Jokyokai20111124

劣モジュラ制約下の凸最適化問題の理論と応用

数理助教会
2011年11月24日

東京大学生産技術研究所
最先端数理モデル連携研究センター

永野清仁

本発表の構成
劣モジュラ関数 f に付随して定義される基多面体 B(f) 上
での凸関数最小化問題に関する理論と応用のはなし

プロローグ Spanning tree のはなし

Part 1 劣モジュラ制約下の凸最適化の理論のはなし
Nagano (IPCO 2007) など

Part 2 劣モジュラ制約下の凸最適化の応用のはなし
Nagano, Kawahara, Aihara, ICML 2011

グラフの spanning tree
Spanning tree :
グラフ G のすべての node をつなげ,
かつサイクルなしの edge 部分集合

Spanning tree はたくさんある

プロローグ

最小重み spanning tree
5 8 各 edge に重みがあるとし, 重み和
7 最小のspanning tree を求めたい
9 3
spanning tree を全列挙する必要なし
最小重み貪欲アルゴリズムで効率的に計算可能
spanning tree Kruskal 1956

プロローグ

Spanning tree が作る多面体
edgeの有無を1と0で表した
Spanning tree の特性ベクトル 0-1ベクトル

a b
c (1,1,1,0,0) (1,0,1,0,1) (0,1,1,1,0) (0,0,1,1,1)

d e

(1,1,0,1,0) (1,1,0,0,1) (0,1,0,1,1) (1,0,0,1,1)

多面体 PG = (これらの凸包)
凸包
実はこの多面体は基多面体B(f)の例になっている
プロローグ

Spanning tree 多面体上の最適化とその一般化

PG = (spanning tree の特性ベクトルの凸包)

 PG 上の線形関数最小化は貪欲アルゴリズムで効率的
に解ける (最小重み spanning tree の計算と同じ)
一般化

 基多面体 B(f) 上での線形関数最小化も同様の貪欲
アルゴリズムで効率的に解ける Edmonds 1970

 基多面体 B(f) 上での(変数分離可能)凸関数最小化
も多項式時間では解ける
今日はこの最適化問題に対する理論と応用のはなし
プロローグ

Part 1 劣モジュラ制約下の凸最適化の理論のはなし

On Convex Minimization
over Base Polytopes
Kiyohito Nagano

Notations:
• V = {1, ..., n}
• x ∈RV は v (∈V) 番目の成分が
x(v) であるような n 次元ベクトル
• x(S) =∑ x(v) (S⊆V)
v∈S

Submodular Functions
V = {1, ... , n } is a finite set

Set function f : 2V→ R is
 monotone if ∀S ,T s. t. S ⊆T , f (S ) ≤ f (T )
 submodular if
V
∀S ,T ⊆V, S
T
f (S ) + f (T ) ≥ f (S ∪T ) + f (S ∩T )
• Graphs, Networks
• Game Theory, Auction Theory, Machine Learning

Part 1

Submodular Functions
V = {1, ... , n } is a finite set

Set function f : 2V→ R is
 monotone if ∀S ,T s. t. S ⊆T , f (S ) ≤ f (T )

∀S ,T s. t. S ⊆T, and ∀v ∉ T
 (equivalently) submodular if
V
v
T
S
f (S +v ) – f (S ) ≥ f (T +v ) – f (T )
Diminishing reterns
• Graphs, Networks
• Game Theory, Auction Theory, Machine Learning

Part 1

An Interpretation of function f
In this part, we assume
f : 2V→ R is monotone & submodular, f (φ ) = 0
Such f can be seen as a natural cost function
For S ⊆V, we incur cost f (S ) S

One extreme example Another example
∑a(v) Monotone & β >0
a ∈RV
2(S ) = 0 if S =φ
>0 v∈S submodular
f 1(S ) = a(S ) functions f
β o. w.
$5
Intermediate
• Customer 1 eats functions of f 1 & f 2 The case where
• Customer 2 eats we can make a
• Customer 3 eats copy for free

Part 1

Base Polytope B (f )
f : 2V→ R is monotone & submodular, f (φ ) = 0

x(S) ≤ f (S ) ( φ ≠S ⊊ V ) 2n − 2 ineq
The base polytope B(f ) ⊆RV : Notations:
• x = (x(v):v ∈V )∈RV

x(V ) = f (V ) 1 equality • x(S) =∑ x(v)
v∈S

x(3)
f 1(S ) = a(S ) f 2(S ) = 0 if S =φ
B(f )⊆RV ≥0 β o.w.

B(f 1) B(f 2)
x(1)
6 inequalities
x(2) n=3 1 equality

Part 1

Our Problem
Let g : RV → R is an n-variable, separable convex function
We mainly consider
x(3)
minx g (x) = ∑ gv (x(v)) 1-variable,
v∈V convex
sub. to x(S) ≤ f (S ) ∀S
x ∊ B (f )
x(V ) = f (V ) x(1)
x(2)

Assumptions :
 f : 2V→ R is monotone & submodular, f (φ ) = 0
 f is given by a value-giving oracle
 Function gv (x(v)) is differentiable & strictly convex

Part 1

Outline of Part 1
 Definitions
 Examples & Our Results
 Equivalence of Problems
 Remark: Efficient algorithm

minx g (x ) over B (f )

Part 1

Examples of minx g (x ) over B (f )

 Lexicographically optimal bases Fujishige 80
• weight vector w ∈ RV>0
• For x ∊ B (f ) ,
x(v ) x(vn) x(v ) x(v )
Inc.Orderw (x) = ( 1 , ... ,
w(v1) w(vn) )
with w(v1 ) ≤ ≤ w(vn )
1 n
the sequence of weighted components of x in nondecreasing order

x = (1,3,4), w = (10,10, 20) Inc.Orderw (x )
x(3)
x(1) 1 3 4 =( 1 , 4 , 3 )
10 10 20 10 20 10
x(2)

• Lexicographically optimal base :
min ∑ x(v)2 over
lex. max Inc.Orderw(x) over B(f ) B(f )
x x v ∈V w(v)
equivalent

Part 1

Examples of minx g (x ) over B (f )

 Lexicographically optimal bases Fujishige 80
• weight vector w ∈ RV>0
• Lexicographically optimal base :
min ∑ x(v)2 over
lex. max Inc.Orderw(x) over B(f ) B(f )
x x v ∈V w(v)
equivalent
Related problems:
Generalization of lex opt flow Megiddo 74
Egalitarian solution in convex game Dutta & Ray 89
Lex opt base x* minimum ratio problem min f (S )
X ⊆V w(S )
Min norm point x* submodular function minimization (SFM)
(w = 1) Fujishige 84 (Now f is not necessarily monotone)
Size-constrained SFM (See Part 2)
Part 1

Examples of minx g (x ) over B (f ) (continued)
 Lex opt bases Fujishige 80 2
min ∑ x(v) over B(f )
• weight vector w ∈ RV >0 x v ∈V w(v)

 Submodular utility allocation (SUA) markets Jain & Vazirani 07
• money vector m ∈ RV >0 max ∑ m (v ) ln x(v ) over B(f )
x v ∈V
Surprisingly, we will see
Corollary (Our Result)
The lex opt base problem and the SUA market problem
have the same solution if w = m
 Subproblems in approximation algorithms
• Facility location with submodular penalty (Chudak & Nagano, 07)
• Submodular cost set cover problem (Iwata & Nagano, 09)

Part 1

α -curve C = { xα∈ RV : α ∈ J }
Let C⊆RV be the set of optimal solutions to
minx { Σ gv (x(v )) : x(V ) = β } ∀β > 0
v ∈V
minimization under a budget constraint

β3
β2
β1
β1 β2 β3 Curve C
β1 β2
β3
parameter
‐1
C = { xα ∈ RV : α ∈ J ⊆ R } where xα (v ) = (gv) (α ),
α -curve J is an interval⊆ R
Since gv is convex, α1 < α2 ⇒ xα1< xα2
component-wise

Part 1

Examples of α - curve C
Positive vectors w, m ∈ RV >0

 gv (x(v )) = x(v )2 / w (v )  gv (x(v )) = - m (v ) ln x(v )
Lex opt base, Fujishige 80 SUA market, Jain-Vazirani 07

α -curve C α -curve C
w m
half line half line

C = { 2α w : α ∈ (0, +∞) } C = { - 1 m : α ∈(- ∞, 0) }
α
xα xα
If w = m, two curves coincide

Part 1

Rationality of optimal solutions

From the correctness of Decomposition Algorithm, we obtain
sufficient conditions for the rationality of optimal solutions :

Theorem (Our Result) y*
∀β ∈Q>0,∀U⊆V , the optimal solution y* to β

min U { Σ gv (y(v )) : y(U ) = β } is rational β
y ∈R v ∈U
then the optimal solution x* to
minx g (x ) over B (f ) is rational

Part 1

Equivalence of problems

Suppose that we are given
α - curve C(1)
(1) (1)
g (x ) = ∑ gv (x(v ))
v ∈V
(2)
α - curve C(2)
(2)
g (x ) = ∑ gv (x(v ))
v ∈V

Theorem (Our Result)
(2)
Curves C(1) and C coincide
minx g(1)(x ) over B (f ) and minx g(2)(x ) over B (f )
have the same optimal solution

Part 1

Equivalence of problems (examples)
Corollary
The lex opt base problem (Fujishige 80) and the SUA market
problem (Jain&Vazirani 07) have the same solution if w = m
gv (x(v )) = x(v )2 / w (v ) gv (x(v )) = - m(v ) ln x(v )

w α -curve C m α -curve C

2 curves coincide
Corollary
Suppose all functions gv are the same (gv = g0)
x ∈ B(f ) minimizes Euclidean norm ||x|| if and only if
x ∈ B(f ) minimizes g(x ) = ∑ g0 (x(v))
v ∈V

Part 1

Remark: Efficient Algorithm
In this work, we generalized the discussion of
Fleischer & Iwata 03 for lex. opt. base problem

An efficient algorithm for minx g(x ) over B (f )
via parametric submodular function minimization
The method is similar to the parametric cut
algorithm by Gallo, Grigoriadis & Tarjan 89

Parametric version of Orlin's algorithm (IPCO07)

Theorem (Nagano 07, manuscript)
minx g(x ) over B (f ) can be solved in O(n6+ n5EO) time,
where EO is the time for one function evaluation

Part 1

Part 2 劣モジュラ制約下の凸最適化の応用のはなし

Size-constrained Submodular Minimization
through Minimum Norm Base

Kiyohito Nagano
Yoshinobu Kawahara
Kazuyuki Aihara


Outline of Part 2
The densest subgraph problem
This problem is an important special case of
 Introduction
the size-constrained submodular minimization
 Size-constrained submodular minimization
 The algorithm through minimum norm base
 Concluding remarks

Part 2

Example: Finding a dense graph
1
G = (V, E) : undirected graph 1 2 1 1 4
node set V = {1, ... , n} 3 2 1 3
edge set E 5 6
2
Edge weights we ≥ 0(e∈E) S={2,3,6}
Integer k (0≤ k ≤ n) I ({2,3,6}) =1+1 =2

Densest k-subgraph problem :
Find a k-subset S ⊆ V (|S|=k ) that maximizes I(S),
where I(S) is sum of weights of edges within S

Part 2 Size-constrained Submodular Minimization through Minimum Norm Base

Example: Finding a dense graph
1 2
Densest k-subgraph problem : 1 1 1 4
maximize I(S) 3 2 1 3
5 6
subject to S⊆{1, ... , n}, |S|= k 2
If k =3, an optimal solution is {1,5,6}
 NP-hard and constant factor approx algorithms are not known
Applications:
 Community detection problem in complex networks
 Identification of functional modules in protein-protein
interaction networks
size-constrained
submodular minimization
In this talk, we deal with
a more general discrete optimization problem



Outline of Part 2
 Introduction
 Concluding remarks min f (S)
sub. to S ⊆{1,..., n}, |S| = k
f : submodular
This problem generalizes
densestsize constraint
a k-subgraph problem
size-constrained minimum
cut problem
Fundamental NP-hard problems

Part 2

Definition: Submodular function

V ={1, ... , n } is a finite set

A real-valued function f defined on 2V = { S : S ⊆ V } is

 submodular if V
S T
∀S, T ⊆V 1
2
3 4
7
f (S)+ f (T )≥ f (S ∪T)+ f (S ∩T)
6
5 8

Discrete analogue of convex function (Lovász 1983)
This function arises in many fields, e.g.,
graph theory, game theory, machine learning, etc.


Examples of submodular functions
Graph G = (V, E) with edge weights we ≥ 0 (e∈E)
V ={1, ... , n }

 Ex. 1. Intensity function I : 2V→ R S
1 2 3
both of two endpoints
Σ
6
I (S) = we : 4
5
7
8 9
of e∈E are in S 10 11 12

– I is submodular (i.e. I is supermodular)

 Ex. 2. Cut function C : 2V→ R S
1 2 3
e∈E has one endpoint in S
C (S) = Σ we :
and one in V – S
4

10
5
6
7
8 9
11 12

C is submodular


Submodular optimization (general form)

min or max f (S) • V ={1, ... , n}
• submodular function f : 2V→ R
subject to S ∈F
• feasible region F ⊆2V

We deal with minimization problems solvable
in poly time
Unconstrained Submodular Minimization (USM)
Size-constrained Submodular Minimization (SSM) NP-hard

Assumption: Computation of each value f (S) is a basic operation
(Given value oracle which answers value queries)


Unconstrained submodular minimization (USM)

Problem (USM) min f (S)
solvable in
s. t. S ⊆V ={1, ... , n} poly time

 polynomial time algorithm:
ellipsoid method: Grotschel, Lovasz & Schrijver (1981, 1988)
combinatorial algorithm: Schrijver (2000), Iwata+ (2001)

 ”practical” algorithm: Fujishige-Wolfe algorithm (2005)
much faster in practice (Fujishige+ 2006)
this algorithm computes the minimum norm base
minimum L2-norm point in the base polyhedron B(f ) ⊆Rn

min norm base x*∈Rn minimizer of f ⊆{1, ... , n}


Size-constrained submodular minimization (SSM)
Problem (SSM) min f (S) integer k (0≤ k ≤ n)

s. t. S ⊆V , |S| = k
NP-hard
 There is no constant factor approximation algorithm
that runs in polynomial time (Svitkina & Fleischer, 2008)
 special cases 1 1 2 1 1 4
3 2 1 3
• f =–I Densest k-subgraph problem 5
2 6

• f =C Size-constrained minimum cut problem
Both of these are fundamental NP-hard problems
Even for these special cases, good approx algorithms are not known
In this work, we propose a new method for (SSM) that
utilizes the minimum norm base
computes ”a portion of exact optimal solutions”


Example of outputs of the proposed method
For all k =0, ... , n, consider the densest k-subgraph problems
max I (S) s. t. S ⊆V, |S| = k

proposed
1 1 2 1 1 4 method 1 2 4
3 2 1 3 3
5 2 6 5 6

k=0 yes Outputs: {},{1,5,6},{1,2,5,6},
k=1 no {1,2,3,4,5,6}
k=2 no
k=3 yes Our method gives optimal
k=4 yes solutions for k = 0, 3, 4, 6
k=5 no
a portion of exact
k=6 yes
optimal solutions

Problem (SSM) in ML min f(S) s. t. S⊆V, |S|= k

Densest k-subgraph problem (f = – I)
This problem naturally formulates a community
detection problem in complex networks
The identification of functional modules in protein-
protein interaction networks is known as an important
application (Dittrich et al., 2008)

Size-constrained minimum cut problem (f = C)
This problem deals with an explicit size constraint
Contrastingly, spectral clustering, which is one of the
most popular clustering algorithms, can deal with a
minimum cut problem with an implicit size constraint



Outline of Part 2
 Introduction
Our algorithm uses the basic polyhedral theory
associated with submodular functions x3 B( f )
f : 2V→ R B(f ) ⊆ Rn x1
submodular func base polyhedron x2

Part 2

Base polyhedron
For a submodular function f : 2V→ R with f ({}) =0,
the base polyhedron B(f ) ⊆ Rn is given by

B(f ) = {x∊Rn : Σ xi ≤ f (S) (∀S ⊆ V), Σ xi = f (V)}
i∈S i∈V
B(f ) is determined by 2n – 2 inequalities and 1 equality

If n = 3, B(f) is determined by x3
B( f )
x1 ≤ f ({1}) , x1 + x2 ≤ f ({1, 2})
x2 ≤ f ({2}) , x1 + x3 ≤ f ({1, 3}) 23 – 2 ineq x1
x3 ≤ f ({3}) , x2 + x3 ≤ f ({2, 3})
x2
x1 + x2 + x3 = f ({1, 2, 3}) 1 equality


Minimum norm base and algorithms
The minimum norm base x*∈ Rn is an optimal solution to
min Σ i∈V xi2 s. t. x∊B(f )

x* can be computed efficiently (Fleischer & Iwata, 2003; Nagano, 2007)
Fujishige-Wolfe algorithm (2005) finds x* much faster in practice

Algorithms through minimum norm base x*∈ Rn
The Fujishige-Wolfe algorithm for (USM) min f(S) s. t. S⊆V
S*={i ∊ V : x*< 0} minimizes f
i The problem can be solved
In FW algorithm, we use partial information about x*

The proposed algorithm for (SSM) min f(S) s. t. S⊆V, |S|= k
In our algorithm, we use full information about x*


min f(S) s. t. S⊆V, |S|= k
Algorithm for (SSM)
Consider the following algorithm: surprisingly simple!
Algorithm SSM
Step1: Compute the minimum norm base x*∈B(f )⊆ Rn
Step2: Let ξ1 < ξ2 < ⋯ < ξ d be distinct values of x*
Return T0 :={} and Tj := {i ∊ V : xi * ξ j }, ∀j = 1,..., d
≤

Example 1 2
1 1 1 4
The densest k-subgraph problems on 3 2 1 3 B(–I ) ⊆ R6
5
2 6
7 7 7
The minimum norm base is x* = (– , –2, –1, –1, – , – ) ∈B(–I )
3 3 3
7
Thus we have ξ1 =– , ξ2 =–2, ξ3 =–1,
3
T0={} , T1={1,5,6}, T2={1,2,5,6}, T3={1,2,3,4,5,6}
optimal solutions for some of size constraints


min f(S) s. t. S⊆V, |S|= k
Algorithm for (SSM)
Consider the following algorithm: surprisingly simple!
Algorithm SSM
Step1: Compute the minimum norm base x*∈B(f )⊆ Rn
Step2: Let ξ1 < ξ2 < ⋯ < ξ d be distinct values of x*
Return T0 :={} and Tj := {i ∊ V : xi * ξ j }, ∀j = 1,..., d
≤

Theorem [This work] For each j ∈{0, 1,... , d}, Tj ⊆ V is
an optimal solution to Problem (SSM) w. r. t. k = |Tj |.

Algorithm SSM computes ”a portion of exact optimal solutions”
Running time = computation of the minimum norm base
E.g., an implementation of the Fujishige-Wolfe algorithm can be found in
a toolbox for submodular function optimization by Krause (2010, JMLR)


Proof of the validity of the algorithm SSM (sketch)
h(λ)
Define a function h : R → R as
λ
h(λ)= min{f (S)–|S|λ : S ⊆ V} (λ ∊ R) 0
linear function in λ
h is the minimum of 2n linear functions
h(λ) λ
Each Sj is an exact optimal
0 solution to problem (SSM)
S0 S1 S w.r.t. some size constraint
2 S3

Furthermore, with the aid of the result of Fujishige
(1980), we can show that Sj=Tj for all j, where each
Tj is the subset returned by the algorithm SSM


Application to artificial data
Networks are randomly generated from the GENRMF generator
For each randomly generated network, we considered
 the size-constrained minimum cut problem
 the size-constrained minimum s-t cut problem
 the densest k-subgraph problem

The number of subsets (= d) found by the algorithm SSM:

dataset 1 (Genrmf-long) dataset 2 (Genrmf-wide)


Application to real data
We applied the algorithm SSM to the densest k-
subgraph problems on the network with 5,000
nodes and 31,664 edges (n = 5000)
This is a sub-network of social network data cnr-2000
(http://law.dsi.unimi.it/webdata/cnr-2000)

X 104 Intensity

3
The algorithm provides
optimal solutions for 57 I(S) 2
out of 5000 size levels
1

0
0 1000 2000 3000 4000 5000
k


Outline of Part 2
 Introduction

Part 2

Concluding remarks
To the size-constrained submodular minimization
problem, we have proposed a new method that
computes a portion of exact optimal solutions

This result contrasts sharply with the NP-hardness of SSM
(see also the result of Nagano, Kawahara & Iwata, NIPS 2010)

Our method is simple. We just utilize the minimum
norm base to the fullest extent.
The Fujishige-Wolfe algorithm does not have worst
time complexity bounds, so its complexity analysis
should be given in future works.


Jokyokai20111124

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