class_docstr: Implements AdaMod algorithm.
  
  It has been proposed in `Adaptive and Momental Bounds for Adaptive
  Learning Rate Methods`__.
  
  Arguments:
      params: iterable of parameters to optimize or dicts defining
          parameter groups
      lr: learning rate (default: 1e-3)
      betas: coefficients used for computing running averages of gradient
          and its square (default: (0.9, 0.999))
      beta3: smoothing coefficient for adaptive learning rates
          (default: 0.9999)
      eps: term added to the denominator to improve numerical stability
          (default: 1e-8)
      weight_decay: weight decay (L2 penalty) (default: 0)
  
  Example:
      >>> import torch_optimizer as optim
      >>> optimizer = optim.AdaMod(model.parameters(), lr=0.1)
      >>> optimizer.zero_grad()
      >>> loss_fn(model(input), target).backward()
      >>> optimizer.step()
  
  __ https://arxiv.org/abs/1910.12249
  
  Note:
      Reference code: https://github.com/lancopku/AdaMod
docstr: Performs a single optimization step.
  
  Arguments:
      closure: A closure that reevaluates the model and returns the loss.
docstr_tok: ["Performs","a","single","optimization","step",".","Arguments",":","closure",":","A","closure","that","reevaluates","the","model","and","returns","the","loss","."]
code: def step(self, closure: OptLossClosure = None) -> OptFloat:
      """Performs a single optimization step.
  
      Arguments:
          closure: 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:
                  msg = "AdaMod does not support sparse gradients"
                  raise RuntimeError(msg)
  
              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, memory_format=torch.preserve_format
                  )
                  # Exponential moving average of squared gradient values
                  state["exp_avg_sq"] = torch.zeros_like(
                      p, memory_format=torch.preserve_format
                  )
                  # Exponential moving average of actual learning rates
                  state["exp_avg_lr"] = torch.zeros_like(
                      p, memory_format=torch.preserve_format
                  )
  
              exp_avg, exp_avg_sq, exp_avg_lr = (
                  state["exp_avg"],
                  state["exp_avg_sq"],
                  state["exp_avg_lr"],
              )
              beta1, beta2 = group["betas"]
  
              state["step"] += 1
  
              # Decay the first and second moment running average coefficient
              exp_avg.mul_(beta1).add_(grad, alpha=1 - beta1)
              exp_avg_sq.mul_(beta2).addcmul_(
                  grad, grad, value=1 - beta2
              )
  
              denom = exp_avg_sq.sqrt().add_(group["eps"])
  
              bias_correction1 = 1 - beta1 ** state["step"]
              bias_correction2 = 1 - beta2 ** state["step"]
              step_size = (
                  group["lr"]
                  * math.sqrt(bias_correction2)
                  / bias_correction1
              )
  
              if group["weight_decay"] != 0:
                  p.data.add_(
                      p.data, alpha=-group["weight_decay"] * group["lr"]
                  )
  
              # Applies momental bounds on actual learning rates
              step_size = torch.full_like(
                  denom, step_size, memory_format=torch.preserve_format
              )
              step_size.div_(denom)
              exp_avg_lr.mul_(group["beta3"]).add_(
                  step_size, alpha=1 - group["beta3"]
              )
              step_size = torch.min(step_size, exp_avg_lr)
              step_size.mul_(exp_avg)
  
              p.data.add_(-step_size)
  
      return loss
  
imports: import math
  import torch
  from torch.optim.optimizer import Optimizer
  from .types import Betas2, OptFloat, OptLossClosure, Params
label_desc: Use image node_id 2 for calling the AdaMod obj's underlying member method code with example usage: obj.step(closure) and returns: loss