Efficient computation of expected hypervolume improvement using box decomposition algorithms

Yang, Kaifeng; Emmerich, Michael; Deutz, André H.; Bäck, Thomas

doi:10.1007/s10898-019-00798-7

Cited by 71 publications

(41 citation statements)

References 38 publications

(118 reference statements)

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“…The computation required to calculate the dominated hypervolume, and therefore the S-metric, scales poorly with the number of objectives and the number of solutions formingP. As observed by Yang et al [20], the expected hypervolume improvement (EHVI) calculation is an NP hard problem in M , but polynomial in |P| for any fixed value of M . Their recent method for decomposition into hyperboxes for EHVI having a complexity of O(2 M −1 • |P| M/2 ).…”

Section: Related Workmentioning

confidence: 99%

Multi-Objective Bayesian Optimisation Using an Exploitative Attainment Front Acquisition Function

Gibson

Everson

Fieldsend

2021

2021 IEEE Congress on Evolutionary Computation (CEC)

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Efficient methods for optimising expensive black-box problems with multiple objectives can often themselves become prohibitively expensive as the number of objectives is increased. We propose an infill criterion based on the distance to the summary attainment front which does not rely on the expensive hypervolume or expected improvement computations, which are the principal causes of poor dimensional scaling in current stateof-the-art approaches. By evaluating performance on the wellknown Walking Fish Group problem set, we show that our method delivers similar performance to the current state-of-theart. We further show that methods based on surrogate mean predictions are more often than not superior to the widely used expected improvement, suggesting that the additional exploration produced by accounting for the uncertainty in the surrogate's prediction of the optimisation landscape is often unnecessary and does not aid convergence towards the Pareto front.

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Section: Related Workmentioning

confidence: 99%

Multi-Objective Bayesian Optimisation Using an Exploitative Attainment Front Acquisition Function

Gibson

Everson

Fieldsend

2021

2021 IEEE Congress on Evolutionary Computation (CEC)

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“…Expected hypervolume improvement (EHVI) (Emmerich et al, 2006) is a natural extension of the popular expected improvement (EI) (Jones et al, 1998) acquisition function to the MOO setting. Recent work has led to efficient computational paradigms using box decomposition algorithms (Yang et al, 2019b) and practical enhancements such as support for parallel candidate generation and gradient-based acquisition optimization (Yang et al, 2019a;Daulton et al, 2020). However, EHVI still suffers from a few limitations including (i) the assumption that observations are noisefree, and (ii) the exponential scaling of its batch variant, qEHVI, in the batch size q, which precludes large-batch optimization.…”

Section: Related Workmentioning

confidence: 99%

“…Computing the hypervolume indicator requires calculating the volume of a typically non-rectangular polytope and is known to have time complexity that is super-polynomial in the number of objectives (Yang et al, 2019b). An efficient approach for computing the hypervolume is to decompose the region that is dominated by the Pareto frontier P and bounded from below by a reference point r into disjoint axisaligned hyperrectangles (Lacour et al, 2017), compute the volume of each hyperrectangle in the decomposition, and sum over all hyperrectangles.…”

Section: Hypervolume Metricsmentioning

confidence: 99%

“…An efficient approach for computing the hypervolume is to decompose the region that is dominated by the Pareto frontier P and bounded from below by a reference point r into disjoint axisaligned hyperrectangles (Lacour et al, 2017), compute the volume of each hyperrectangle in the decomposition, and sum over all hyperrectangles. So-called box decomposition algorithms have also been applied to partition the region that is not dominated by the Pareto frontier P, which can be used to compute the hypervolume improvement from a set of new points (Dächert et al, 2017;Yang et al, 2019b). See Appendix A for further details.…”

Section: Hypervolume Metricsmentioning

confidence: 99%

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Parallel Bayesian Optimization of Multiple Noisy Objectives with Expected Hypervolume Improvement

Daulton¹,

Balandat²,

Bakshy³

2021

Preprint

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Optimizing multiple competing black-box objectives is a challenging problem in many fields, including science, engineering, and machine learning. Multi-objective Bayesian optimization is a powerful approach for identifying the optimal trade-offs between the objectives with very few function evaluations. However, existing methods tend to perform poorly when observations are corrupted by noise, as they do not take into account uncertainty in the true Pareto frontier over the previously evaluated designs. We propose a novel acquisition function, NEHVI, that overcomes this important practical limitation by applying a Bayesian treatment to the popular expected hypervolume improvement criterion to integrate over this uncertainty in the Pareto frontier. We further argue that, even in the noiseless setting, the problem of generating multiple candidates in parallel reduces that of handling uncertainty in the Pareto frontier. Through this lens, we derive a natural parallel variant of NEHVI that can efficiently generate large batches of candidates. We provide a theoretical convergence guarantee for optimizing a Monte Carlo estimator of NEHVI using exact sample-path gradients. Empirically, we show that NEHVI achieves state-of-the-art performance in noisy and large-batch environments.

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“…We exemplarily refer to Auger et al (2009); Yang et al (2019) as two examples from this active research field. The hypervolume indicator also plays a prominent role as a quality indicator in the subset selection problem (see, for example, Guerreiro and Fonseca (2020)), i.e., when a subset of usually bounded cardinality is sought to represent the Pareto front.…”

Section: Introductionmentioning

confidence: 99%

Hypervolume scalarization for shape optimization to improve reliability and cost of ceramic components

et al. 2021

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In engineering applications one often has to trade-off among several objectives as, for example, the mechanical stability of a component, its efficiency, its weight and its cost. We consider a biobjective shape optimization problem maximizing the mechanical stability of a ceramic component under tensile load while minimizing its volume. Stability is thereby modeled using a Weibull-type formulation of the probability of failure under external loads. The PDE formulation of the mechanical state equation is discretized by a finite element method on a regular grid. To solve the discretized biobjective shape optimization problem we suggest a hypervolume scalarization, with which also unsupported efficient solutions can be determined without adding constraints to the problem formulation. FurthIn this section, general properties of the hypervolumeermore, maximizing the dominated hypervolume supports the decision maker in identifying compromise solutions. We investigate the relation of the hypervolume scalarization to the weighted sum scalarization and to direct multiobjective descent methods. Since gradient information can be efficiently obtained by solving the adjoint equation, the scalarized problem can be solved by a gradient ascent algorithm. We evaluate our approach on a 2 D test case representing a straight joint under tensile load.

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Efficient computation of expected hypervolume improvement using box decomposition algorithms

Cited by 71 publications

References 38 publications

Multi-Objective Bayesian Optimisation Using an Exploitative Attainment Front Acquisition Function

Multi-Objective Bayesian Optimisation Using an Exploitative Attainment Front Acquisition Function

Parallel Bayesian Optimization of Multiple Noisy Objectives with Expected Hypervolume Improvement

Hypervolume scalarization for shape optimization to improve reliability and cost of ceramic components

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