appeared in AAAI 2017.

1. Regret Ratio Minimization in
Multi-objective Submodular
Function Maximization
P(S)
a
Tasuku Soma (U. Tokyo)
with Yuichi Yoshida (NII & PFI)
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2. Submodular Func. Maximization
f : 2E
→ R+ is submodular:
f(X + e) − f(X) ≥ f(Y + e) − f(Y) (X ⊆ Y, e ∈ E Y)
“diminishing return”
max f(X)
s.t. X ∈ C
Applications
• Influence Maximization
• Data Summarization,
etc
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3. Multiple Criteria
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4. Multiple Criteria
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5. Multiple Criteria
1. coverage
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6. Multiple Criteria
1. coverage
2. diversity
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7. Multi-objective Optimization?
× Exponentially Many Pareto Solutions!
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8. Multi-objective Optimization?
× Exponentially Many Pareto Solutions!
“Good” Subsets of Pareto Solutions
• k-representative skyline queries [Lin et al. 07,
Tao et al. 09]
• top-k dominating queries [Yiu and Mamoulis 09]
• regret minimizing database [Nanongkai et al. 10]
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9. Multi-objective Optimization?
× Exponentially Many Pareto Solutions!
“Good” Subsets of Pareto Solutions
• k-representative skyline queries [Lin et al. 07,
Tao et al. 09]
• top-k dominating queries [Yiu and Mamoulis 09]
• regret minimizing database [Nanongkai et al. 10]
Issue These assume data points are explicitly given...
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10. Our Results
Extend regret ratio framework to submodular maximization
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11. Our Results
Extend regret ratio framework to submodular maximization
Upper Bound
Given an α-approx algorithm for (weighted) single
objective problem,
• regret ratio 1 − α/d for any d
• regret ratio 1 − α + O(1/k) for any k and d = 2.
d = # objectives, k = # of feasible solutions
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12. Our Results
Extend regret ratio framework to submodular maximization
Upper Bound
Given an α-approx algorithm for (weighted) single
objective problem,
• regret ratio 1 − α/d for any d
• regret ratio 1 − α + O(1/k) for any k and d = 2.
d = # objectives, k = # of feasible solutions
Lower Bound
• Even if α = 1 and d = 2, it is impossible to achieve
regret ratio o(1/k2
).
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13. Regret Ratio
Single Objective
The regret ratio for S ⊆ C and f is
rr(S) = 1 −
maxX∈S f(X)
maxX∈C f(X)
.
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14. Regret Ratio
Single Objective
The regret ratio for S ⊆ C and f is
rr(S) = 1 −
maxX∈S f(X)
maxX∈C f(X)
.
Multi Objective
The regret ratio for S ⊆ C and f1, . . ., fd is
rrf1,...,fd,C(S) = max
a∈Rd
+
rrfa,C(S),
where fa := a1f1 + · · · + adfd. (linear weighting)
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15. Geometry of Regret Ratio
f1
f2
Pareto opt point
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16. Geometry of Regret Ratio
f1
f2
Pareto opt point
P(S)
point in S
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17. Geometry of Regret Ratio
f1
f2
Pareto opt point
P(S)
point in S
ε−1
P(S)
rr(S) ≤ 1 − ε ⇐⇒ f(X) ∈ ε−1
P(S) (X ∈ C).
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18. Regret Ratio Minimization
Given: f1, . . ., fd: submodular, C ⊆ 2E
, k > 0
minimize rr(S)
subject to S ⊆ C, |S| ≤ k.
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19. Algorithm 1: Coordinate
f1
f2
approx solution to
maxX∈C f1(X)
approx solution to
maxX∈C f2(X)
Scoord: α-approx. solutions to maxX∈C fi(X) (i = 1, . . ., d)
=⇒ rr(Scoord) ≤ 1 − α/d.
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20. Algorithm 2: Polytope
f1
f2
a: normal vector
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21. Algorithm 2: Polytope
f1
f2
approx solution to
maxX∈C fa(X)
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22. Algorithm 2: Polytope
f1
f2
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23. Algorithm 2: Polytope
f1
f2
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24. Algorithm 2: Polytope
f1
f2
S: output of Coordinate and d = 2,
=⇒ rr(S) ≤ 1 − α − O(1/k).
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25. Lower Bound
f1(X) = cos
π|X|
2n
, f2(X) = sin
π|X|
2n
> π
2k
f1
f2
distance = O(1/k2
)
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26. Experiment
Algorithms
• Coordinate
• Polytope
• Random: Pick k random directions a1, . . ., ak and
output the family {X1, . . ., Xk } of solutions, where Xi
is an approx solution to max
X∈C
fai
(X).
Machine
• Intel Xeon E5-2690 (2.90 GHz) CPU, 256 GB RAM
• implemented in C#
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27. Data Summarization
Dataset: MovieLens
E: set of movies, si,j: similarities of movies i and j
f1(X) =
i∈E j∈X
si,j, coverage
f2(X) = λ
i∈E j∈E
si,j − λ
i∈X j∈X
si,j diversity
C = 2E
(unconstrained), 1 ≤ k ≤ 20, λ > 0,
single-objective algorithm: double greedy (1/2-approx)
[Buchbinder et al. 12]
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28. Result
0 5 10 15 20
k
10−3
10−2
10−1
100
101
Estimatedregretratio
Polytope
Random
Coordinate

29. Result
0 5 10 15 20
k
10−3
10−2
10−1
100
101
Estimatedregretratio
Polytope
Random
Coordinate
regret ratio
decreases dramatically
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30. Our Results
Extend regret ratio framework to submodular maximization
Upper Bound
Given an α-approx algorithm for (weighted) single
objective problem,
• regret ratio 1 − α/d for any d
• regret ratio 1 − α + O(1/k) for any k and d = 2.
d = # objectives, k = # of feasible solutions
Lower Bound
• Even if α = 1 and d = 2, it is impossible to achieve
regret ratio o(1/k2
).
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