tfp.optimizer.nelder_mead_minimize

Minimum of the objective function using the Nelder Mead simplex algorithm.

tfp.optimizer.nelder_mead_minimize(
    objective_function,
    initial_simplex=None,
    initial_vertex=None,
    step_sizes=None,
    objective_at_initial_simplex=None,
    objective_at_initial_vertex=None,
    batch_evaluate_objective=False,
    func_tolerance=1e-08,
    position_tolerance=1e-08,
    parallel_iterations=1,
    max_iterations=None,
    reflection=None,
    expansion=None,
    contraction=None,
    shrinkage=None,
    name=None
)

Performs an unconstrained minimization of a (possibly non-smooth) function using the Nelder Mead simplex method. Nelder Mead method does not support univariate functions. Hence the dimensions of the domain must be 2 or greater. For details of the algorithm, see [Press, Teukolsky, Vetterling and Flannery(2007)][1].

Points in the domain of the objective function may be represented as a Tensor of general shape but with rank at least 1. The algorithm proceeds by modifying a full rank simplex in the domain. The initial simplex may either be specified by the user or can be constructed using a single vertex supplied by the user. In the latter case, if v0 is the supplied vertex, the simplex is the convex hull of the set:

S = {v0} + {v0 + step_i * e_i}

Here e_i is a vector which is 1 along the i-th axis and zero elsewhere and step_i is a characteristic length scale along the i-th axis. If the step size is not supplied by the user, a unit step size is used in every axis. Alternately, a single step size may be specified which is used for every axis. The most flexible option is to supply a bespoke step size for every axis.

Usage:

The following example demonstrates the usage of the Nelder Mead minimzation on a two dimensional problem with the minimum located at a non-differentiable point.

  # The objective function
  def sqrt_quadratic(x):
    return tf.sqrt(tf.reduce_sum(x ** 2, axis=-1))

  start = tf.constant([6.0, -21.0])  # Starting point for the search.
  optim_results = tfp.optimizer.nelder_mead_minimize(
      sqrt_quadratic, initial_vertex=start, func_tolerance=1e-8,
      batch_evaluate_objective=True)

  with tf.Session() as session:
    results = session.run(optim_results)
    # Check that the search converged
    assert(results.converged)
    # Check that the argmin is close to the actual value.
    np.testing.assert_allclose(results.position, np.array([0.0, 0.0]),
                               atol=1e-7)
    # Print out the total number of function evaluations it took.
    print ("Function evaluations: %d" % results.num_objective_evaluations)

References:

[1]: William Press, Saul Teukolsky, William Vetterling and Brian Flannery. Numerical Recipes in C++, third edition. pp. 502-507. (2007). http://numerical.recipes/cpppages/chap0sel.pdf

[2]: Jeffrey Lagarias, James Reeds, Margaret Wright and Paul Wright. Convergence properties of the Nelder-Mead simplex method in low dimensions, Siam J. Optim., Vol 9, No. 1, pp. 112-147. (1998). http://www.math.kent.edu/~reichel/courses/Opt/reading.material.2/nelder.mead.pdf

[3]: Fuchang Gao and Lixing Han. Implementing the Nelder-Mead simplex algorithm with adaptive parameters. Computational Optimization and Applications, Vol 51, Issue 1, pp 259-277. (2012). https://pdfs.semanticscholar.org/15b4/c4aa7437df4d032c6ee6ce98d6030dd627be.pdf

Args:

objective_function: A Python callable that accepts a point as a real Tensor and returns a Tensor of real dtype containing the value of the function at that point. The function to be minimized. If batch_evaluate_objective is True, the callable may be evaluated on a Tensor of shape [n+1] + s where n is the dimension of the problem and s is the shape of a single point in the domain (so n is the size of a Tensor representing a single point). In this case, the expected return value is a Tensor of shape [n+1]. Note that this method does not support univariate functions so the problem dimension n must be strictly greater than 1.
initial_simplex: (Optional) Tensor of real dtype. The initial simplex to start the search. If supplied, should be a Tensor of shape [n+1] + s where n is the dimension of the problem and s is the shape of a single point in the domain. Each row (i.e. the Tensor with a given value of the first index) is interpreted as a vertex of a simplex and hence the rows must be affinely independent. If not supplied, an axes aligned simplex is constructed using the initial_vertex and step_sizes. Only one and at least one of initial_simplex and initial_vertex must be supplied.
initial_vertex: (Optional) Tensor of real dtype and any shape that can be consumed by the objective_function. A single point in the domain that will be used to construct an axes aligned initial simplex.
step_sizes: (Optional) Tensor of real dtype and shape broadcasting compatible with initial_vertex. Supplies the simplex scale along each axes. Only used if initial_simplex is not supplied. See description above for details on how step sizes and initial vertex are used to construct the initial simplex.
objective_at_initial_simplex: (Optional) Rank 1 Tensor of real dtype of a rank 1 Tensor. The value of the objective function at the initial simplex. May be supplied only if initial_simplex is supplied. If not supplied, it will be computed.
objective_at_initial_vertex: (Optional) Scalar Tensor of real dtype. The value of the objective function at the initial vertex. May be supplied only if the initial_vertex is also supplied.
batch_evaluate_objective: (Optional) Python bool. If True, the objective function will be evaluated on all the vertices of the simplex packed into a single tensor. If False, the objective will be mapped across each vertex separately. Evaluating the objective function in a batch allows use of vectorization and should be preferred if the objective function allows it.
func_tolerance: (Optional) Scalar Tensor of real dtype. The algorithm stops if the absolute difference between the largest and the smallest function value on the vertices of the simplex is below this number.
position_tolerance: (Optional) Scalar Tensor of real dtype. The algorithm stops if the largest absolute difference between the coordinates of the vertices is below this threshold.
parallel_iterations: (Optional) Positive integer. The number of iterations allowed to run in parallel.
max_iterations: (Optional) Scalar positive Tensor of dtype int32. The maximum number of iterations allowed. If None then no limit is applied.
reflection: (Optional) Positive Scalar Tensor of same dtype as initial_vertex. This parameter controls the scaling of the reflected vertex. See, [Press et al(2007)][1] for details. If not specified, uses the dimension dependent prescription of [Gao and Han(2012)][3].
expansion: (Optional) Positive Scalar Tensor of same dtype as initial_vertex. Should be greater than 1 and reflection. This parameter controls the expanded scaling of a reflected vertex. See, [Press et al(2007)][1] for details. If not specified, uses the dimension dependent prescription of [Gao and Han(2012)][3].
contraction: (Optional) Positive scalar Tensor of same dtype as initial_vertex. Must be between 0 and 1. This parameter controls the contraction of the reflected vertex when the objective function at the reflected point fails to show sufficient decrease. See, [Press et al(2007)][1] for more details. If not specified, uses the dimension dependent prescription of [Gao and Han(2012][3].
shrinkage: (Optional) Positive scalar Tensor of same dtype as initial_vertex. Must be between 0 and 1. This parameter is the scale by which the simplex is shrunk around the best point when the other steps fail to produce improvements. See, [Press et al(2007)][1] for more details. If not specified, uses the dimension dependent prescription of [Gao and Han(2012][3].
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. 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. final_simplex: The last simplex constructed before stopping. final_objective_values: The objective function evaluated at the vertices of the final simplex. initial_simplex: The starting simplex. initial_objective_values: The objective function evaluated at the vertices of the initial simplex. num_iterations: The number of iterations of the main algorithm body.

Raises:

ValueError: If any of the following conditions hold
1. If none or more than one of initial_simplex and initial_vertex are supplied.
2. If initial_simplex and step_sizes are both specified.