code: def smooth_amax(
      X: Tensor,
      dim: Union[int, Tuple[int, ...]] = -1,
      keepdim: bool = False,
      tau: Union[float, Tensor] = 1.0,
  ) -> Tensor:
      """Computes a smooth approximation to `max(X, dim=dim)`, i.e the maximum value of
      `X` over dimension `dim`, using the logarithm of the `l_(1/tau)` norm of `exp(X)`.
      Note that when `X = log(U)` is the *logarithm* of an acquisition utility `U`,
  
      `logsumexp(log(U) / tau) * tau = log(sum(U^(1/tau))^tau) = log(norm(U, ord=(1/tau))`
  
      Args:
          X: A Tensor from which to compute the smoothed amax.
          dim: The dimensions to reduce over.
          keepdim: If True, keeps the reduced dimensions.
          tau: Temperature parameter controlling the smooth approximation
              to max operator, becomes tighter as tau goes to 0. Needs to be positive.
  
      Returns:
          A Tensor of smooth approximations to `max(X, dim=dim)`.
      """
      # consider normalizing by log_n = math.log(X.shape[dim]) to reduce error
      return logsumexp(X / tau, dim=dim, keepdim=keepdim) * tau
  
imports: from __future__ import annotations
  import math
  from typing import Callable, Tuple, Union
  import torch
  from botorch.exceptions import UnsupportedError
  from botorch.utils.constants import get_constants_like
  from torch import finfo, Tensor
  from torch.nn.functional import softplus
label_desc: Use image node_id 17 for calling a global function with example usage: smooth_amax(X, dim, keepdim, tau) and returns: unknown