tf_quant_finance.math.optimizer.lbfgs_minimize

tf_quant_finance.math.optimizer.lbfgs_minimize#

Applies the L-BFGS algorithm to minimize a differentiable function.

tf_quant_finance.math.optimizer.lbfgs_minimize(
    value_and_gradients_function, initial_position, previous_optimizer_results=None,
    num_correction_pairs=10, tolerance=1e-08, x_tolerance=0, f_relative_tolerance=0,
    initial_inverse_hessian_estimate=None, max_iterations=50, parallel_iterations=1,
    stopping_condition=None, max_line_search_iterations=50, f_absolute_tolerance=0,
    name=None
)

Performs unconstrained minimization of a differentiable function using the L-BFGS scheme. See [Nocedal and Wright(2006)][1] for details of the algorithm.

Usage:#

The following example demonstrates the L-BFGS optimizer attempting to find the minimum for a simple high-dimensional quadratic objective function.

  # A high-dimensional quadratic bowl.
  ndims = 60
  minimum = np.ones([ndims], dtype='float64')
  scales = np.arange(ndims, dtype='float64') + 1.0

  # 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 = np.arange(ndims, 0, -1, dtype='float64')
  optim_results = tfp.optimizer.lbfgs_minimize(
      quadratic_loss_and_gradient,
      initial_position=start,
      num_correction_pairs=10,
      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)

References:#

[1] Jorge Nocedal, Stephen Wright. Numerical Optimization. Springer Series in Operations Research. pp 176-180. 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 Tensors of real dtype containing the value of the function and its gradient at that point. The function to be minimized. The input is 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 is a real Tensor of matching shape [...]. The second component (the gradient) is also 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. Exactly one of initial_position and previous_optimizer_results can be non-None.
previous_optimizer_results: An LBfgsOptimizerResults namedtuple to intialize the optimizer state from, instead of an initial_position. This can be passed in from a previous return value to resume optimization with a different stopping_condition. Exactly one of initial_position and previous_optimizer_results can be non-None.
num_correction_pairs: Positive integer. Specifies the maximum number of (position_delta, gradient_delta) correction pairs to keep as implicit approximation of the Hessian matrix.
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: None. Option currently not supported.
max_iterations: Scalar positive int32 Tensor. The maximum number of iterations for L-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.
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: Scalar boolean tensor indicating whether the minimum was found within tolerance. failed: Scalar boolean tensor indicating 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 containing the last argument value found during the search. If the search converged, then this value is the argmin of the objective function. objective_value: A tensor containing 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 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. position_deltas: A tensor encoding information about the latest changes in position during the algorithm execution. gradient_deltas: A tensor encoding information about the latest changes in objective_gradient during the algorithm execution.