# Gaussian Smoothing (GS)¶

class pypop7.optimizers.rs.gs.GS(problem, options)

Gaussian Smoothing (GS).

Note

In 2017, Nesterov published state-of-the-art theoretical results on convergence rate of GS for a class of convex functions in the gradient-free context (see Foundations of Computational Mathematics).

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):
• ’n_individuals’ - number of individuals/samples (int, default: 100),

• ’lr’ - learning rate (float, default: 0.001),

• ’c’ - factor of finite-difference gradient estimate (float, default: 0.1),

• ’x’ - initial (starting) point (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’].

Examples

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

``` 1>>> import numpy
2>>> from pypop7.benchmarks.base_functions import rosenbrock  # function to be minimized
3>>> from pypop7.optimizers.rs.gs import GS
4>>> problem = {'fitness_function': rosenbrock,  # define problem arguments
5...            'ndim_problem': 100,
6...            'lower_boundary': -2*numpy.ones((100,)),
7...            'upper_boundary': 2*numpy.ones((100,))}
8>>> options = {'max_function_evaluations': 10000*101,  # set optimizer options
9...            'seed_rng': 2022,
10...            'n_individuals': 10,
11...            'c': 0.1,
12...            'lr': 0.000001}
13>>> gs = GS(problem, options)  # initialize the optimizer class
14>>> results = gs.optimize()  # run the optimization process
15>>> # return the number of used function evaluations and found best-so-far fitness
16>>> print(f"GS: {results['n_function_evaluations']}, {results['best_so_far_y']}")
17GS: 1010000, 99.99696650242736
```

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

c

Type:

float

lr

learning rate of (estimated) gradient update.

Type:

float

n_individuals

number of individuals/samples.

Type:

int

x

initial (starting) point.

Type:

array_like

References

Gao, K. and Sener, O., 2022, June. Generalizing Gaussian Smoothing for Random Search. In International Conference on Machine Learning (pp. 7077-7101). PMLR. https://proceedings.mlr.press/v162/gao22f.html https://icml.cc/media/icml-2022/Slides/16434.pdf

Nesterov, Y. and Spokoiny, V., 2017. Random gradient-free minimization of convex functions. Foundations of Computational Mathematics, 17(2), pp.527-566. https://link.springer.com/article/10.1007/s10208-015-9296-2