The History of Hypervolume

1. The History of Hypervolume Lyndon While Walking Fish Group School of Computer Science & Software Engineering The University of Western Australia wfg.csse.uwa.edu.au (Work performed by Luigi Barone, Lucas Bradstreet, Phil Hingston, Simon Huband, and Lyndon While)

2. Background • An optimisation problem is one where the performance of a solution is measured on a continuous scale • usually don’t expect to find an optimal solution • A multi-objective optimisation problem (MOOP) is one where in addition the performance of a solution is measured by more than one objective • e.g. for vehicles: safety vs. acceleration • An algorithm for solving a MOOP returns a set of solutions offering varying trade-offs between the objectives • e.g. a Hummer vs. a Volvo vs. a Porsche • How can we compare such sets? • i.e. how can we compare algorithms? The History of Hypervolume

3. A A’ B B’ C C’ D D’ E A 2-objective MOOP – objective space Objective 2 Maximising both objectives 5 Improving in one objective means downgrading in at least one other 4 3 NB: C’dominatesD 2 1 0 Objective 1 0 1 2 3 4 5 The History of Hypervolume

4. Hypervolume – a metric for comparing sets • The hypervolume of a set is the size of the portion of objective space dominated by the members of the set • also called the S-metric, or the Lebesgue measure, or the Klee’s measure • Hypervolume captures in one scalar both the convergence and the spread of the set • Hypervolume has nicer mathematical properties than many other metrics • Buthypervolume is expensive to calculate The History of Hypervolume

5. Hypervolume {A, B, C, D, E} = 11 A Hypervolume {A’, B’, C’, D’} = 12 A’ B B’ C C’ D D’ Reference point E Hypervolume in 2D Objective 2 5 4 3 2 1 0 Objective 1 0 1 2 3 4 5 The History of Hypervolume

6. Hypervolume in 3D The History of Hypervolume

7. Algorithms for calculating hypervolume • Inclusion-exclusion O(n2m): impractical • LebMeasure • HSO • optimised HSO ← WFG • IHSO ← WFG • IIHSO ← WFG • FPL • BROY • Approximation algorithms notdiscussed today The History of Hypervolume

8. Hypervolume by slicing objectives (HSO, 2001) Each slice has known thickness The kth slice has k points But some of them are dominated in the remaining objectives = The History of Hypervolume

9. LebMeasure (LM, 2003) A dominates exclusively the yellow shape A lops off the pink hyper-cuboid A is replaced by three “spawns” But A2 is dominated The History of Hypervolume

10. Timeline: 2003 — 2006 • HSO known to be O(mn) • LM believed to be O(n2m3) • “proof” published in 2003 • confusion between LM’s space complexity (which is amazingly good) and its time complexity • Proof that LM is O(mn) published by While at EMO in 2005 • Empirical demonstration that HSO substantiallyoutperforms LM published by While et al. in IEEE TEC in 2006 The History of Hypervolume

11. Making HSO faster – reordering objectives • If we process the last objective… • A dominates B, which dominates C, etc • Every slice will have exactly one point • Best-case performance! • If we process the first objective… • No point dominates any other point in the remaining objectives • The kth slice will have k points • Worst-case performance! The History of Hypervolume

12. Timeline: 2005 • Objective-reordering heuristics which improve the performance of HSO by 25–98% published by While et al. at CEC in 2005 • i.e. up to 50x speed-up! • The best heuristic MWW (“minimising worst-case work”) works by estimating for each objective the amount of work that will be required if that objective is processed The History of Hypervolume

13. A A p B B C C D D E E In-line hypervolume • Hypervolume is also used within the operation of multi-objective EAs: • to promote diversity • to aid in selection • for archiving purposes • What we need to calculate now is how much hypervolume a new solution adds to an existing set The History of Hypervolume

14. Incremental HSO (IHSO, 2008) The kth slice has k points But some of them are dominated in the remaining objectives Or p itself may be dominated in the remaining objectives = The History of Hypervolume

15. Timeline: 2006 – 2008 • Use of HSO for in-line hypervolume calculation published by Bradstreet et al. at CEC in 2006 • IHSO published by Bradstreet et al. in IEEE TEC in 2008 • including point- and objective-reordering optimisations • won a UWA best paper award this year! • Use of IHSO for in-line hypervolume calculation published by Bradstreet et al. at CEC in 2007 • substantiallyfaster than the 2006 work The History of Hypervolume

16. Iterated IHSO (IIHSO, 2009) – back to the metric = Need to calculate only the “bottom part” of each slice The History of Hypervolume

17. Timeline: 2008 – 2009 • IIHSO paper currently under review by IEEE TEC • with good heuristics, IIHSO substantiallyoutperforms all previously known algorithms on typical data in 5+D • Publication held up partly by philosophical differences about empirical vs. theoretical analyses • how important is it to know the complexity of a heuristic-based algorithm? • Should we prefer • the algorithm with the best worst-case complexity? • the algorithm with the best “average” performance? The History of Hypervolume

18. We now have worldwide competition! • Remember in 2003 they thought it was all over! The History of Hypervolume

19. IIHSO vs. BROY vs. FPL: performance • Random data in 7D (averages of 200 distinct fronts) The History of Hypervolume

20. IIHSO vs. BROY vs. FPL: variation • Random data with 640 points in 6D (1,000 distinct fronts) The History of Hypervolume

21. IIHSO vs. BROY vs. FPL vs. earlier HSOs • How many points can be processed in 10s? (random data) The History of Hypervolume

22. Future plans • Further adaptation of IHSO for in-line calculations • Possibility of reducing duplicate calculations • Possibility of optimising BROY • Get Lucas to complete his PhD! The History of Hypervolume

23. Any questions?