Hypervolume

As explained in Bayesian Optimization in Action

Hypervolume is the measure of objective space dominated by a set of multiobjective solutions relative to a reference point.

Hypervolume measures the quality of a set of solutions when objectives conflict. Rather than asking for one best point, it measures how much objective space is covered by the set’s nondominated outcomes. A larger dominated region indicates a better approximation to the trade-offs available between objectives. section 11.2 section 11.5

For two objectives being maximized, first identify the nondominated points: no other observed point improves one objective without worsening the other. These points form the Pareto frontier. Choose a reference point below the objective values of interest; hypervolume is the area between that point and the frontier. In more objectives, the same idea becomes a volume in higher-dimensional objective space. section 11.2 section 11.5

Comparing two search result sets

Consider two conflicting one-dimensional objective functions whose maxima lie near opposite ends of the search space. A dense grid of evaluations traces much more of the trade-off boundary than 20 random evaluations, so its dominated hypervolume is larger. Hypervolume therefore compares complete datasets, rewarding both strong objective values and coverage of the frontier rather than performance on only one objective. section 11.2

Figure 11.8 The reference point in a multiobjective optimization problem, which sets a lower bound for the dominated space. The hypervolume is computed to be the volume of the region between the reference point and the Pareto frontier.

Use the reference point as the lower corner and the Pareto frontier as the upper boundary of the region being measured. section 11.2

Hypervolume can also guide Bayesian optimization. Expected Hypervolume Improvement (EHVI) scores candidate queries by their potential to extend the current frontier, balancing competing objectives instead of alternating between separate single-objective searches. Across repeated query budgets, average hypervolume provides a direct way to compare such policies. section 11.3 section 11.5

In BoTorch, construct a DominatedPartitioning object from the reference point and observed objective values, then call compute_hypervolume(). The reference point is commonly set to the lowest values of the objectives being optimized, so it fixes the endpoint of the measured dominated space. section 11.2 section 11.5

Airplane design exercise

For a four-dimensional airplane-design search space, flight utility and negated flight cost form two objectives. Evaluate EHVI and alternating Expected Improvement over repeated 50-query experiments, then plot their average hypervolumes and error bars to compare how effectively they discover trade-offs. section 11.4

How Bayesian Optimization in Action covers it

How deep it goes

Defines hypervolume through nondominance, the Pareto frontier, and a reference point, then implements its BoTorch computation.

Examples

Contrasts dense-grid and random evaluations, then uses airplane utility and cost to compare EHVI with alternating Expected Improvement.

What it assumes

Assumes familiarity with objective functions and Bayesian optimization while explaining nondominance, Pareto frontiers, and reference points directly.

What you can do after

Apply it via BoTorch to compute dataset hypervolume and evaluate multiobjective Bayesian-optimization policies across query budgets.

How it connects

Connects hypervolume to multiobjective optimization, nondominated points, Pareto frontiers, Bayesian optimization, Expected Improvement, and EHVI.

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Related concepts

• baselineconfigured by · A baseline is a reference value, model, condition, or performance level used to evaluate or compare measurements, outcomes, or methods.
• ExpectedHypervolumeImprovementused by · ExpectedHypervolumeImprovement is a BoTorch acquisition-function class that computes analytic expected hypervolume improvement for multiobjective Bayesian optimization.
• Pareto frontiermeasured by · A Pareto frontier is the set of feasible solutions that are not dominated by any other solution across the objectives being optimized.
• multi-parameter optimizationmeasures · Multi-parameter optimization is the simultaneous optimization of multiple objectives, generally seeking solutions that balance trade-offs among them.
• dominated regionmeasures · A dominated region is the portion of objective space whose points are no better than at least one solution in the objective set.

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.