Acquisition function

As explained in Bayesian Optimization in Action

An acquisition score is a numerical measure of the expected usefulness of evaluating a candidate point or batch for identifying an objective-function optimum.

An acquisition function is a decision rule for Bayesian optimization: it turns a probabilistic model of an expensive objective into a numerical score for each candidate query. A high score means that evaluating that location is expected to be useful for finding the global optimum. Choose the candidate that maximizes the score, evaluate the objective there, add the result to the observed data, retrain the model, and repeat. section 1.2.2 section 4.1.1

The score should not merely chase the largest predicted value. It can value a location because it looks promising, because its outcome is uncertain, or because both signals together make it worth testing. This is the exploration–exploitation trade-off: exploitation concentrates evaluations in high-performing regions, while exploration learns about uncertain regions that may contain a better result. section 1.2.2 section 4.5

Figure 1.4 A BayesOpt policy scores each individual data point by its usefulness in locating the global optimum. The policy prefers high predictive values (where the payoff is more likely) as well as high uncertainty (where the payoff may be large).

Compare the predictive distribution with the score curve: useful candidates can arise from high predicted payoff or from uncertainty with substantial upside. section 1.2.2

A practical acquisition-driven optimization loop is:

• Fit a Gaussian process (GP) to the objective evaluations collected so far. section 1.2.2
• Score candidate inputs with a policy or acquisition function using the GP’s predictive distribution. section 4.1.1
• Maximize the acquisition score to select the next input, then evaluate the expensive objective. section 4.1.1
• Add the observed label to the training data and continue until the evaluation budget is exhausted. section 4.5

Selecting the next one-dimensional query

For a one-dimensional objective over a bounded search space, plot the GP mean and confidence interval above an acquisition-score curve. The peak of the lower curve identifies the next query. In BoTorch, optimize_acqf() takes a policy and bounds and returns a point that maximizes its acquisition score. Expected Improvement is one policy whose scores can balance improving on the current best value with investigating uncertainty; Probability of Improvement can be more exploitative and risk-averse. section 4.1.1 section 4.5

The score’s meaning depends on the policy. Improvement-based policies value prospective improvement over the best observed value, while information-theoretic policies can value information gained about the objective’s maximum. No single policy is consistently best across all objectives, so inspect optimization progress and adjust the policy when its recommendations waste evaluations or converge poorly. section 4.1.1 section 4.5 section 7.4.1

Figure 7.4 Queries made in the batch setting if we were to simply pick out the points with the highest acquisition scores, denoted by the vertical ticks on the lower curve. These queries are close to each other and are less useful than if they were more spread out.

Selecting the individually highest sequential scores for a batch can cluster queries near one location, wasting parallel evaluations. section 7.1.2

Batch optimization changes what a useful score must represent. When several objective evaluations can run in parallel, selecting the top individual scores may produce nearly identical queries. Batch policies instead need to assess the usefulness of the collection, favoring points that provide complementary value. For batch maximum-value entropy search, the joint information score is well-defined but costly to compute and difficult to optimize; cyclic refinement offers a way to improve batch members in turn. section 7.1.1 section 7.1.2 section 7.4.1

How Bayesian Optimization in Action covers it

How deep it goes

Defines scoring and develops iterative, visual, implementation, batch, and information-theoretic uses, while deferring policy-specific details to other treatments.

Examples

Uses a one-dimensional objective, GP confidence intervals, Expected Improvement score curves, BoTorch optimization, and clustered parallel queries.

What it assumes

Introduces acquisition scoring through GP predictive means and uncertainty without requiring advanced probability or GP knowledge.

What you can do after

Apply it via BoTorch optimize_acqf() to select score-maximizing Bayesian-optimization queries and inspect policy progress.

How it connects

Connects Gaussian processes, Bayesian optimization, Expected Improvement, Probability of Improvement, exploration–exploitation, reinforcement learning policies, and maximum-value entropy search.

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Also explained in

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  Experimentation for EngineersAn acquisition function is a criterion that scores candidate evaluations according to their expected value for selecting the next experiment in Bayesian optimization.
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Related concepts

• lower confidence boundincludes · A lower confidence bound is an acquisition function that combines a predicted response with a multiple of predictive uncertainty to favor promising or informative evaluations.
• active learningused by · Active learning is a machine-learning technique in which a learner selectively chooses data examples for labeling or feedback to improve a model.
• upper confidence boundused by · Upper confidence bound is a decision rule that combines an estimated value with an uncertainty bonus to select actions or candidates optimistically.
• Bayesian optimizationproduced by · Bayesian optimization is a sequential method for optimizing an expensive or unknown objective by using a probabilistic model to guide evaluations.
• optimizationused by · Optimization is the systematic selection or adjustment of values, actions, or system designs to improve a specified objective, often subject to constraints.
• expected improvementincludes · Expected improvement is a Bayesian optimization acquisition function that estimates a candidate’s expected gain over the best observed objective value while accounting for predictive uncertainty.

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.