Obtain working parameters for benchmark problems

[hkayabilisim]

Using trial-and-error, it would be nice to find a working set of parameters for all benchmark functions.

rastgrin: -m 3 sample_size 1000, ... --> HDMR finds the optima. ....

[erdemyelken]

HDMR-OPT was evaluated on various benchmark functions using different combinations of parameters. The benchmark functions used were camel3_2d, camel16_2d, trecanni_2d, goldstein_2d, branin_2d, rosenbrock_2d, ackley_2d, rosenbrock_10d, griewank_10d, and rastrigin_10d. The combinations tested included N (Dimension): 50, 100, 500, and m (HDMR Parameter): 3, 5, 7. The results were obtained and can be found at the following link: Results

[hkayabilisim]

Thank you @erdemyelken! It is nice to see the results all at once. I have one remark though! The parameter N represents the number of samples taken from the input space; it doesn't represent the dimension. The dimension in our case is the number of variables.

[erdemyelken]

I apologize for the confusion in my previous response. You are absolutely right. In our case, the dimension represents the number of hyperparameters we will use in TFT, isn't it? I will fix that. Thank you.

[hkayabilisim]

Yes, you are right. The number of hyperparameters of TFT will be our dimension. In the context of analytical benchmark functions, dimension corresponds to the number of variables which is mostly 2. Please note that we sometimes use "input" and "variable" to refer to the same thing. Only the context is different. For instance, when we are dealing with analytical functions, we prefer "variable" and we say "variable space" or "dimension of the variable space". On the other hand when the analytical form of the function is not known then we usually use "input". I guess the reason is that we are inclined to think that kind of functions as a black box which has some inputs and outputs obviously.

[erdemyelken]

Upon analyzing the performed operations and the obtained results, it is evident that the adaptive HDMR optimization successfully identified the optimal point and exhibited a short processing time. Furthermore, it was observed that the cosine basis set outperformed the Legendre basis set, particularly in certain benchmark functions.

The achieved results have been documented in the HDMR-OPT Results Summary table.