code: """
  Adapted from the pytorch-lamb library at https://github.com/cybertronai/pytorch-lamb
  """
  
  import torch
  from torch.optim import Optimizer
  
  from colossalai.registry import OPTIMIZERS
  
  
  @OPTIMIZERS.register_module
  class Lamb(Optimizer):
      r"""Implements Lamb algorithm.
      It has been proposed in `Large Batch Optimization for Deep Learning: Training BERT in 76 minutes`_.
  
      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)
          adam (bool, optional): always use trust ratio = 1, which turns this into
              Adam. Useful for comparison purposes.
  
      .. _Large Batch Optimization for Deep Learning\: Training BERT in 76 minutes:
          https://arxiv.org/abs/1904.00962
      """
  
      def __init__(self, params, lr=1e-3, betas=(0.9, 0.999), eps=1e-6,
                   weight_decay=0, adam=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]))
          defaults = dict(lr=lr, betas=betas, eps=eps,
                          weight_decay=weight_decay)
          self.adam = adam
          super(Lamb, 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(
                          'Lamb does not support sparse gradients, consider SparseAdam instad.')
  
                  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
  
                  # Decay the first and second moment running average coefficient
                  # m_t
                  exp_avg.mul_(beta1).add_(grad, alpha=1 - beta1)
                  # v_t
                  exp_avg_sq.mul_(beta2).addcmul_(grad, grad, value=1 - beta2)
  
                  # Paper v3 does not use debiasing.
                  # bias_correction1 = 1 - beta1 ** state['step']
                  # bias_correction2 = 1 - beta2 ** state['step']
                  # Apply bias to lr to avoid broadcast.
                  # * math.sqrt(bias_correction2) / bias_correction1
                  step_size = group['lr']
  
                  weight_norm = p.data.pow(2).sum().sqrt()
  
                  adam_step = exp_avg / exp_avg_sq.sqrt().add(group['eps'])
                  if group['weight_decay'] != 0:
                      adam_step.add_(p.data, alpha=group['weight_decay'])
  
                  adam_norm = adam_step.pow(2).sum().sqrt()
                  if weight_norm == 0 or adam_norm == 0:
                      trust_ratio = 1
                  else:
                      trust_ratio = weight_norm / adam_norm
                  state['weight_norm'] = weight_norm
                  state['adam_norm'] = adam_norm
                  state['trust_ratio'] = trust_ratio
                  if self.adam:
                      trust_ratio = 1
  
                  p.data.add_(adam_step, alpha=-step_size * trust_ratio)
  
          return loss
  
apis: torch.zeros_like