Utility function

As explained in Bayesian Optimization in Action

Utility is the value assigned to an outcome according to how desirable or useful it is for an optimization goal.

A utility function assigns value to an outcome according to how useful it is for an optimization goal. In Bayesian optimization, utility measures whether a query helps find a better objective value, rather than rewarding its raw value alone. section 4.2.1

Improvement over the best-seen point

Suppose one evaluation returns 2 and another returns 10. The second is better until a previous observation has already achieved 20; then neither adds useful improvement. An observation returning 21 does add utility because it exceeds the incumbent, the best value found so far. section 4.2.1

Utility can encode different priorities when selecting the next query:

• Improvement-based utility values a point according to how much it may exceed the incumbent, using the Gaussian-process belief about the objective. section 4.2.1
• Cost-aware utility values information gained per unit of query cost, following a return-on-investment calculation. section 9.3.2
• The cost-aware trade-off can favor inexpensive low-fidelity queries, but high-fidelity queries become preferable when low-fidelity data or prices make them more valuable. section 9.3.2

Figure 9.10 The formula of the ROI acquisition score for multifidelity optimization. This score quantifies the amount of information a query provides about the objective’s optimum for each unit of cost.

Use the ROI acquisition-score formula to see how information gain is divided by query cost in multifidelity optimization. section 9.3.2

How Bayesian Optimization in Action covers it

How deep it goes

Defines utility through incumbents, Gaussian-process predictions, information gain, and cost-aware ROI, but does not derive the full underlying models.

Examples

Uses objective values 2, 10, 20, and 21, plus ROI scores and low- versus high-fidelity query comparisons.

What it assumes

Assumes objective-function evaluations and introduces the incumbent before using Gaussian-process beliefs and information gain.

What you can do after

Apply it via Bayesian optimization to judge candidate queries by expected improvement or information gained per unit cost.

How it connects

Connects utility to Bayesian optimization, acquisition functions, Gaussian processes, incumbents, multifidelity optimization, information gain, and return on investment.

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Related concepts

• expected utilityused by · Expected utility is the probability-weighted average of the utility values of possible outcomes associated with an action or decision.
• objective functioncontrasts with · An objective function assigns values to candidate solutions, predictions, or parameter settings according to a criterion to be minimized or maximized.
• costequivalent to · Cost is the monetary expenditure, computational or other resource consumption, effort, or burden incurred in acquiring, producing, operating, or performing something.

This explainer was generated by Manning's knowledge graph from the text of Bayesian Optimization in Action. The section links point to the parts of the book it draws on.