code: def minimize(
      value_and_gradients_function,
      initial_position,
      tolerance=1e-8,
      x_tolerance=0,
      f_relative_tolerance=0,
      initial_inverse_hessian_estimate=None,
      scale_initial_inverse_hessian=True,
      max_iterations=50,
      parallel_iterations=1,
      stopping_condition=None,
      validate_args=True,
      max_line_search_iterations=50,
      f_absolute_tolerance=0,
      name=None,
  ):
      """Applies the BFGS algorithm to minimize a differentiable function.
  
      Performs unconstrained minimization of a differentiable function using the
      BFGS scheme. For details of the algorithm, see [Nocedal and Wright(2006)][1].
  
      ### Usage:
  
      The following example demonstrates the BFGS optimizer attempting to find the
      minimum for a simple two dimensional quadratic objective function.
  
      ```python
        minimum = np.array([1.0, 1.0])  # The center of the quadratic bowl.
        scales = np.array([2.0, 3.0])  # The scales along the two axes.
  
        # The objective function and the gradient.
        def quadratic_loss_and_gradient(x):
          return tfp.math.value_and_gradient(
              lambda x: tf.reduce_sum(
                  scales * tf.math.squared_difference(x, minimum), axis=-1),
              x)
  
        start = tf.constant([0.6, 0.8])  # Starting point for the search.
        optim_results = tfp.optimizer.bfgs_minimize(
            quadratic_loss_and_gradient, initial_position=start, tolerance=1e-8)
  
        # Check that the search converged
        assert(optim_results.converged)
        # Check that the argmin is close to the actual value.
        np.testing.assert_allclose(optim_results.position, minimum)
        # Print out the total number of function evaluations it took. Should be 5.
        print ("Function evaluations: %d" % optim_results.num_objective_evaluations)
      ```
  
      ### References:
      [1]: Jorge Nocedal, Stephen Wright. Numerical Optimization. Springer Series in
        Operations Research. pp 136-140. 2006
        http://pages.mtu.edu/~struther/Courses/OLD/Sp2013/5630/Jorge_Nocedal_Numerical_optimization_267490.pdf
  
      Args:
        value_and_gradients_function:  A Python callable that accepts a point as a
          real `Tensor` and returns a tuple of `Tensor`s of real dtype containing
          the value of the function and its gradient at that point. The function
          to be minimized. The input should be of shape `[..., n]`, where `n` is
          the size of the domain of input points, and all others are batching
          dimensions. The first component of the return value should be a real
          `Tensor` of matching shape `[...]`. The second component (the gradient)
          should also be of shape `[..., n]` like the input value to the function.
        initial_position: real `Tensor` of shape `[..., n]`. The starting point, or
          points when using batching dimensions, of the search procedure. At these
          points the function value and the gradient norm should be finite.
        tolerance: Scalar `Tensor` of real dtype. Specifies the gradient tolerance
          for the procedure. If the supremum norm of the gradient vector is below
          this number, the algorithm is stopped.
        x_tolerance: Scalar `Tensor` of real dtype. If the absolute change in the
          position between one iteration and the next is smaller than this number,
          the algorithm is stopped.
        f_relative_tolerance: Scalar `Tensor` of real dtype. If the relative change
          in the objective value between one iteration and the next is smaller
          than this value, the algorithm is stopped.
        initial_inverse_hessian_estimate: Optional `Tensor` of the same dtype
          as the components of the output of the `value_and_gradients_function`.
          If specified, the shape should broadcastable to shape `[..., n, n]`; e.g.
          if a single `[n, n]` matrix is provided, it will be automatically
          broadcasted to all batches. Alternatively, one can also specify a
          different hessian estimate for each batch member.
          For the correctness of the algorithm, it is required that this parameter
          be symmetric and positive definite. Specifies the starting estimate for
          the inverse of the Hessian at the initial point. If not specified,
          the identity matrix is used as the starting estimate for the
          inverse Hessian.
        scale_initial_inverse_hessian: If overridden to False, we skip scaling the
          initial inverse Hessian (Chapter 6 of Nocedal and Wright suggests scaling
          this).
        max_iterations: Scalar positive int32 `Tensor`. The maximum number of
          iterations for BFGS updates.
        parallel_iterations: Positive integer. The number of iterations allowed to
          run in parallel.
        stopping_condition: (Optional) A Python function that takes as input two
          Boolean tensors of shape `[...]`, and returns a Boolean scalar tensor.
          The input tensors are `converged` and `failed`, indicating the current
          status of each respective batch member; the return value states whether
          the algorithm should stop. The default is tfp.optimizer.converged_all
          which only stops when all batch members have either converged or failed.
          An alternative is tfp.optimizer.converged_any which stops as soon as one
          batch member has converged, or when all have failed.
        validate_args: Python `bool`, default `True`. When `True` optimizer
          parameters are checked for validity despite possibly degrading runtime
          performance. When `False` invalid inputs may silently render incorrect
          outputs.
        max_line_search_iterations: Python int. The maximum number of iterations
          for the `hager_zhang` line search algorithm.
        f_absolute_tolerance: Scalar `Tensor` of real dtype. If the absolute change
          in the objective value between one iteration and the next is smaller
          than this value, the algorithm is stopped.
        name: (Optional) Python str. The name prefixed to the ops created by this
          function. If not supplied, the default name 'minimize' is used.
  
      Returns:
        optimizer_results: A namedtuple containing the following items:
          converged: boolean tensor of shape `[...]` indicating for each batch
            member whether the minimum was found within tolerance.
          failed:  boolean tensor of shape `[...]` indicating for each batch
            member whether a line search step failed to find a suitable step size
            satisfying Wolfe conditions. In the absence of any constraints on the
            number of objective evaluations permitted, this value will
            be the complement of `converged`. However, if there is
            a constraint and the search stopped due to available
            evaluations being exhausted, both `failed` and `converged`
            will be simultaneously False.
          num_objective_evaluations: The total number of objective
            evaluations performed.
          position: A tensor of shape `[..., n]` containing the last argument value
            found during the search from each starting point. If the search
            converged, then this value is the argmin of the objective function.
          objective_value: A tensor of shape `[...]` with the value of the
            objective function at the `position`. If the search converged, then
            this is the (local) minimum of the objective function.
          objective_gradient: A tensor of shape `[..., n]` containing the gradient
            of the objective function at the `position`. If the search converged
            the max-norm of this tensor should be below the tolerance.
          inverse_hessian_estimate: A tensor of shape `[..., n, n]` containing the
            inverse of the estimated Hessian.
      """
      with tf.name_scope(name or "minimize"):
          initial_position = tf.convert_to_tensor(
              initial_position, name="initial_position"
          )
          dtype = dtype_util.base_dtype(initial_position.dtype)
          tolerance = tf.convert_to_tensor(
              tolerance, dtype=dtype, name="grad_tolerance"
          )
          f_relative_tolerance = tf.convert_to_tensor(
              f_relative_tolerance,
              dtype=dtype,
              name="f_relative_tolerance",
          )
          x_tolerance = tf.convert_to_tensor(
              x_tolerance, dtype=dtype, name="x_tolerance"
          )
          max_iterations = tf.convert_to_tensor(
              max_iterations, name="max_iterations"
          )
  
          input_shape = ps.shape(initial_position)
          batch_shape, domain_size = input_shape[:-1], input_shape[-1]
  
          if stopping_condition is None:
              stopping_condition = bfgs_utils.converged_all
  
          # Control inputs are an optional list of tensors to evaluate before
          # the start of the search procedure. These can be used to assert the
          # validity of inputs to the search procedure.
          control_inputs = None
  
          if initial_inverse_hessian_estimate is None:
              # Create a default initial inverse Hessian.
              initial_inv_hessian = tf.eye(
                  domain_size,
                  batch_shape=batch_shape,
                  dtype=dtype,
                  name="initial_inv_hessian",
              )
          else:
              # If an initial inverse Hessian is supplied, compute some control inputs
              # to ensure that it is positive definite and symmetric.
              initial_inv_hessian = tf.convert_to_tensor(
                  initial_inverse_hessian_estimate,
                  dtype=dtype,
                  name="initial_inv_hessian",
              )
              if validate_args:
                  control_inputs = _inv_hessian_control_inputs(
                      initial_inv_hessian
                  )
              else:
                  control_inputs = None
              hessian_shape = ps.concat(
                  [batch_shape, [domain_size, domain_size]], 0
              )
              initial_inv_hessian = tf.broadcast_to(
                  initial_inv_hessian, hessian_shape
              )
  
          # The `state` here is a `BfgsOptimizerResults` tuple with values for the
          # current state of the algorithm computation.
          def _cond(state):
              """Continue if iterations remain and stopping condition is not met."""
              return (
                  state.num_iterations < max_iterations
              ) & tf.logical_not(
                  stopping_condition(state.converged, state.failed)
              )
  
          def _body(state):
              """Main optimization loop."""
              state = bfgs_utils.terminate_if_not_finite(state)
              search_direction = _get_search_direction(
                  state.inverse_hessian_estimate,
                  state.objective_gradient,
              )
              derivative_at_start_pt = tf.reduce_sum(
                  state.objective_gradient * search_direction, axis=-1
              )
  
              # If the derivative at the start point is not negative, recompute the
              # search direction with the initial inverse Hessian.
              needs_reset = (
                  ~state.failed
                  & ~state.converged
                  & (derivative_at_start_pt >= 0)
              )
  
              search_direction_reset = _get_search_direction(
                  initial_inv_hessian, state.objective_gradient
              )
  
              actual_search_direction = tf.where(
                  needs_reset[..., tf.newaxis],
                  search_direction_reset,
                  search_direction,
              )
              actual_inv_hessian = tf.where(
                  needs_reset[..., tf.newaxis, tf.newaxis],
                  initial_inv_hessian,
                  state.inverse_hessian_estimate,
              )
  
              # Replace the hessian estimate in the state, in case it had to be reset.
              current_state = bfgs_utils.update_fields(
                  state, inverse_hessian_estimate=actual_inv_hessian
              )
  
              next_state = bfgs_utils.line_search_step(
                  current_state,
                  value_and_gradients_function,
                  actual_search_direction,
                  tolerance,
                  f_relative_tolerance,
                  x_tolerance,
                  stopping_condition,
                  max_line_search_iterations,
                  f_absolute_tolerance,
              )
  
              # Update the inverse Hessian if needed and continue.
              return [_update_inv_hessian(current_state, next_state)]
  
          kwargs = bfgs_utils.get_initial_state_args(
              value_and_gradients_function,
              initial_position,
              tolerance,
              control_inputs,
          )
          kwargs["inverse_hessian_estimate"] = initial_inv_hessian
          kwargs[
              "scale_initial_inverse_hessian"
          ] = scale_initial_inverse_hessian
          initial_state = BfgsOptimizerResults(**kwargs)
          return tf.while_loop(
              cond=_cond,
              body=_body,
              loop_vars=[initial_state],
              parallel_iterations=parallel_iterations,
          )[0]
  
file_path: probability/tensorflow_probability/python/optimizer/bfgs.py
imports: import collections
  import tensorflow.compat.v2
  from tensorflow_probability.python.internal import distribution_util
  from tensorflow_probability.python.internal import dtype_util
  from tensorflow_probability.python.internal import prefer_static
  from tensorflow_probability.python.optimizer import bfgs_utils
label_desc: Use image node_id 1 for calling a global function with example usage: minimize(value_and_gradients_function, initial_position, tolerance, x_tolerance, f_relative_tolerance, initial_inverse_hessian_estimate, scale_initial_inverse_hessian, max_iterations, parallel_iterations, stopping_condition, validate_args, max_line_search_iterations, f_absolute_tolerance, name) and returns: tf, unknown, list