BayesianOptimizer

class BayesianOptimizer(search_space: dict[str, list], initialize: dict[Literal['grid', 'vertices', 'random', 'warm_start'], int | list[dict]] = None, constraints: list[callable] = None, random_state: int = None, rand_rest_p: float = 0, nth_process: int = None, warm_start_smbo=None, max_sample_size: int = 10000000, sampling: dict[Literal['random'], int] = None, replacement: bool = True, gpr=None, xi: float = 0.03)[source]

Sequential model-based optimizer using Gaussian Process surrogate models.

Bayesian Optimization is a powerful technique for optimizing expensive black-box functions. It builds a probabilistic surrogate model (Gaussian Process) of the objective function and uses an acquisition function to determine the most promising points to evaluate next. This approach is sample-efficient, requiring fewer function evaluations than many other methods to find good solutions.

The algorithm works by: (1) fitting a Gaussian Process to observed data, (2) using the GP to predict mean and uncertainty at unobserved points, (3) selecting the next point to evaluate based on an acquisition function (Expected Improvement), and (4) updating the model with the new observation.

The algorithm is well-suited for:

  • Expensive objective functions (e.g., ML model training, simulations)

  • Low to moderate dimensional problems (typically < 20 dimensions)

  • Problems where sample efficiency is critical

  • Hyperparameter optimization of machine learning models

The xi parameter controls the exploration-exploitation trade-off in the Expected Improvement acquisition function. Higher values encourage more exploration of uncertain regions.

Parameters:
search_spacedict[str, list]

The search space to explore, defined as a dictionary mapping parameter names to arrays of possible values.

Each key is a parameter name (string), and each value is a numpy array or list of discrete values that the parameter can take. The optimizer will only evaluate positions that are on this discrete grid.

Example: A 2D search space with 100 points per dimension:

search_space = {
    "x": np.linspace(-10, 10, 100),
    "y": np.linspace(-10, 10, 100),
}

The resolution of each dimension (number of points in the array) directly affects optimization quality and speed. More points give finer resolution but increase the search space size exponentially.

initializedict[str, int], default={“vertices”: 4, “random”: 2}

Strategy for generating initial positions before the main optimization loop begins. Initialization samples are evaluated first, and the best one becomes the starting point for the optimizer.

Supported keys:

  • "grid": int – Number of positions on a regular grid.

  • "vertices": int – Number of corner/edge positions of the search space.

  • "random": int – Number of uniformly random positions.

  • "warm_start": list[dict] – Specific positions to evaluate, each as a dict mapping parameter names to values.

Multiple strategies can be combined:

initialize = {"vertices": 4, "random": 10}
initialize = {"warm_start": [{"x": 0.5, "y": 1.0}], "random": 5}

More initialization samples improve the starting point but consume iterations from n_iter. For expensive objectives, a few targeted warm-start points are often more efficient than many random samples.

constraintslist[callable], default=[]

A list of constraint functions that restrict the search space. Each constraint is a callable that receives a parameter dictionary and returns True if the position is valid, False if it should be rejected.

Rejected positions are discarded and regenerated: the optimizer resamples a new candidate position (up to 100 retries per step). During initialization, positions that violate constraints are filtered out entirely.

Example: Constrain the search to a circular region:

def circular_constraint(para):
    return para["x"]**2 + para["y"]**2 <= 25

constraints = [circular_constraint]

Multiple constraints are combined with AND logic (all must return True).

random_stateint or None, default=None

Seed for the random number generator to ensure reproducible results.

  • None: Use a new random state each run (non-deterministic).

  • int: Seed the random number generator for reproducibility.

Setting a fixed seed is recommended for debugging and benchmarking. Different seeds may lead to different optimization trajectories, especially for stochastic optimizers.

rand_rest_pfloat, default=0

Probability of performing a random restart instead of the normal algorithm step. At each iteration, a uniform random number is drawn; if it falls below rand_rest_p, the optimizer jumps to a random position instead of following its strategy.

  • 0.0: No random restarts (pure algorithm behavior).

  • 0.01-0.05: Light diversification, helps escape shallow local optima.

  • 0.1-0.3: Aggressive restarts, useful for highly multi-modal landscapes.

  • 1.0: Equivalent to random search.

This is especially useful for local search optimizers (Hill Climbing, Simulated Annealing) that can get trapped. For population-based optimizers, the effect is less pronounced since they already maintain diversity through multiple agents.

warm_start_smboobject or None, default=None

Previous SMBO state for warm-starting the surrogate model. Allows continuing optimization from a previous run by reusing the fitted model state, avoiding the cost of refitting from scratch.

Pass None to start fresh without warm-starting.

max_sample_sizeint, default=10000000

Maximum number of candidate points to consider when optimizing the acquisition function. The surrogate model predicts scores for these candidates, and the best one according to the acquisition function is selected for evaluation.

Larger values improve acquisition optimization quality but increase memory usage and computation time. For most problems, the default is more than sufficient. Reduce this if memory is a concern.

samplingdict, default={“random”: 1000000}

Configuration for how candidate points are generated for acquisition function optimization. The key specifies the sampling strategy and the value specifies the number of samples.

Currently supported: {"random": N} for uniform random sampling of N candidate points from the search space.

Example:

sampling = {"random": 500000}  # Fewer candidates, faster
replacementbool, default=True

Whether to sample candidate points with replacement when generating candidates for acquisition function optimization. When True, the same point can appear multiple times. When False, each candidate is unique, ensuring diversity but potentially slower for very large sample sizes.

gprobject or None, default=None

The Gaussian Process Regressor used as the surrogate model. Accepts three forms:

  • None: Uses the default GPR implementation.

  • Class: A GPR class that will be instantiated automatically.

  • Instance: A pre-configured GPR instance.

Custom GPR implementations should follow the scikit-learn API with fit(X, y) and predict(X, return_std=True) methods. This allows using different kernels, noise models, or entirely different GP libraries.

xifloat, default=0.03

Exploration-exploitation trade-off parameter for the Expected Improvement (EI) acquisition function. Controls how much the optimizer values uncertain regions over predicted-good regions.

  • 0.0: Pure exploitation, always samples where the GP predicts the best score. Tends to converge fast but may miss the global optimum.

  • 0.01-0.05: Mild exploration (default region). Good balance for most problems.

  • 0.1-0.3: Moderate exploration, favors uncertain regions.

  • 1.0+: Strong exploration, heavily prefers unexplored areas.

Higher xi is useful early in optimization or when the search space is large relative to the number of evaluations.

Attributes:
best_para

Return the best parameters found as a dictionary.

best_value

Return the best values found (raw parameter values).

search_data

Lazily construct and return the search results DataFrame.

Methods

eval_time

init_stats

iter_time

search

See also

ForestOptimizer

SMBO using tree ensembles, scales better.

TreeStructuredParzenEstimators

SMBO using kernel density estimation.

EnsembleOptimizer

Combines multiple surrogate model types for robustness.

LipschitzOptimizer

Global optimization using Lipschitz continuity bounds.

Notes

The algorithm follows a sequential loop:

  1. Fit a Gaussian Process to all observed (position, score) pairs.

  2. For each candidate point, predict mean \(\\mu(x)\) and standard deviation \(\\sigma(x)\).

  3. Compute Expected Improvement:

\[\begin{split}EI(x) = (\\mu(x) - f_{\\text{best}} - \\xi) \\cdot \\Phi(Z) + \\sigma(x) \\cdot \\phi(Z)\end{split}\]

where \(Z = (\\mu(x) - f_{\\text{best}} - \\xi) / \\sigma(x)\), \(\\Phi\) is the standard normal CDF, and \(\\phi\) is the standard normal PDF.

  1. Evaluate the point with the highest EI.

The GP surrogate provides a principled balance between exploration (sampling where \(\\sigma(x)\) is high) and exploitation (sampling where \(\\mu(x)\) is high). Fitting the GP has complexity \(O(n^3)\) where n is the number of observations, which can become a bottleneck for large evaluation budgets.

For visual explanations and tuning guides, see the Bayesian Optimization user guide.

Examples

>>> import numpy as np
>>> from gradient_free_optimizers import BayesianOptimizer

>>> def expensive_function(para):
...     # Simulating an expensive evaluation
...     x, y = para["x"], para["y"]
...     return -((x - 0.5) ** 2 + (y - 0.5) ** 2)

>>> search_space = {
...     "x": np.linspace(0, 1, 100),
...     "y": np.linspace(0, 1, 100),
... }

>>> opt = BayesianOptimizer(search_space, xi=0.01)
>>> opt.search(expensive_function, n_iter=50)
property best_para[source]

Return the best parameters found as a dictionary.

Uses the Converter to transform the best position into user-friendly parameter names and values.

Returns:
dict or None

Dictionary mapping parameter names to their best values, or None if no evaluation has been performed yet.

property best_value[source]

Return the best values found (raw parameter values).

Returns:
list or None

List of best values in parameter order, or None if no evaluation has been performed yet.

property search_data: pd.DataFrame[source]

Lazily construct and return the search results DataFrame.

The DataFrame is only built when this property is accessed, avoiding a large memory spike at the end of high-dimensional optimizations. The result is cached so subsequent accesses don’t rebuild it.