Acquisition Functions and Infill Strategies

How spotoptim selects the next point to evaluate: acquisition functions, optimizers, and infill configuration.

After fitting the surrogate model, spotoptim must decide where to evaluate the (expensive) objective function next. This decision is governed by the acquisition function and the acquisition optimizer.

Acquisition Functions

The acquisition parameter controls the strategy:

Value	Name	Behavior
`"y"`	Predicted Value	Minimizes the surrogate’s prediction (pure exploitation)
`"ei"`	Expected Improvement	Balances exploitation and exploration via the EI formula
`"pi"`	Probability of Improvement	Probability of finding a value better than the current best

The default is "y", which simply picks the point where the surrogate predicts the lowest value. This is fast and works well when the surrogate is accurate, but it can get stuck in local minima.

Expected Improvement

Expected Improvement is the most popular acquisition function for Bayesian optimization. It accounts for both the predicted value and the surrogate’s uncertainty:

\[ \text{EI}(\mathbf{x}) = (y_{\min} - \mu(\mathbf{x})) \, \Phi(Z) + \sigma(\mathbf{x}) \, \phi(Z) \]

where \(Z = (y_{\min} - \mu(\mathbf{x})) / \sigma(\mathbf{x})\), \(\Phi\) is the standard normal CDF, and \(\phi\) is the PDF.

from spotoptim import SpotOptim
from spotoptim.function import ackley

opt = SpotOptim(
    fun=ackley,
    bounds=[(-5, 5), (-5, 5)],
    acquisition="ei",
    max_iter=25,
    n_initial=10,
    seed=0,
)
result = opt.optimize()

print(f"Best x    : {result.x}")
print(f"Best f(x) : {result.fun:.6f}")

Best x    : [-0.02052553  0.01131928]
Best f(x) : 0.080871

Acquisition Optimizers

The acquisition optimizer determines how spotoptim searches for the point that maximizes the acquisition function. Set via acquisition_optimizer:

Value	Description
`"differential_evolution"`	Differential Evolution (default) — global optimizer, robust
`"tricands"`	Triangulation candidates — geometric infill based on Delaunay triangulation
`"de_tricands"`	Hybrid: randomly switches between DE and tricands per iteration
`"L-BFGS-B"`, `"Nelder-Mead"`, etc.	Any `scipy.optimize.minimize` method

Differential Evolution (Default)

DE is a global optimizer that works well in most situations. It searches the entire bounds for the best acquisition value.

from spotoptim import SpotOptim
from spotoptim.function import rosenbrock

opt = SpotOptim(
    fun=rosenbrock,
    bounds=[(-2, 2), (-2, 2)],
    acquisition="ei",
    acquisition_optimizer="differential_evolution",
    max_iter=25,
    n_initial=10,
    seed=0,
)
result = opt.optimize()

print(f"Best f(x) : {result.fun:.6f}")

Best f(x) : 0.864057

Triangulation Candidates

Tricands generates candidate points from the Delaunay triangulation of existing data. This is a purely geometric approach that does not require gradient information.

from spotoptim import SpotOptim
from spotoptim.function import sphere

opt = SpotOptim(
    fun=sphere,
    bounds=[(-5, 5), (-5, 5)],
    acquisition_optimizer="tricands",
    max_iter=20,
    n_initial=10,
    seed=0,
)
result = opt.optimize()

print(f"Best f(x) : {result.fun:.6f}")

Best f(x) : 0.210910

Hybrid: DE + Tricands

The prob_de_tricands parameter (default 0.8) controls the probability of using DE vs. tricands when acquisition_optimizer="de_tricands":

from spotoptim import SpotOptim
from spotoptim.function import ackley

opt = SpotOptim(
    fun=ackley,
    bounds=[(-5, 5), (-5, 5)],
    acquisition_optimizer="de_tricands",
    prob_de_tricands=0.7,
    max_iter=25,
    n_initial=10,
    seed=0,
)
result = opt.optimize()

print(f"Best f(x) : {result.fun:.6f}")

Best f(x) : 0.005149

See Triangulation Candidates for details on the underlying algorithm.

Multiple Infill Points

By default, spotoptim proposes one new point per iteration. Set n_infill_points to propose multiple candidates at once, which is useful for batch-parallel evaluation:

from spotoptim import SpotOptim
from spotoptim.function import sphere

opt = SpotOptim(
    fun=sphere,
    bounds=[(-5, 5), (-5, 5)],
    n_infill_points=3,
    max_iter=25,
    n_initial=10,
    seed=0,
)
result = opt.optimize()

print(f"Best f(x) : {result.fun:.6f}")
print(f"Evaluations: {result.nfev}")

Best f(x) : 0.000000
Evaluations: 25

Handling Acquisition Failures

When the acquisition optimizer fails to find a valid new point (e.g., due to a flat surrogate surface), the fallback strategy is controlled by acquisition_failure_strategy:

"random" (default) — generate a random point within bounds

Surrogate Point Budget

Use max_surrogate_points to limit how many data points the surrogate is fitted on. This keeps surrogate fitting fast when the evaluation budget is large:

from spotoptim import SpotOptim
from spotoptim.function import sphere

opt = SpotOptim(
    fun=sphere,
    bounds=[(-5, 5), (-5, 5)],
    max_surrogate_points=30,
    selection_method="distant",
    max_iter=25,
    n_initial=10,
    seed=0,
)
result = opt.optimize()

print(f"Best f(x) : {result.fun:.6f}")

Best f(x) : 0.000001

The selection_method="distant" (default) selects the most spatially diverse subset of points for surrogate fitting.