Limited Memory Covariance Matrix Adaptation (LMCMA)

class pypop7.optimizers.es.lmcma.LMCMA(problem, options)

Limited-Memory Covariance Matrix Adaptation (LMCMA).

Note

Currently LMCMA is a State-Of-The-Art (SOTA) variant of CMA-ES designed especially for large-scale black-box optimization. Inspired by L-BFGS (a well-established second-order gradient-based optimizer), it stores only m direction vectors to reconstruct the covariance matirx on-the-fly, resulting in O(mn) time complexity w.r.t. each sampling, where m=O(log(n)) and n is the dimensionality of objective function.

Parameters:

• problem (dict) –

  problem arguments with the following common settings (keys):

  • ’fitness_function’ - objective function to be minimized (func),

  • ’ndim_problem’ - number of dimensionality (int),

  • ’upper_boundary’ - upper boundary of search range (array_like),

  • ’lower_boundary’ - lower boundary of search range (array_like).

• options (dict) –

  optimizer options with the following common settings (keys):

  • ’max_function_evaluations’ - maximum of function evaluations (int, default: np.Inf),

  • ’max_runtime’ - maximal runtime to be allowed (float, default: np.Inf),

  • ’seed_rng’ - seed for random number generation needed to be explicitly set (int);

  and with the following particular settings (keys):

  • ’sigma’ - initial global step-size, aka mutation strength (float),

  • ’mean’ - initial (starting) point, aka mean of Gaussian search distribution (array_like),

    • if not given, it will draw a random sample from the uniform distribution whose search range is bounded by problem[‘lower_boundary’] and problem[‘upper_boundary’].

  • ’m’ - number of direction vectors (int, default: 4 + int(3*np.log(problem[‘ndim_problem’]))),

  • ’base_m’ - base number of direction vectors (int, default: 4),

  • ’period’ - update period (int, default: int(np.maximum(1, np.log(problem[‘ndim_problem’])))),

  • ’n_steps’ - target number of generations between vectors (int, default: problem[‘ndim_problem’]),

  • ’c_c’ - learning rate for evolution path update (float, default: 0.5/np.sqrt(problem[‘ndim_problem’])),

  • ’c_1’ - learning rate for covariance matrix adaptation (float, default: 1.0/(10.0*np.log(problem[‘ndim_problem’] + 1.0))),

  • ’c_s’ - learning rate for population success rule (float, default: 0.3),

  • ’d_s’ - changing rate for population success rule (float, default: 1.0),

  • ’z_star’ - target success rate for population success rule (float, default: 0.3),

  • ’n_individuals’ - number of offspring, aka offspring population size (int, default: 4 + int(3*np.log(problem[‘ndim_problem’]))),

  • ’n_parents’ - number of parents, aka parental population size (int, default: int(options[‘n_individuals’]/2)).

Examples

Use the optimizer LMCMA to minimize the well-known test function Rosenbrock:

>>> import numpy
>>> from pypop7.benchmarks.base_functions import rosenbrock  # function to be minimized
>>> from pypop7.optimizers.es.lmcma import LMCMA
>>> problem = {'fitness_function': rosenbrock,  # define problem arguments
...            'ndim_problem': 1000,
...            'lower_boundary': -5*numpy.ones((1000,)),
...            'upper_boundary': 5*numpy.ones((1000,))}
>>> options = {'max_function_evaluations': 1e5*1000,  # set optimizer options
...            'fitness_threshold': 1e-8,
...            'seed_rng': 2022,
...            'mean': 3*numpy.ones((1000,)),
...            'sigma': 0.1}  # the global step-size may need to be tuned for better performance
>>> lmcma = LMCMA(problem, options)  # initialize the optimizer class
>>> results = lmcma.optimize()  # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"LMCMA: {results['n_function_evaluations']}, {results['best_so_far_y']}")
LMCMA: 2482983, 9.998653039116044e-09

For its correctness checking of coding, refer to this code-based repeatability report for more details.

base_m

base number of direction vectors.

Type:

int

c_c

learning rate for evolution path update.

Type:

float

c_s

learning rate for population success rule.

Type:

float

c_1

learning rate for covariance matrix adaptation.

Type:

float

d_s

changing rate for population success rule.

Type:

float

m

number of direction vectors.

Type:

int

mean

initial (starting) point, aka mean of Gaussian search distribution.

Type:

array_like

n_individuals

number of offspring, aka offspring population size.

Type:

int

n_parents

number of parents, aka parental population size.

Type:

int

n_steps

target number of generations between vectors.

Type:

int

period

update period.

Type:

int

sigma

final global step-size, aka mutation strength.

Type:

float

z_star

target success rate for population success rule.

Type:

float

References

Loshchilov, I., 2017. LM-CMA: An alternative to L-BFGS for large-scale black box optimization. Evolutionary Computation, 25(1), pp.143-171. https://direct.mit.edu/evco/article-abstract/25/1/143/1041/LM-CMA-An-Alternative-to-L-BFGS-for-Large-Scale (See Algorithm 7 for details.)

See the official C++ version from Loshchilov, which provides an interface for Matlab users: https://sites.google.com/site/ecjlmcma/ (Unfortunately, this website link appears to be not available now.)