Download presentation

Presentation is loading. Please wait.

Published bySantos Mull Modified over 2 years ago

1
A New Analysis of the LebMeasure Algorithm for Calculating Hypervolume Lyndon While Walking Fish Group School of Computer Science & Software Engineering The University of Western Australia

2
A New Analysis of LebMeasure Page 2 of 25 Overview l Metrics for MOEAs l Hypervolume l LebMeasure and its behaviour l Empirical data on the performance of LebMeasure l A lower-bound on the complexity of LebMeasure l The general case l Conclusions and future work

3
A New Analysis of LebMeasure Page 3 of 25 Metrics for MOEAs l A MOEA produces a front of mutually non-dominating solutions to a given problem u m points in n objectives l To compare the performance of MOEAs, we need metrics to compare fronts l Many metrics have been proposed, of several types u cardinality-based metrics u convergence-based metrics u spread-based metrics u volume-based metrics

4
A New Analysis of LebMeasure Page 4 of 25 Hypervolume (S-metric, Lebesgue measure) l The hypervolume of a front is the size of the portion of objective space collectively dominated by the points on the front l Hypervolume captures in one scalar both the convergence and the spread of the front l Hypervolume has nicer mathematical properties than many other metrics l Hypervolume can be sensitive to scaling of objectives and to extremal values l Hypervolume is expensive to calculate u enter LebMeasure

5
A New Analysis of LebMeasure Page 5 of 25 LebMeasure (LM) l Given a mutually non-dominating front S, LM u calculates the hypervolume dominated exclusively by the first point p, then u discards p and processes the rest of S l If the hypervolume dominated exclusively by p is not “hyper-cuboid”, LM u lops off a hyper-cuboid that is dominated exclusively by p, and u replaces p with up to n “spawns” that collectively dominate the remainder of p’s exclusive hypervolume l A spawn is discarded immediately if it dominates no exclusive hypervolume, either because u it has a “zero” objective, or u it is dominated by an unprocessed point

6
A New Analysis of LebMeasure Page 6 of 25 LebMeasure in action A dominates exclusively the yellow shape A lops off the pink hyper-cuboid A has three potential spawns: A1 = (4,9,4) A2 = (6,7,4) A3 = (6,9,3) But A2 is dominated by B, so it is discarded immediately

7
A New Analysis of LebMeasure Page 7 of 25 A boost for LebMeasure Some “spawns of spawns” are guaranteed to be dominated, so LM doesn’t need to generate them at all l This limits the maximum depth of the stack to m + n – 1 (6, 9, 4) (9, 7, 5) (1,12, 3) (4, 2, 9) (4, 9, 4) (6, 9, 3) (9, 7, 5) (1,12, 3) (4, 2, 9) (1, 9, 4) (4, 7, 4) (4, 9, 3) (6, 9, 3) (9, 7, 5) (1,12, 3) (4, 2, 9) A13 A12 A11 guaranteed to be dominated } A3 B C DDD CC BB A A1

8
A New Analysis of LebMeasure Page 8 of 25 But… l This boost greatly reduces the space complexity of LM u the maximum depth of the stack is linear in both m and n l But it does far less for the time complexity of LM u note that the time complexity depends not only on the number of stack slots used, but also on how many times each slot is used l We shall measure the time complexity of LM in terms of the number of points (and spawns, and spawns of spawns, etc) that actually contribute to the hypervolume u i.e. the number of hyper-cuboids that must be summed

9
A New Analysis of LebMeasure Page 9 of 25 Running LebMeasure 1 5 5 2 4 4 3 3 3 4 2 2 5 1 1 nm = 2m = 5m = 8m = 10 225810 342564100 481255121,000 5166254,09610,000 6323,12532,768100,000 76415,625262,1441,000,000 812878,1252,097,15210,000,000 9256390,62516,777,216100,000,000 No. of hyper-cuboids = m n−1 m points in n objectives

10
A New Analysis of LebMeasure Page 10 of 25 Running LebMeasure (in reverse order) 1 5 5 2 4 4 3 3 3 4 2 2 5 1 1 No. of hyper-cuboids = m nm = 2m = 5m = 8m = 10 225810 3258 4258 5258 6258 7258 8258 9258 m points in n objectives

11
A New Analysis of LebMeasure Page 11 of 25 Running LebMeasure (in optimal order) 1 1 2 3 4 5 1 5 2 2 3 4 5 1 2 4 3 3 4 5 1 2 3 3 4 4 5 1 2 3 4 2 5 5 1 2 3 4 5 1 nm = 2m = 3m = 4m = 5 22345 32345 446810 54172329 681788112 783588549 816105180549 9161055581,115 10322132,2483,421 11326412,24814,083 12646414,52870,899 13641,28913,70870,889 141283,87354,976142,309 151283,87354,976428,449 162567,761110,1601,721,605 1725623,297331,1288,618,577 No. of hyper-cuboids m(m!) ((n−2)div m) ((n – 2)mod m)! m points in n objectives

12
A New Analysis of LebMeasure Page 12 of 25 Running LebMeasure (first point only) 1 5 5 2 4 4 3 3 3 4 2 2 5 1 1 nm = 2m = 5m = 8m = 10 21111 3391519 4761169271 5153691,6953,439 6312,10115,96140,951 76311,529144,495468,559 812761,7411,273,6095,217,031 9255325,08911,012,41556,953,279 No. of hyper-cuboids = m n−1 – (m – 1) n−1, i.e. O(m n−2 ) m points in n objectives

13
A New Analysis of LebMeasure Page 13 of 25 A lower-bound on the complexity of LebMeasure l We can determine a lower-bound on the worst-case complexity of LM by considering a single example l We will derive a recurrence for the number of hyper-cuboids summed for this example, then prove that the recurrence equals 2 n−1 1 2 2 2 2 2 1 1 1 1

14
A New Analysis of LebMeasure Page 14 of 25 The simple picture 12222 11222121221221212221 11112 112211221112121112121211211122 112111211111121

15
A New Analysis of LebMeasure Page 15 of 25 The recursive picture 12222 1121212112 11112 11122 112221212212212 12111 11211 12211 11121 1122112121 12221

16
A New Analysis of LebMeasure Page 16 of 25 A recurrence h(n,k) gives the number of hyper-cuboids summed for a point (or spawn) with n 2s, of which we can reduce k and still generate points that aren’t dominated by their relatives hcs(n) gives the total number of hyper-cuboids summed for the example, with n objectives

17
A New Analysis of LebMeasure Page 17 of 25 The recurrence in action [ h(4,4) ] (1,2,2,2,2) (1,2,2,1,2) [ h(3,2) ] (1,2,2,2,1) [ h(3,3) ] (1,2,1,2,2) [ h(3,1) ] (1,1,2,2,2) [ h(3,0) ]

18
A New Analysis of LebMeasure Page 18 of 25 The recurrence solved l Simple expansion shows that l The paper gives a formal proof using mathematical induction

19
A New Analysis of LebMeasure Page 19 of 25 The general case l It is difficult to be certain what patterns of points will perform worst for LM l We will describe the behaviour of an illegal “beyond worst case” pattern l Illegal because some points dominate others mmm m−1m−1m−1m−1m−1m−1 11 1

20
A New Analysis of LebMeasure Page 20 of 25 m points in 2 objectives x i denotes the i th best value in objective x l Each vertical list has length m l Total size m 2 u1v1u1v1

21
A New Analysis of LebMeasure Page 21 of 25 m points in 3 objectives l Each vertical list has length m l Each 2-way sub-tree has size m 2 l Total size m 3 u1v1w1u1v1w1

22
A New Analysis of LebMeasure Page 22 of 25 m points in 4 objectives l denotes a k-way sub-tree l Each k-way sub-tree has size m k l Total size m 4 k 1 2 3 1 2 3 1 2 31 2 3 u1v1w1x1u1v1w1x1

23
A New Analysis of LebMeasure Page 23 of 25 A recurrence and its solution l Again, we can capture this behaviour as a recurrence By simple expansion (and proved formally in the paper)

24
A New Analysis of LebMeasure Page 24 of 25 Conclusions l LM is exponential in the number of objectives, in the worst case l Re-ordering the points often makes LM go faster, but the worst case is still exponential u the proof technique used for the “simple” case will also work for the “unreorderable” case

25
A New Analysis of LebMeasure Page 25 of 25 Future work l Try to make LM faster u re-order the points u re-order the objectives l Develop and refine other algorithms (e.g. HSO) u possibly develop a hybrid algorithm l Prove that no polynomial-time algorithm exists for calculating hypervolume

Similar presentations

Presentation is loading. Please wait....

OK

Addition Facts 1 - 20.

Addition Facts 1 - 20.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on biodegradable and non biodegradable materials Ppt on colour tv transmitter and receiver Ppt on blue ocean strategy Ppt on marketing management introduction Ppt on articles of association of private Download ppt on social impacts of it Ppt on nestle india ltd Ppt on 9 11 attack Ppt on c programming language Ppt on traffic light controller using verilog