lq-CMA-ES on a linear function

[Nikolaus Hansen]

Seems to work fine in 4-D (confirmed in plots) even though the model optimum seems to be (in some runs) in the wrong position (the first coordinate should be negative here as is the one of the incumbent).

import cma
x, es = cma.fmin_lq_surr2(lambda x: x[0], 4 * [1], 1, {'tolfacupx': 1e7})
es.surrogate.model.xopt

(4_w,8)-aCMA-ES (mu_w=2.6,w_1=52%) in dimension 4 (seed=173331, Mon Jan 30 14:28:41 2023)
Iterat #Fevals   function value  axis ratio  sigma  min&max std  t[m:s]
    1      5 6.215994990973674e-01 1.0e+00 1.01e+00  1e+00  1e+00 0:00.0
    2      6 -2.432495786087054e+00 1.4e+00 1.35e+00  1e+00  2e+00 0:00.0
    3      7 -4.432495786087054e+00 1.5e+00 1.79e+00  2e+00  2e+00 0:00.0
   51     55 -3.130000098985985e+07 7.4e+00 6.51e+06  3e+06  1e+07 0:00.1
termination on tolfacupx=10000000.0 (Mon Jan 30 14:28:41 2023)
final/bestever f-value = -3.130000e+07 -3.130000e+07 after 55/57 evaluations
incumbent solution: [-30184337.2435357, -12389244.93773907, 10474158.540809048, 3434412.5840689354]
std deviation: [10736839.080872687, 5123362.03878537, 4678639.153572443, 2990627.555213554]
array([ 2.22130972e+12,  8.98455545e+11, -6.58257448e+11, -3.16643176e+11])

The sorting of sampled solutions is almost always correct (and surrogate fitness values are not too far off):

X = es.ask()
F = [es.surrogate.model.eval(x) for x in X]
np.argsort([x[0] for x in X]), np.argsort(F)

(array([7, 4, 0, 2, 3, 6, 1, 5]), array([7, 4, 0, 2, 3, 6, 1, 5]))

However in 14-D

x, es = cma.fmin_lq_surr2(lambda x: x[0], 14 * [1], 1, {'tolfacupx': 1e17})
cma.plot()

divergence becomes much slower after the switch to the full model at evaluation ~185

X = es.ask()
F = [es.surrogate.model.eval(x) for x in X]
np.argsort([x[0] for x in X]), np.argsort(F)

(array([ 8,  0,  7,  9,  6,  4,  1,  2,  3, 10,  5]),
 array([ 5,  7,  4,  3, 10,  8,  0,  6,  1,  2,  9]))

[Nikolaus Hansen]

Rerunning after activating the logging like

cma.fitness_models.Logger = cma.logger.Logger
[...]
es.surrogate.logger.plot()

reveals that the evaluated solutions have a large enough Kendall tau most of the time.