Is there one universally best acquisition function?

No. Noise, evaluation cost, constraints, parallelism and the decision objective can change which approach is suitable.

Must an acquisition score be a probability?

No. Expected improvement has objective units, while probability of improvement is dimensionless.

Can acquisitions select several experiments together?

Yes. Batch formulations account for the joint value and dependence of multiple proposed evaluations.

AI Glossary / Probabilistic machine learning / Advanced

Bayesian optimization acquisition function

Definition

A Bayesian optimization acquisition function scores possible evaluations using the current probabilistic model of an objective. It guides the choice of where to evaluate next, balancing considerations such as predicted performance, uncertainty, expected improvement, information value and, in some formulations, evaluation cost or feasibility.

What it means

When each experiment is expensive, choosing the next experiment becomes a decision problem of its own. The surrogate model describes what is currently believed about the objective. The acquisition function translates those beliefs into a preference among candidate evaluations.
Its score is not the actual outcome of the experiment. A candidate can have a high acquisition value because it offers useful uncertainty or upside, even if its predicted average performance is lower than another candidate's. Different acquisition rules can reasonably value those possibilities differently.

How it works

Fit or update a probabilistic surrogate using completed evaluations. Compute an acquisition value for each candidate, optimize that acquisition over the allowed domain, then evaluate the selected candidate on the actual objective. Add the result and repeat within the budget.
For noise-free maximization, expected improvement is EI(x)=E[max(f(x)−f_best,0)]. With a normal predictive distribution of mean μ and standard deviation σ>0, EI=(μ−f_best)Φ(z)+σφ(z), where z=(μ−f_best)/σ. Here Φ and φ are the standard normal cumulative distribution and density. Other rules value improvement probability or information about the optimum. Noise, constraints and parallel evaluations require suitable formulations.

Three practical examples

A simulation team could choose which machine setting to test next when each run takes several hours. Acquisition would help allocate simulation effort while final conclusions still depend on evaluated outcomes.
A model-development team could select a hyperparameter configuration based on expected improvement in a validation metric. It would retain a separate final test procedure rather than repeatedly optimizing on the test data.
A materials researcher could consider candidate experiments with different costs or feasibility requirements. The acquisition rule would need to reflect those constraints instead of assuming every point is equally cheap and usable.

Think of it this way

Imagine deciding which unfamiliar route to explore before choosing a regular journey. One route looks reliably good; another is uncertain but might be much better. A rule favoring the greatest chance of a small improvement may choose differently from a rule valuing the expected size of improvement. Neither rule turns the map's predictions into observed travel times.

Worked example

Consider an original noise-free maximization problem whose best measured objective value is 10. Suppose the surrogate predicts candidate A with a normal distribution of mean 11 and standard deviation 0.20, and candidate B with mean 10 and standard deviation 3. These are stipulated predictive distributions, not measured outcomes.
For A, z=(11−10)/0.20=5. Its expected improvement is 1×Φ(5)+0.20×φ(5), approximately 1.000000 objective units. For B, z=0, so the mean-difference term is zero and EI=3×φ(0)=3/√(2π), approximately 1.196827 units.
Expected improvement therefore selects B, even though A has the higher predicted mean. B's uncertain upper tail offers enough potential improvement to outweigh its lower mean under this criterion.
Now use probability of improvement instead. The probability of exceeding 10 is Φ(5), approximately 0.9999997 for A, and 0.50 for B. That rule selects A. It values the chance of beating the incumbent, whereas expected improvement also values how much the candidate might beat it by.
B can still fail to improve: the stipulated probability of that outcome is 0.50. The acquisition calculation makes a choice under uncertainty; it does not certify the true best candidate or guarantee an improvement on the next evaluation.

Illustrative case study

Fictional engineering firm Willow Systems is tuning a simulator with a limited allowance of runs. Its analysts define a single performance objective and the feasible range of each setting. They compare an expected-improvement strategy with a simple baseline under the same evaluation budget.
After each run, they update the surrogate and inspect whether its uncertainty descriptions remain plausible. They keep the experiment-selection score separate from the table of actually measured results, and independently recheck promising settings before a final recommendation. This fictional case illustrates disciplined experimentation; it makes no claim of achieved optimization gains or a verified global optimum.

Common misconceptions and mistakes

“The acquisition function is the expensive objective.” It is a model-based rule used to decide which objective evaluation to perform.
“The highest predicted mean must have the highest acquisition score.” Candidate B disproves that for expected improvement.
“Expected improvement is an improvement probability.” Their units and preferences differ, as the example shows.
“More uncertainty is always better.” The result depends on the rule, predicted values, costs and constraints; uncertainty alone does not establish useful evaluation value.
“A noise-free formula can be applied unchanged to noisy measurements.” Uncertainty in the incumbent and in observations can change the appropriate decision calculation.
“A proposed experiment is the final recommended solution.” Exploration may deliberately evaluate a candidate that would not currently be chosen for deployment.

Business use

Use acquisition functions to make limited experimental budgets more deliberate. Define the objective, allowed settings, evaluation costs and stopping conditions before comparing methods. Record which surrogate and acquisition rule generated each proposal, along with the measured result. Judge the entire optimization process against a credible baseline, and account for the cost of fitting and optimizing the surrogate as well as running experiments. Validate any final recommendation separately when the operational decision warrants it.

Check your understanding

Which candidate does expected improvement select in the example?

Answer: B, with approximately 1.196827 expected improvement units.

Which does probability of improvement select?

Answer: A, whose probability of exceeding 10 is much higher.

Does selecting B guarantee an improvement?

Answer: no; its modeled chance of failing to exceed the incumbent is 0.50.

Related terms

Learn next

Calculate expected improvement for two normal predictive distributions. Then examine how noisy observations or unequal evaluation costs alter the experiment-selection problem.

Sources

Frazier, P. I. A Tutorial on Bayesian Optimization, 2018, especially Sections 2, 4 and 5: https://arxiv.org/pdf/1807.02811 The two-candidate calculation and fictional engineering scenario are original illustrations.

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