Monotone 𝑘-Submodular Function

Maximization with Size Constraints

Naoto Ohsaka (UTokyo)

Joint work with Yuichi Yoshida (NII & PFI)

1

2016/3/23

My research interests:

Graph algorithms in practice & theoryWork in [Ohsaka,Akiba,Yoshida,Kawarabayashi. AAAI'14]

Fast algorithm for influence maximization[Kempe,Kleinberg,Tardos. KDD'03]

Q. How to distribute free items to 𝐵 peopleto maximize the spread of influence?

(e.g., rumors & product adoptions)

Instance of monotone submodular maximization

■Introduction

Item0.2

0.3

0.4

0.5

0.9

0.10.7

0.2

0.80.6

Monotone submodular maximization

Monotonicity:

𝑓 𝑆 ≤ 𝑓 𝑇 for 𝑆 ⊆ 𝑇

Submodularity (diminishing return):

𝑓 𝑆 + 𝑒 − 𝑓 𝑆 ≥ 𝑓 𝑇 + 𝑒 − 𝑓 𝑇

for 𝑆 ⊆ 𝑇 & 𝑒 ∉ 𝑇

Goal: max 𝑓 𝑆 s.t. 𝑆 = 𝐵

Naturally arise & 0.63-approx. in poly time[Nemhauser,Wolsey,Fisher. Math. Program. '78]

3

■Introduction

Document summarization [Lin,Bilmes. ACL-HLT'11]

Network inference [Gomez-Rodriguez,Leskovec,Krause. KDD'10]

Sensor placement [Krause,Singh,Guestrin. J. Mach. Learn. Res.'08]

Invest.

Retu

rn

⇨

Motivation of this research

We often have multiple kinds of items 1, 2, 3

Each has a different topic ⇨ different cascades

Q. How to distribute 𝒌 kinds of items to 𝐵 people?

Can be modeled as 𝒌-submodular functions

4

■Introduction

13

2

⋃ ⋃

3

Our contributionsPresented in [Ohsaka,Yoshida. NIPS'15]

5

■Introduction

Monotone 𝑘-submodular function maximization under

Total size constraint

#1 + #2 + #3 = 𝐵

Approx. ratio = 1/2# function eval. =Greedy: Quadratic

Randomized: Almost linear

Influence maximization

with 𝑘 topics

Individual size constraint

#1 = 𝐵1 , #2 = 𝐵2, #3 = 𝐵3

Approx. ratio = 1/3# function eval. =Greedy: Quadratic

Randomized: Almost linear

Sensor placement

with kinds of sensors

Approximation algorithms

Experimental evaluations

Related work

Theoretical results under NO constraint[Iwata,Tanigawa,Yoshida. SODA'16]

Non-monotone: 1/2-approx. alg.

Monotone:𝑘

2𝑘−1-approx. alg.

𝑘+1

2𝑘+ 𝜖 -approx. needs exponentially many queries

Application of bi(2-)submodular functions[Singh,Guillory,Bilmes. AISTATS'12]

Sensor placement & feature selection

6

■Introduction

Function formWe consider a function of form

7

■𝑘-submodular functions

𝑿𝟏 𝑿𝟐

𝑿𝟑

𝑒 𝒂 𝒃 𝒄 𝒅 𝒆 𝒇

𝒙(𝑒) 1 3 2 1 0 3

𝒂

𝒄

𝒃 𝒇 𝒆𝒅

one-to-one

correspondence

the family of

𝑘 disjoint subsets of 𝑉

𝑓: 0,1, … , 𝑘 𝑉 → ℝ𝑉 = 𝑛

𝑓: 𝑘 + 1 𝑉 → ℝ

Another interpretation

𝑓 is monotone 𝒌-submodular iff① & ②

Characterization of 𝑘-submodular functions [Huber,Kolmogorov. ISCO'12] [Ward,Živný. ACM Trans. Algor. '15]

8

■𝑘-submodular functions

0 1 1 3 ≥ 0

Δ𝑒,𝑖𝑓 𝒙 ≥ 0 for 𝒙 s.t. 𝒙 𝑒 = 0

①Monotone

Δ𝑒,𝑖𝑓 𝒙 = change of 𝑓 by assigning 𝑖 to 𝑒th element

② Orthant submodular

0 1 0 0 2 1 0 3≥

Δ𝑒,𝑖𝑓 𝒙 ≥ Δ𝑒,𝑖𝑓 𝒚 for 𝒙 ≼ 𝒚 s.t. 𝒚 𝑒 = 0

0 1 0 3−

0 1 2 0 2 1 2 3− −𝒚𝒙

Modeling influence maximization with 𝑘 topics

9

■𝑘-submodular functions

𝑓 𝒔 ≔Exp. # vertices influenced

in one of the 𝑘 graphs

We have 𝑘 graphs 𝐺1, … , 𝐺𝑘 and a strategy 𝒔 ∈ 𝑘 + 1 𝑉

a b c d e f

1 0 3 3 0 2

Influence from 𝑣 (𝒔 𝑣 = 𝑖) stochastically spreads in 𝐺𝑖

ac d

be f

ac d

be f

ac d

be f

⋃ ⋃ → 5

ac d

be f

ac d

be f

ac d

be f

⋃ ⋃ → 4

…

Event 1

Event 𝑟 𝑓 is

Monotone & 𝑘-submodular

Monotone submodular maximization by

greedy algorithm with lazy evaluations[Nemhauser,Wolsey,Fisher. Math. Program. '78] [Minoux. '78]

■Preliminaries

𝑗 𝑆𝑗 Δ𝑎(𝑆𝑗−1) Δ𝑏(𝑆𝑗−1) Δ𝑐(𝑆𝑗−1) Δ𝑑(𝑆𝑗−1) Δ𝑒(𝑆𝑗−1)

1 𝑏 7 10 3 8 6

2 𝑏, 𝑒 ― ― ≤ 3 2 5

3 ― ― ≤ 3 ≤ 2 ―

𝐟𝐨𝐫 𝑗 = 1 𝐭𝐨 𝐵

𝑒𝑗 ← argmax𝑒

Δ𝑒 𝑆𝑗−1 ≔ 𝑓 𝑆𝑗−1 + 𝑒 − 𝑓 𝑆𝑗−1

𝑆𝑗 ← 𝑆𝑗−1 + 𝑒𝑗

Greedy algorithm

for the total size constraint

Constraint: #1 + #2 + #3 + ⋯ + #k = 𝐵

𝑘-Greedy-TS (total size)

# function eval. = 𝑶 𝒌𝒏𝑩

Approx. ratio = 𝟏/𝟐11

■Proposed algorithms

𝒔𝟎 ← 𝟎𝐟𝐨𝐫 𝑗 = 1 𝐭𝐨 𝐵

𝑒𝑗 , 𝑖𝑗 ← argmax𝑒:𝒔𝑗−1 𝑒 =0, 𝑖

Δ𝑒,𝑖𝑓(𝒔𝑗−1)

𝒔𝑗 ← assign 𝑖𝑗 to the 𝑒𝑗st element in 𝒔𝑗−1

Proof sketch

12

■Proposed algorithms

𝒐0 = 𝒐∗

𝒐1

𝒐2

𝒐3

𝒐4 = 𝒔4

𝒔3

𝒔2

𝒔1

𝒔0 = 𝟎 0 0 0 0 0 0

3 0 0 0 0 0

3 1 0 0 0 0

3 1 4 1 0 0

2 0 4 2 3 0

3 1 4 0 0 0

Sw

ap

Gre

ed

y

optimal

output

𝑎 𝑏 𝑐 𝑑 𝑒 𝑓

3

3 1

3 1 4

Construct a sequence

from 𝒐∗ to 𝒔𝐵 preserving

(#nonzero in 𝒐𝑖) = 𝐵

zero

𝑓 𝒔𝑗 − 𝑓 𝒔𝑗−1

𝑓 𝒐𝑗−1 − 𝑓 𝒐𝑗

≥

Proof sketch

13

■Proposed algorithms

𝑓 𝒐∗ − 𝑓 𝒔

𝑓 𝒔 − 𝑓 𝟎

𝑓 𝒔

≤≤ 𝒇 𝒔 ≥

𝟏

𝟐𝒇(𝒐∗)

𝒐0 = 𝒐∗

𝒐1

𝒐2

𝒐3

𝒐4 = 𝒔4

𝒔3

𝒔2

𝒔1

𝒔0 = 𝟎

Sw

ap

Gre

ed

y

optimal

output

zero

Speeding-up by random sampling

ー 𝑂(𝑘𝑛𝐵) may be quadratic (e.g., 𝐵 = 𝑛/2)

Can be reduced?

Combining random sampling[Mirzasoleiman,Badanidiyuru,Karbasi,Vondrák,Krause. AAAI'15]

At step 𝑗:

Looking at all elements a single random element

W.p. ≥𝐵−𝑗+1

𝑛−𝑗+1, 𝑓 𝒔𝑗 − 𝑓 𝒔𝑗−1 ≥ 𝑓 𝒐𝑗−1 − 𝑓 𝒐𝑗

Looking more elements gains success probability

14

■Proposed algorithms

Almost linear-time algorithm

for the total size constraint

𝑘-Stochastic-Greedy-TS

# function eval. = 𝑶 𝒌𝒏 𝐥𝐨𝐠 𝑩 ⋅ 𝐥𝐨𝐠 𝑩 𝜹

Approx. ratio = 𝟏/𝟐 w.p. 𝟏 − 𝜹15

■Proposed algorithms

𝒔0 ← 𝟎𝐟𝐨𝐫 𝑗 = 1 𝐭𝐨 𝐵

𝑅𝑗 ← a random subset of size min𝑛−𝑗+1

𝐵−𝑗+1log

𝐵

𝛿, 𝑛

𝑒𝑗 , 𝑖𝑗 ← argmax𝑒∈𝑅𝑗:𝒔𝑗−1 𝑒 =0,𝑖

Δ𝑒,𝑖𝑓(𝒔𝒋−𝟏)

𝒔𝒋 ← assign 𝑖𝑗 to the 𝑒𝑗st element in 𝒔𝑗−1

Influence maximization with 𝑘 topics

16

■Experimental evaluations

𝑓 𝒔 ≔ Exp. # vertices influenced in one of the 𝑘 graphs

Exact computation of 𝑓(⋅) is #P-hard [Chen,Wang,Wang. KDD'10]

Averaging 100 simulations

Given: 𝑘 graphs 𝐺1, … , 𝐺𝑘 & a budget 𝐵

Goal: max𝑓(𝒔) s.t. #1 + #2 + #3 + ⋯ + #k = 𝐵

a b c d e f

1 0 3 3 0 2

ac d

be f

ac d

be f

ac d

be f

⋃ ⋃ → 5

Settings

Digg dataset (a social news website)http://www.isi.edu/~lerman/downloads/digg2009.html

A friendship network

𝑛 = 3,522 vertices & 90,244 edges

A log of user votes for stories

Learning parameters

Using the method by [Barbieri,Bonchi,Manco. ICDM'12]

Edge probabilities in 𝑘 = 10 graphs

17

■Experimental evaluations

Algorithms

𝒌-Greedy-TS w/ lazy evaluations

𝒌-Stochastic-Greedy-TS 𝛿 = 0.1 w/ lazy evaluations

Baselines

Single 𝒊

Greedy strategy that considers only the 𝑖th topic

Degree

Select vertices of high degree & assign random topics

Random

Select vertices randomly & assign random topics

18

■Experimental evaluations

Result: Objective function value

19Environment: Intel Xeon E5-2690 2.90GHz CPU with 256GB memory

■Experimental evaluations

outperformed baselines

Result: Objective function value

20Environment: Intel Xeon E5-2690 2.90GHz CPU with 256GB memory

■Experimental evaluations

Single(𝟑) was worse than Degree(Effectiveness of assigning multiple kinds of items)

3 3 0 3 0 1 5 3

Result: # function evaluations

21Environment: Intel Xeon E5-2690 2.90GHz CPU with 256GB memory

Random sampling works well

𝒌-Greedy-TS rapidly grows

∵ 𝑒, 𝑖 gives similar marginal gains at 40th iteration

⇨ we need to check Δ𝑒,𝑖𝑓(𝒔) for most of the 𝑒, 𝑖 's

■Experimental evaluations

Summary

22

Constraint Approx. ratio # function eval.

Total size

#1 + #2 + #3 = 𝐵

1/2 𝑂 𝑘𝑛𝐵

1/2 w.p. 1 − 𝛿 𝑂 𝑘𝑛

Individual size

#1 = 𝐵1 , #2 = 𝐵2, #3 = 𝐵3

1/3 𝑂 𝑘𝑛 𝑖 𝐵𝑖

1/3 w.p. 1 − 𝛿 𝑂 𝑘2𝑛

𝑨𝑪 𝑫

𝑩𝑬 𝑭

𝑨𝑪 𝑫

𝑩𝑬 𝑭

𝑨𝑪 𝑫

𝑩𝑬 𝑭

⋃ ⋃

Influence maximization with 𝑘 topics Sensor placement with 𝑘 kinds of sensors

Monotone 𝑘-submodular function maximization