tf_quant_finance.math.optimizer.differential_evolution_minimize

Last updated: 2023-03-16.

tf_quant_finance.math.optimizer.differential_evolution_minimize#

Applies the Differential evolution algorithm to minimize a function.

tf_quant_finance.math.optimizer.differential_evolution_minimize(
    objective_function, initial_population=None, initial_position=None,
    population_size=50, population_stddev=1.0, max_iterations=100, func_tolerance=0,
    position_tolerance=1e-08, differential_weight=0.5, crossover_prob=0.9,
    seed=None, name=None
)

Differential Evolution is an evolutionary optimization algorithm which works on a set of candidate solutions called the population. It iteratively improves the population by applying genetic operators of mutation and recombination. The objective function f supplies the fitness of each candidate. A candidate s_1 is considered better than s_2 if f(s_1) < f(s_2).

This method allows the user to either specify an initial population or a single candidate solution. If a single solution is specified, a population of the specified size is initialized by adding independent normal noise to the candidate solution.

The implementation also supports a multi-part specification of the state. For example, consider the objective function:

# x is a tensor of shape [n, m] while y is of shape [n].
def objective(x, y):
  return tf.math.reduce_sum(x ** 2, axis=-1) + y ** 2

The state in this case is specified by two input tensors x and y. To apply the algorithm to this objective function, one would need to specify either an initial population as a list of two tensors of shapes [population_size, k] and [population_size]. The following code shows the complete example:

  population_size = 40
  # With an initial population and a multi-part state.
  initial_population = (tf.random.normal([population_size]),
                        tf.random.normal([population_size]))
  def easom_fn(x, y):
    return -(tf.math.cos(x) * tf.math.cos(y) *
             tf.math.exp(-(x-np.pi)**2 - (y-np.pi)**2))

  optim_results = tfp.optimizer.differential_evolution_minimize(
      easom_fn,
      initial_population=initial_population,
      seed=43210)

  print(optim_results.converged)
  print(optim_results.position)  # Should be (close to) [pi, pi].
  print(optim_results.objective_value)    # Should be -1.


  # With a single starting point
  initial_position = (tf.constant(1.0), tf.constant(1.0))

  optim_results = tfp.optimizer.differential_evolution_minimize(
      easom_fn,
      initial_position=initial_position,
      population_size=40,
      population_stddev=2.0,
      seed=43210)

Args:#

  • objective_function: A Python callable that accepts a batch of possible solutions and returns the values of the objective function at those arguments as a rank 1 real Tensor. This specifies the function to be minimized. The input to this callable may be either a single Tensor or a Python list of Tensors. The signature must match the format of the argument population. (i.e. objective_function(*population) must return the value of the function to be minimized).

  • initial_population: A real Tensor or Python list of Tensors. If a list, each Tensor must be of rank at least 1 and with a common first dimension. The first dimension indexes into the candidate solutions while the rest of the dimensions (if any) index into an individual solution. The size of the population must be at least 4. This is a requirement of the DE algorithm.

  • initial_position: A real Tensor of any shape. The seed solution used to initialize the population of solutions. If this parameter is specified then initial_population must not be specified.

  • population_size: A positive scalar int32 Tensor greater than 4. The size of the population to evolve. This parameter is ignored if initial_population is specified. Default value: 50.

  • population_stddev: A positive scalar real Tensor of the same dtype as initial_position. This parameter is ignored if initial_population is specified. Used to generate the population from the initial_position by adding random normal noise with zero mean and the specified standard deviation. Default value: 1.0

  • max_iterations: Positive scalar int32 Tensor. The maximum number of generations to evolve the population for. Default value: 100

  • func_tolerance: Scalar Tensor of the same dtype as the output of the objective_function. The algorithm stops if the absolute difference between the largest and the smallest objective function value in the population is below this number. Default value: 0

  • position_tolerance: Scalar Tensor of the same real dtype as initial_position or initial_population. The algorithm terminates if the largest absolute difference between the coordinates of the population members is below this threshold. Default value: 1e-8

  • differential_weight: Real scalar Tensor. Must be positive and less than 2.0. The parameter controlling the strength of mutation in the algorithm. Default value: 0.5

  • crossover_prob: Real scalar Tensor. Must be between 0 and 1. The probability of recombination per site. Default value: 0.9

  • seed: int or None. The random seed for this Op. If None, no seed is applied. Default value: None.

  • name: (Optional) Python str. The name prefixed to the ops created by this function. If not supplied, the default name ‘differential_evolution_minimize’ is used. Default value: None

Returns:#

  • optimizer_results: An object containing the following attributes: converged: Scalar boolean Tensor indicating whether the minimum was found within the specified tolerances. num_objective_evaluations: The total number of objective evaluations performed. position: A Tensor containing the best point found during the search. If the search converged, then this value is the argmin of the objective function within the specified tolerances. 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_population: The final state of the population. final_objective_values: The objective function evaluated at the final population. initial_population: The starting population. initial_objective_values: The objective function evaluated at the initial population. num_iterations: The number of iterations of the main algorithm body.

Raises:#

  • ValueError: If neither the initial population, nor the initial position are specified or if both are specified.

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