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Introduction

Gradient-Free-Optimizers provides a collection of easy to use optimization techniques, whose objective function only requires an arbitrary score that gets maximized. This makes gradient-free methods capable of solving various optimization problems, including:

• Optimizing arbitrary mathematical functions.
• Fitting multiple gauss-distributions to data.
• Hyperparameter-optimization of machine-learning methods.

Gradient-Free-Optimizers is the optimization backend of <a href="https://github.com/SimonBlanke/Hyperactive">Hyperactive</a> (in v3.0.0 and higher) but it can also be used by itself as a leaner and simpler optimization toolkit.

Main features

• Easy to use:

  <details> <summary><b> Simple API-design</b></summary> <br>

  You can optimize anything that can be defined in a python function. For example a simple parabola function:

  def objective_function(para):
      score = para["x1"] * para["x1"]
      return -score
  

  Define where to search via numpy ranges:

  search_space = {
      "x": np.arange(0, 5, 0.1),
  }
  

  That`s all the information the algorithm needs to search for the maximum in the objective function:

  from gradient_free_optimizers import RandomSearchOptimizer
  
  opt = RandomSearchOptimizer(search_space)
  opt.search(objective_function, n_iter=100000)
  

  </details> <details> <summary><b> Receive prepared information about ongoing and finished optimization runs</b></summary> <br>

  During the optimization you will receive ongoing information in a progress bar:

  • current best score
  • the position in the search space of the current best score
  • the iteration when the current best score was found
  • other information about the progress native to tqdm
  </details>

• High performance:

  <details> <summary><b> Modern optimization techniques</b></summary> <br>

  Gradient-Free-Optimizers provides not just meta-heuristic optimization methods but also sequential model based optimizers like bayesian optimization, which delivers good results for expensive objetive functions like deep-learning models.

  </details> <details> <summary><b> Lightweight backend</b></summary> <br>

  Even for the very simple parabola function the optimization time is about 60% of the entire iteration time when optimizing with random search. This shows, that (despite all its features) Gradient-Free-Optimizers has an efficient optimization backend without any unnecessary slowdown.

  </details> <details> <summary><b> Save time with memory dictionary</b></summary> <br>

  Per default Gradient-Free-Optimizers will look for the current position in a memory dictionary before evaluating the objective function.

  • If the position is not in the dictionary the objective function will be evaluated and the position and score is saved in the dictionary.

  • If a position is already saved in the dictionary Gradient-Free-Optimizers will just extract the score from it instead of evaluating the objective function. This avoids reevaluating computationally expensive objective functions (machine- or deep-learning) and therefore saves time.

  </details>

• High reliability:

  <details> <summary><b> Extensive testing</b></summary> <br>

  Gradient-Free-Optimizers is extensivly tested with more than 400 tests in 2500 lines of test code. This includes the testing of:

  • Each optimization algorithm
  • Each optimization parameter
  • All attributes that are part of the public api
  </details> <details> <summary><b> Performance test for each optimizer</b></summary> <br>

  Each optimization algorithm must perform above a certain threshold to be included. Poorly performing algorithms are reworked or scraped.

  </details>

Optimization algorithms:

Gradient-Free-Optimizers supports a variety of optimization algorithms, which can make choosing the right algorithm a tedious endeavor. The gifs in this section give a visual representation how the different optimization algorithms explore the search space and exploit the collected information about the search space for a convex and non-convex objective function. More detailed explanations of all optimization algorithms can be found in the official documentation.

Local Optimization

Evaluates the score of n neighbours in an epsilon environment and moves to the best one.

Adds a probability to the hill climbing to move to a worse position in the search-space to escape local optima.

Hill climbing algorithm with the addition of increasing epsilon by a factor if no better neighbour was found.

Adds a probability to the hill climbing to move to a worse position in the search-space to escape local optima with decreasing probability over time.

Constructs a simplex from multiple positions that moves through the search-space by reflecting, expanding, contracting or shrinking.

Global Optimization

Moves to random positions in each iteration.

Grid-search that moves through search-space diagonal (with step-size=1) starting from a corner.

Hill climbingm, that moves to a random position after n iterations.

Hill Climbing, that has large epsilon at the start of the search decreasing over time.

Creates cross-shaped collection of positions that move through search-space by moving as a whole towards optima or shrinking the cross.

Optimizes each search-space dimension at a time with a hill-climbing algorithm.

Population-Based Optimization

Population of n simulated annealers, which occasionally swap transition probabilities.

Population of n particles attracting each other and moving towards the best particle.

Population of n particles moving in a spiral pattern around the best position.

Evolutionary algorithm selecting the best individuals in the population, mixing their parameters to get new solutions.

Population of n hill climbers occasionally mixing positional information and removing worst positions from population.

Improves a population of candidate solutions by creating trial vectors through the differential mutation of three randomly selected individuals.

Sequential Model-Based Optimization

Gaussian process fitting to explored positions and predicting promising new positions.

Calculates an upper bound from the distances of the previously explored positions to find new promising positions.

Separates search space into subspaces. It evaluates the center position of each subspace to decide which subspace to sepate further.

Kernel density estimators fitting to good and bad explored positions and predicting promising new positions.

Ensemble of decision trees fitting to explored positions and predicting promising new positions.

Sideprojects and Tools

The following packages are designed to support Gradient-Free-Optimizers and expand its use cases.

If you want news about Gradient-Free-Optimizers and related projects you can follow me on twitter.

Installation

The most recent version of Gradient-Free-Optimizers is available on PyPi:

pip install gradient-free-optimizers

Examples

import numpy as np
from gradient_free_optimizers import RandomSearchOptimizer


def parabola_function(para):
    loss = para["x"] * para["x"]
    return -loss


search_space = {"x": np.arange(-10, 10, 0.1)}

opt = RandomSearchOptimizer(search_space)
opt.search(parabola_function, n_iter=100000)

import numpy as np
from gradient_free_optimizers import RandomSearchOptimizer


def ackley_function(pos_new):
    x = pos_new["x1"]
    y = pos_new["x2"]

    a1 = -20 * np.exp(-0.2 * np.sqrt(0.5 * (x * x + y * y)))
    a2 = -np.exp(0.5 * (np.cos(2 * np.pi * x) + np.cos(2 * np.pi * y)))
    score = a1 + a2 + 20
    return -score


search_space = {
    "x1": np.arange(-100, 101, 0.1),
    "x2": np.arange(-100, 101, 0.1),
}

opt = RandomSearchOptimizer(search_space)
opt.search(ackley_function, n_iter=30000)

import numpy as np
from sklearn.model_selection import cross_val_score
from sklearn.ensemble import GradientBoostingClassifier
from sklearn.datasets import load_wine

from gradient_free_optimizers import HillClimbingOptimizer


data = load_wine()
X, y = data.data, data.target


def model(para):
    gbc = GradientBoostingClassifier(
        n_estimators=para["n_estimators"],
        max_depth=para["max_depth"],
        min_samples_split=para["min_samples_split"],
        min_samples_leaf=para["min_samples_leaf"],
    )
    scores = cross_val_score(gbc, X, y, cv=3)

    return scores.mean()


search_space = {
    "n_estimators": np.arange(20, 120, 1),
    "max_depth": np.arange(2, 12, 1),
    "min_samples_split": np.arange(2, 12, 1),
    "min_samples_leaf": np.arange(1, 12, 1),
}

opt = HillClimbingOptimizer(search_space)
opt.search(model, n_iter=50)

import numpy as np
from gradient_free_optimizers import RandomSearchOptimizer


def convex_function(pos_new):
    score = -(pos_new["x1"] * pos_new["x1"] + pos_new["x2"] * pos_new["x2"])
    return score


search_space = {
    "x1": np.arange(-100, 101, 0.1),
    "x2": np.arange(-100, 101, 0.1),
}


def constraint_1(para):
    # only values in 'x1' higher than -5 are valid
    return para["x1"] > -5


# put one or more constraints inside a list
constraints_list = [constraint_1]


# pass list of constraints to the optimizer
opt = RandomSearchOptimizer(search_space, constraints=constraints_list)
opt.search(convex_function, n_iter=50)

search_data = opt.search_data

# the search-data does not contain any samples where x1 is equal or below -5
print("\n search_data \n", search_data, "\n")

Roadmap

• add sampling parameter to Bayesian optimizer
• add warnings parameter to Bayesian optimizer
• improve access to parameters of optimizers within population-based-optimizers (e.g. annealing rate of simulated annealing population in parallel tempering)

• add early stopping parameter

• add grid-search to optimizers
• impoved performance testing for optimizers

• Finalize API (1.0.0)
• add Downhill-simplex algorithm to optimizers
• add Pattern search to optimizers
• add Powell's method to optimizers
• add parallel random annealing to optimizers
• add ensemble-optimizer to optimizers

• add Spiral Optimization
• add Lipschitz Optimizer
• print the random seed for reproducibility

• add DIRECT algorithm
• automatically add random initial positions if necessary (often requested)

• add support for constrained optimization

• add Grid search parameter that changes direction of search
• add SMBO parameter that enables to avoid replacement of the sampling

• add Genetic Algorithm
• add Differential evolution

• add support for numpy v2
• add support for pandas v2
• add support for python 3.12
• transfer setup.py to pyproject.toml
• change project structure to src-layout

• add Ant-colony optimization
• add Harmonic-serch
• add API, testing and doc to (better) use GFO as backend-optimization package
• add Random search parameter that enables to avoid replacement of the sampling
• add other acquisition functions to smbo (Probability of improvement, Entropy search, ...)

Gradient Free Optimizers <=> Hyperactive

Gradient-Free-Optimizers was created as the optimization backend of the Hyperactive package. Therefore the algorithms are exactly the same in both packages and deliver the same results. However you can still use Gradient-Free-Optimizers as a standalone package. The separation of Gradient-Free-Optimizers from Hyperactive enables multiple advantages:

• Even easier to use than Hyperactive
• Separate and more thorough testing
• Other developers can easily use GFOs as an optimizaton backend if desired
• Better isolation from the complex information flow in Hyperactive. GFOs only uses positions and scores in a N-dimensional search-space. It returns only the new position after each iteration.
• a smaller and cleaner code base, if you want to explore my implementation of these optimization techniques.

While Gradient-Free-Optimizers is relatively simple, Hyperactive is a more complex project with additional features to make optimization of computationally expensive models (like engineering simulation or machine-/deep-learning models) more convenient.

Citation

@Misc{gfo2020,
  author =   {{Simon Blanke}},
  title =    {{Gradient-Free-Optimizers}: Simple and reliable optimization with local, global, population-based and sequential techniques in numerical search spaces.},
  howpublished = {\url{https://github.com/SimonBlanke}},
  year = {since 2020}
}

License

Gradient-Free-Optimizers is licensed under the following License: