code: def newton_step_1d(loss, x, trust_radius=None):
      """
      Performs a Newton update step to minimize loss on a batch of 1-dimensional
      variables, optionally regularizing to constrain to a trust region.
  
      See :func:`newton_step` for details.
  
      :param torch.Tensor loss: A scalar function of ``x`` to be minimized.
      :param torch.Tensor x: A dependent variable with rightmost size of 1.
      :param float trust_radius: An optional trust region trust_radius. The
          updated value ``mode`` of this function will be within
          ``trust_radius`` of the input ``x``.
      :return: A pair ``(mode, cov)`` where ``mode`` is an updated tensor
          of the same shape as the original value ``x``, and ``cov`` is an
          esitmate of the covariance 1x1 matrix with
          ``cov.shape == x.shape[:-1] + (1,1)``.
      :rtype: tuple
      """
      if loss.shape != ():
          raise ValueError(
              "Expected loss to be a scalar, actual shape {}".format(
                  loss.shape
              )
          )
      if x.dim() < 1 or x.shape[-1] != 1:
          raise ValueError(
              "Expected x to have rightmost size 1, actual shape {}".format(
                  x.shape
              )
          )
  
      # compute derivatives
      g = grad(loss, [x], create_graph=True)[0]
      H = grad(g.sum(), [x], create_graph=True)[0]
      warn_if_nan(g, "g")
      warn_if_nan(H, "H")
      Hinv = H.clamp(min=1e-8).reciprocal()
      dx = -g * Hinv
      dx[~(dx == dx)] = 0
      if trust_radius is not None:
          dx.clamp_(min=-trust_radius, max=trust_radius)
  
      # apply update
      x_new = x.detach() + dx
      assert x_new.shape == x.shape
      return x_new, Hinv.unsqueeze(-1)
  
file_path: pyro/pyro/ops/newton.py
imports: import torch
  from torch.autograd import grad
  from pyro.ops.linalg import eig_3d, rinverse
  from pyro.util import warn_if_nan
label_desc: Use image node_id 2 for calling a global function with example usage: newton_step_1d(loss, x, trust_radius) and returns: x_new,  Hinv