ExpectedHypervolumeImprovement

As explained in Bayesian Optimization in Action

An expected hypervolume improvement policy selects evaluations according to their expected increase in the dominated hypervolume of a multiobjective solution set.

ExpectedHypervolumeImprovement (EHVI) chooses the next evaluation by estimating how much a candidate is expected to increase the dominated hypervolume of a multiobjective solution set. It generalizes Expected Improvement (EI): instead of improving one objective, it seeks points that may extend several parts of the Pareto frontier. section 11.3 section 11.5

The acquisition score reflects both predicted objective values and uncertainty. EHVI evaluates the expected hypervolume increase probabilistically; its calculation partitions the dominated region into hyperrectangles. Compared with alternating single-objective EI policies, it can assign meaningful scores to multiple regions that might extend the Pareto frontier. section 11.3

BoTorch implementation

Fit one Gaussian process per objective, combine them with ModelListGP, and define a reference point representing the lower endpoint of the dominated region. FastNondominatedPartitioning supplies the current partitioning. Construct ExpectedHypervolumeImprovement with the combined model, reference point, and partitioning, then optimize it with optimize_acqf. A practical comparison alternates EI between two objectives and tracks average hypervolume; EHVI consistently performs better in the demonstrated experiment. section 11.3 section 11.5

Airplane-design exercise

Apply EHVI to a four-dimensional airplane design problem with flight utility and negated flight cost as objectives. Use the reference point [-1.5, -2], run 10 experiments with 50 queries each, and compare EHVI with alternating EI using average hypervolume and error bars. section 11.4

Figure 11.11 The current GP belief about each of the two objective functions (top), the corresponding EI acquisition scores (bottom left), and the EHVI acquisition scores (bottom right). EHVI balances the two objectives, assigning high acquisition scores to multiple regions that likely extend the Pareto frontier.

Compare separate EI scores with EHVI scores to see how multiobjective acquisition balances regions likely to extend the Pareto frontier. section 11.3

How Bayesian Optimization in Action covers it

How deep it goes

Defines expected hypervolume improvement and demonstrates implementation, while deferring hyperrectangle-partitioning mathematics to research literature.

Examples

Uses two-objective acquisition plots, BoTorch code, alternating-EI comparisons, and airplane aerostructural design experiments.

What it assumes

Assumes Gaussian-process modeling and prior GP fitting code, while introducing multiobjective hypervolume acquisition through examples.

What you can do after

Apply it via BoTorch by fitting objective GPs, constructing partitioning, optimizing EHVI, and comparing hypervolume outcomes.

How it connects

Connects Expected Improvement, Gaussian processes, Pareto frontiers, dominated hypervolume, reference points, and non-dominated partitioning.

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Related concepts

• hypervolumeuses · Hypervolume is the measure of objective space dominated by a set of multiobjective solutions relative to a reference point.
• multi-parameter optimizationuses · Multi-parameter optimization is the simultaneous optimization of multiple objectives, generally seeking solutions that balance trade-offs among them.
• expected improvementkind of · 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.