An objective function maps a candidate input to a value that defines success. Optimization seeks the input with the best value—maximization in the examples here. In expensive black-box optimization, its formula is unknown: each query only reveals the value at the chosen input. Because evaluations may require costly training, simulation, or other experiments, the aim is to locate a high-value input with few queries. section 1.1.2
Figure 1.2 The framework of a black box optimization problem. We repeatedly query the function values at various locations to find the global optimum.
View optimization as a repeated loop: choose an input, query the black-box objective, and use the observed values to move toward the global optimum. section 1.1.2
Hyperparameter tuning is a typical objective-function problem. For example, model accuracy can be the output while settings such as the number of layers are inputs. Since the response to changing a setting is not known in advance, exhaustive search can be impractical. Random search and grid search provide baselines, but both can spend limited evaluations in unpromising regions. section 1.1.2 section 1.2.4
A one-dimensional hidden landscape
Consider an objective on the interval from −5 to 5 with several local maxima and a global maximum near 4.3. With only 10 evaluations, random search finds a high point near x = 4 in one run, but it can cluster samples by chance. Bayesian optimization instead uses observations to decide where to evaluate next, eventually locating the global optimum in the illustrated run. section 1.2.4
An objective can be continuous and real-valued. This distinguishes Bayesian optimization from a multi-armed bandit formulation with finitely many actions and cumulative reward: Bayesian optimization seeks a single best observed objective value, while nearby inputs are assumed to have related values under a Gaussian-process model. section 5.1.2
Constraints change what counts as a usable optimum. A cost function can mark inputs as feasible or infeasible; the target becomes the feasible input with the greatest objective value. Even when an unconstrained global maximum is highest, its cost may rule it out and force a lower-valued or different local solution. section 8.1.1 section 8.1.2
Simulated airplane design
A four-parameter airplane-design simulator can serve as an objective function: scaled design settings are inputs, and its output is a utility score for performance. Treating the simulator as a black box, Bayesian optimization searches the four-dimensional unit hypercube for settings that maximize that score. section 7.5
For constrained Bayesian optimization, model observed objective values and observed cost values separately. Constrained expected improvement combines expected improvement with the probability of feasibility, prioritizing queries that can improve on the best feasible result. BoTorch provides this policy while retaining separate models for utility and cost. section 8.3 section 8.4
