ScaleCoordinates gives an unexpected outcomes

[Landau1908]

Hi,
Below is a toy example that confuses me. If no ScaleCoordinates is implemented, the result is right, but if it is implemented, the result has a bit difference from the expected.
Regards

import numpy as np
import cma
from cma.fitness_transformations import ScaleCoordinates

def objective_function(x):
    return sum(xi**2 for xi in x)

min_values = [-5., -3.0, -90.0]
max_values = [1.0, 10.0, 1000.0]

x0 = [1.]*3
sigma = 0.5
popsize = (4 + 3 * int(np.log(len(x0))))*2**0

scaled_objective = ScaleCoordinates(objective_function, upper=max_values, lower=min_values)
result = cma.fmin(objective_function, x0, sigma, options={"bounds":[min_values,max_values]})
#result = cma.fmin(scaled_objective, x0, sigma, options={'popsize':popsize})

While use non-ScaleCoordinates, executing result = cma.fmin(objective_function, x0, sigma, options={"bounds":[min_values,max_values]}), the outcomes as follows:

result = cma.fmin(objective_function, x0, sigma, options={"bounds":[min_values,max_values]})
(3_w,7)-aCMA-ES (mu_w=2.3,w_1=58%) in dimension 3 (seed=920413, Sat Nov 16 20:21:12 2024)
Iterat #Fevals   function value  axis ratio  sigma  min&max std  t[m:s]
    1      7 9.256576457964325e-01 1.0e+00 5.16e-01  5e-01  6e-01 0:00.0
    2     14 8.942005501052922e-01 1.3e+00 5.37e-01  5e-01  6e-01 0:00.0
    3     21 1.149100036298669e-01 1.4e+00 5.63e-01  5e-01  7e-01 0:00.0
   87    609 3.175444797697971e-15 1.6e+00 1.10e-05  5e-08  8e-08 0:00.3
termination on tolfun=1e-11 (Sat Nov 16 20:21:13 2024)
final/bestever f-value = 2.709817e-15 2.709817e-15 after 610/610 evaluations
incumbent solution: [-4.350586266957847e-08, -7.639240333223374e-11, 2.858409961559774e-08]
std deviation: [5.516615851554868e-08, 7.651271053286152e-08, 4.661592773435972e-08]

While use ScaleCoordinates, executing result = cma.fmin(scaled_objective, x0, sigma, options={'popsize':popsize}), the outcomes as follows:

result = cma.fmin(scaled_objective, x0, sigma, options={'popsize':popsize})
(3_w,7)-aCMA-ES (mu_w=2.3,w_1=58%) in dimension 3 (seed=893013, Sat Nov 16 20:35:37 2024)
Iterat #Fevals   function value  axis ratio  sigma  min&max std  t[m:s]
    1      7 2.922935931957261e+04 1.0e+00 6.02e-01  6e-01  7e-01 0:00.0
    2     14 2.582756871627401e+04 1.6e+00 6.02e-01  6e-01  7e-01 0:00.0
    3     21 8.082952119227459e+02 1.4e+00 5.54e-01  5e-01  6e-01 0:00.0
  100    700 2.865720965914941e-09 2.1e+02 2.14e-04  7e-08  1e-05 0:00.3
  137    959 1.141555637256842e-14 1.9e+02 2.49e-06  7e-11  1e-08 0:00.5
termination on tolfun=1e-11 (Sat Nov 16 20:35:39 2024)
final/bestever f-value = 4.063595e-15 4.012026e-15 after 960/938 evaluations
incumbent solution: [0.8333333278186403, 0.2307692269918536, 0.08256880736110907]
std deviation: [1.3134116534238395e-08, 5.417763022950125e-09, 6.802938878850222e-11]

[nikohansen]

the outcomes as follows:
[...]

What is the expected outcome?

[Landau1908]

the outcomes as follows:
[...]

What is the expected outcome?

What I interseted is the best solutions. If I understand correctly, the best solution from the outputs is the incumbent solution. non-ScaleCoordinates gives incumbent solution: [-4.350586266957847e-08, -7.639240333223374e-11, 2.858409961559774e-08], and ScaleCoordinates gives incumbent solution: [0.8333333278186403, 0.2307692269918536, 0.08256880736110907], and there is a bit different in between although the final/bestever f-value ~ 1e-15 is extremely good for both the cases. I don't know why this happens.

[nikohansen]

ScaleCoordinates gives incumbent solution: [0.8333333278186403, 0.2307692269918536, 0.08256880736110907], and there is a bit different in between although the final/bestever f-value ~ 1e-15 is extremely good for both the cases. I don't know why this happens.

I suspect the incumbent is what you are interested in ("the best solution"): if you evaluate [0.8333333278186403, 0.2307692269918536, 0.08256880736110907] on the scaled function, scaled_objective, you will get about $10^{-15}$, which is what I would expect should happen, because you have run CMA-ES on scaled_objective.

The respective solution to the original function can be obtained as

scaled_objective.transform([0.8333333278186403, 0.2307692269918536, 0.08256880736110907])

giving the output

array([-3.30881575e-08, -4.91059035e-08,  2.36088783e-08])