) + (1-s)*1
  
  
  def warmup_linear(x, warmup=0.002):
      s = 1 if x <= warmup else 0
      return (s*(x/warmup) + (1-s))*(1-x)
  
  
  SCHEDULES = {
      'warmup_cosine':warmup_cosine,
      'warmup_constant':warmup_constant,
      'warmup_linear':warmup_linear,
  }
  
  
  class OpenAIAdam(Optimizer):
      """Implements Open AI version of Adam algorithm with weight decay fix.
      """
      def __init__(self, params, lr, schedule, warmup, t_total,
                   b1=0.9, b2=0.999, e=1e-8, l2=0,
                   vector_l2=False, max_grad_norm=-1, **kwargs):
          if not 0.0 <= lr:
              raise ValueError("Invalid learning rate: {}".format(lr))
          if schedule not in SCHEDULES:
              raise ValueError("Invalid schedule parameter: {}".format(schedule))
          if not 0 <= warmup:
              raise ValueError("Invalid warmup: {}".format(warmup))
          if not 0.0 <= b1 < 1.0:
              raise ValueError("Invalid b1 parameter: {}".format(b1))
          if not 0.0 <= b2 < 1.0:
              raise ValueError("Invalid b2 parameter: {}".format(b2))
          if not 0.0 <= e:
              raise ValueError("Invalid epsilon value: {}".format(e))
          defaults = dict(lr=lr, schedule=schedule, warmup=warmup, t_total=t_total,
                          b1=b1, b2=b2, e=e, l2=l2, vector_l2=vector_l2,
                          max_grad_norm=max_grad_norm)
          super(OpenAIAdam, self).__init__(params, defaults)
  
      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('Adam 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['b1'], group['b2']
  
                  state['step'] += 1
  
                  # Add grad clipping
                  if group['max_grad_norm'] > 0:
                      clip_grad_norm_(p, group['max_grad_norm'])
  
                  # Decay the first and second moment running average coefficient
                  exp_avg.mul_(beta1).add_(1 - beta1, grad)
                  exp_avg_sq.mul_(beta2).addcmul_(1 - beta2, grad, grad)
                  denom = exp_avg_sq.sqrt().add_(group['e'])
  
                  bias_correction1 = 1 - beta1 ** state['step']
                  bias_correction2 = 1 - beta2 ** state['step']
  
                  schedule_fct = SCHEDULES[group['schedule']]
                  lr_scheduled = group['lr'] * schedule_fct(state['step']/group['t_total'], group['warmup'])
                  step_size = lr_scheduled * math.sqrt(bias_correction2) / bias_correction1
  
                  p.data.addcdiv_(-step_size, exp_avg, denom)
  
                  # Add weight decay at the end (fixed version)
                  if (len(p.size()) > 1 or group['vector_l2']) and group['l2'] > 0:
                      p.data.add_(-lr_scheduled * group['l2'], p.data)
  
          return loss
  