class_docstr: Implements a pure pytorch variant of FuseLAMB (NvLamb variant) optimizer from apex.optimizers.FusedLAMB
  reference: https://github.com/NVIDIA/DeepLearningExamples/blob/master/PyTorch/LanguageModeling/Transformer-XL/pytorch/lamb.py
  
  LAMB was 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 norm. (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)
      grad_averaging (bool, optional): whether apply (1-beta2) to grad when
          calculating running averages of gradient. (default: True)
      max_grad_norm (float, optional): value used to clip global grad norm (default: 1.0)
      trust_clip (bool): enable LAMBC trust ratio clipping (default: False)
      always_adapt (boolean, optional): Apply adaptive learning rate to 0.0
          weight decay parameter (default: False)
  
  .. _Large Batch Optimization for Deep Learning - Training BERT in 76 minutes:
      https://arxiv.org/abs/1904.00962
  .. _On the Convergence of Adam and Beyond:
      https://openreview.net/forum?id=ryQu7f-RZ
docstr: Performs a single optimization step.
  Arguments:
      closure (callable, optional): A closure that reevaluates the model
          and returns the loss.
docstr_tok: ["Performs","a","single","optimization","step",".","Arguments",":","closure","(","callable",",","optional",")",":","A","closure","that","reevaluates","the","model","and","returns","the","loss","."]
code: 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:
          with torch.enable_grad():
              loss = closure()
  
      device = self.param_groups[0]["params"][0].device
      one_tensor = torch.tensor(
          1.0, device=device
      )  # because torch.where doesn't handle scalars correctly
      global_grad_norm = torch.zeros(1, device=device)
      for group in self.param_groups:
          for p in group["params"]:
              if p.grad is None:
                  continue
              grad = p.grad
              if grad.is_sparse:
                  raise RuntimeError(
                      "Lamb does not support sparse gradients, consider SparseAdam instad."
                  )
              global_grad_norm.add_(grad.pow(2).sum())
  
      global_grad_norm = torch.sqrt(global_grad_norm)
      # FIXME it'd be nice to remove explicit tensor conversion of scalars when torch.where promotes
      # scalar types properly https://github.com/pytorch/pytorch/issues/9190
      max_grad_norm = torch.tensor(
          self.defaults["max_grad_norm"], device=device
      )
      clip_global_grad_norm = torch.where(
          global_grad_norm > max_grad_norm,
          global_grad_norm / max_grad_norm,
          one_tensor,
      )
  
      for group in self.param_groups:
          bias_correction = 1 if group["bias_correction"] else 0
          beta1, beta2 = group["betas"]
          grad_averaging = 1 if group["grad_averaging"] else 0
          beta3 = 1 - beta1 if grad_averaging else 1.0
  
          # assume same step across group now to simplify things
          # per parameter step can be easily support by making it tensor, or pass list into kernel
          if "step" in group:
              group["step"] += 1
          else:
              group["step"] = 1
  
          if bias_correction:
              bias_correction1 = 1 - beta1 ** group["step"]
              bias_correction2 = 1 - beta2 ** group["step"]
          else:
              bias_correction1, bias_correction2 = 1.0, 1.0
  
          for p in group["params"]:
              if p.grad is None:
                  continue
              grad = p.grad.div_(clip_global_grad_norm)
              state = self.state[p]
  
              # State initialization
              if len(state) == 0:
                  # Exponential moving average of gradient valuesa
                  state["exp_avg"] = torch.zeros_like(p)
                  # Exponential moving average of squared gradient values
                  state["exp_avg_sq"] = torch.zeros_like(p)
  
              exp_avg, exp_avg_sq = (
                  state["exp_avg"],
                  state["exp_avg_sq"],
              )
  
              # Decay the first and second moment running average coefficient
              exp_avg.mul_(beta1).add_(grad, alpha=beta3)  # m_t
              exp_avg_sq.mul_(beta2).addcmul_(
                  grad, grad, value=1 - beta2
              )  # v_t
  
              denom = (
                  exp_avg_sq.sqrt() / math.sqrt(bias_correction2)
              ).add_(group["eps"])
              update = (exp_avg / bias_correction1).div_(denom)
  
              weight_decay = group["weight_decay"]
              if weight_decay != 0:
                  update.add_(p, alpha=weight_decay)
  
              if weight_decay != 0 or group["always_adapt"]:
                  # Layer-wise LR adaptation. By default, skip adaptation on parameters that are
                  # excluded from weight decay, unless always_adapt == True, then always enabled.
                  w_norm = p.norm(2.0)
                  g_norm = update.norm(2.0)
                  # FIXME nested where required since logical and/or not working in PT XLA
                  trust_ratio = torch.where(
                      w_norm > 0,
                      torch.where(
                          g_norm > 0, w_norm / g_norm, one_tensor
                      ),
                      one_tensor,
                  )
                  if group["trust_clip"]:
                      # LAMBC trust clipping, upper bound fixed at one
                      trust_ratio = torch.minimum(
                          trust_ratio, one_tensor
                      )
                  update.mul_(trust_ratio)
  
              p.add_(update, alpha=-group["lr"])
  
      return loss
  
imports: import math
  import torch
  from torch.optim import Optimizer
label_desc: Use image node_id 2 for calling the Lamb obj's underlying member method code with example usage: obj.step(closure) and returns: loss