StochasticHillClimbingOptimizer

class StochasticHillClimbingOptimizer(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, epsilon: float = 0.03, distribution: Literal['normal', 'laplace', 'gumbel', 'logistic'] = 'normal', n_neighbours: int = 3, p_accept: float = 0.5)

Hill climbing variant that accepts worse solutions to escape local optima.

Stochastic Hill Climbing extends the basic hill climbing algorithm by introducing a probability of accepting solutions that are worse than the current one. This stochastic acceptance mechanism helps the optimizer escape local optima and explore a broader region of the search space. Unlike standard hill climbing, which always moves to better positions, this variant can temporarily accept inferior solutions, enabling it to “climb down” from local peaks.

The algorithm is well-suited for:

• Multimodal optimization problems with multiple local optima

• Problems where standard hill climbing gets stuck frequently

• Situations requiring a balance between local refinement and exploration

• Optimization landscapes with many plateaus or ridges

The p_accept parameter controls the probability of accepting worse solutions. Higher values increase exploration but may slow convergence, while lower values make the algorithm behave more like standard hill climbing. A value of 0.0 reduces this to deterministic hill climbing.

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.

epsilonfloat, default=0.03

Step size for generating neighbor positions, expressed as a fraction of each dimension’s range. Controls how far the optimizer looks from the current position when sampling neighbors.

• 0.01-0.02: Fine-grained local search, slow convergence.

• 0.03-0.05: Moderate step size, good default range.

• 0.1-0.3: Large steps, broader exploration.

• 0.5-1.0: Very large steps, nearly global jumps.

Example: For a dimension with np.linspace(0, 100, 1000) (range = 100):

• epsilon=0.03 leads to neighbors within ~3 units

• epsilon=0.1 leads to neighbors within ~10 units

Smaller values are better for fine-tuning near a known good solution. Larger values help escape local optima but may overshoot narrow peaks.

distribution{“normal”, “laplace”, “gumbel”, “logistic”}, default=”normal”

Probability distribution used to sample neighbor offsets. Each distribution produces different exploration patterns:

• "normal": Gaussian distribution. Most neighbors are close to the current position with rare far jumps. Best general-purpose choice.

• "laplace": Sharper peak than normal with heavier tails. Good for landscapes where occasional large jumps help.

• "gumbel": Asymmetric, skewed distribution. Can bias exploration in one direction.

• "logistic": Similar to normal but with slightly heavier tails. A middle ground between normal and Laplace.

The distribution interacts with epsilon: heavy-tailed distributions (Laplace) effectively increase the chance of large steps beyond what epsilon alone suggests.

n_neighboursint, default=3

Number of neighbor positions to sample and evaluate per iteration. The optimizer moves to the best among these neighbors (if it improves on the current position).

• 1: Minimal sampling, fast iterations but may miss good directions. Good for very cheap objective functions.

• 3-5: Moderate sampling, good balance of quality and speed.

• 10-20: Thorough neighborhood evaluation, better directional choices but more function evaluations per step.

Higher values act like a local beam search, giving the optimizer more information about the local landscape at each iteration.

p_acceptfloat, default=0.5

Probability of accepting a solution that is worse than the current one. This flat acceptance probability applies uniformly regardless of how much worse the candidate is.

• 0.0: Never accept worse solutions (equivalent to standard Hill Climbing).

• 0.1-0.3: Mild exploration, occasionally escapes shallow local optima.

• 0.5: Balanced exploration and exploitation (default).

• 0.7-1.0: Strong exploration, frequently accepts worse solutions, slow convergence.

Unlike Simulated Annealing where acceptance depends on score difference and temperature, here the probability is constant throughout the search.

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

HillClimbingOptimizer

Deterministic variant that only accepts improvements.

SimulatedAnnealingOptimizer

Score-dependent acceptance with temperature cooling.

RepulsingHillClimbingOptimizer

Escapes local optima by increasing step size.

Notes

At each step, the best neighbor is found (same as Hill Climbing). If the best neighbor is better, it is always accepted. If worse, it is accepted with probability p_accept:

\[\begin{split}P(\\text{accept}) = \\begin{cases} 1 & \\text{if } f(x_{\\text{neighbor}}) > f(x_{\\text{current}}) \\\\ p_{\\text{accept}} & \\text{otherwise} \\end{cases}\end{split}\]

This is simpler than Simulated Annealing’s Metropolis criterion but lacks the adaptive cooling behavior. The constant acceptance rate means exploration intensity does not change over time.

For visual explanations and tuning guides, see the Stochastic Hill Climbing user guide.

Examples

>>> import numpy as np
>>> from gradient_free_optimizers import StochasticHillClimbingOptimizer

>>> def rastrigin(para):
...     x, y = para["x"], para["y"]
...     return -(20 + x**2 + y**2 - 10 * (np.cos(2*np.pi*x) + np.cos(2*np.pi*y)))

>>> search_space = {
...     "x": np.linspace(-5.12, 5.12, 100),
...     "y": np.linspace(-5.12, 5.12, 100),
... }

>>> opt = StochasticHillClimbingOptimizer(search_space, p_accept=0.3)
>>> opt.search(rastrigin, n_iter=200)

property best_para

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

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

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.