def get_cosine_schedule_with_warmup(
    optimizer: Optimizer,
    num_warmup_steps: int,
    num_training_steps: int,
    min_lr_ratio: float = 0.0,
    num_cycles: float = 0.5,
    last_epoch: int = -1,
):
    """
    Create a schedule with a learning rate that decreases following the values of the cosine function between the
    initial lr set in the optimizer to 0, after a warmup period during which it increases linearly between 0 and the
    initial lr set in the optimizer.
    Args:
        optimizer (:class:`~torch.optim.Optimizer`):
            The optimizer for which to schedule the learning rate.
        num_warmup_steps (:obj:`int`):
            The number of steps for the warmup phase.
        num_training_steps (:obj:`int`):
            The total number of training steps.
        min_lr_ratio (:obj:`float`, `optional`, defaults to 0.0):
            The minimum lr ratio w.r.t the maximum.
        num_cycles (:obj:`float`, `optional`, defaults to 0.5):
            The number of waves in the cosine schedule (the defaults is to just decrease from the max value to 0
            following a half-cosine).
        last_epoch (:obj:`int`, `optional`, defaults to -1):
            The index of the last epoch when resuming training.
    Return:
        :obj:`torch.optim.lr_scheduler.LambdaLR` with the appropriate schedule.
    """
    assert min_lr_ratio >= 0 and min_lr_ratio <= 1.
    coef = (1 - min_lr_ratio) * 0.5
    intercept = (1 + min_lr_ratio) * 0.5

    def lr_lambda(current_step):
        if current_step < num_warmup_steps:
            return float(current_step) / float(max(1, num_warmup_steps))
        progress = float(current_step - num_warmup_steps) / float(max(1, num_training_steps - num_warmup_steps))
        x = math.cos(math.pi * float(num_cycles) * 2.0 * progress)
        return max(0.0, x * coef + intercept)

    return LambdaLR(optimizer, lr_lambda, last_epoch)

    Create a schedule with a learning rate that decreases following the values of the cosine function between the
    initial lr set in the optimizer to 0, after a warmup period during which it increases linearly between 0 and the
    initial lr set in the optimizer.
    Args:
        optimizer (:class:`~torch.optim.Optimizer`):
            The optimizer for which to schedule the learning rate.
        num_warmup_steps (:obj:`int`):
            The number of steps for the warmup phase.
        num_training_steps (:obj:`int`):
            The total number of training steps.
        min_lr_ratio (:obj:`float`, `optional`, defaults to 0.0):
            The minimum lr ratio w.r.t the maximum.
        num_cycles (:obj:`float`, `optional`, defaults to 0.5):
            The number of waves in the cosine schedule (the defaults is to just decrease from the max value to 0
            following a half-cosine).
        last_epoch (:obj:`int`, `optional`, defaults to -1):
            The index of the last epoch when resuming training.
    Return:
        :obj:`torch.optim.lr_scheduler.LambdaLR` with the appropriate schedule.
    
verl/utils/torch_functional.py