regret ratio minimization in multi-objective submodular ... · 自己紹介...

Regret Ratio Minimization inMulti-objective SubmodularFunction Maximization

a相馬輔 (東京大学)

with吉田悠一 (NII & PFI)

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自己紹介

相馬輔（そうまたすく）• 東大情報理工数理 7研,助教• 指導教員: 岩田覚教授• 通信・機械学習に最適化手法からアプローチ

最近の興味• テンソル分解• 劣モジュラ最適化の近似アルゴリズム• 圧縮センシング・スパース最適化

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劣モジュラ最適化劣モジュラ関数: “効率的”に解ける離散構造に現れる関数

Xf (X) = 3

ネットワークフロー全域木

f : 2S → Rが劣モジュラ関数:

f (X ∪ s) − f (X) ≥ f (Y ∪ s) − f (Y ) (X ⊆ Y , s < Y )“限界効用逓減 (diminishing return)”

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劣モジュラ最適化劣モジュラ関数: “効率的”に解ける離散構造に現れる関数

Xf (X) = 3

ネットワークフロー全域木

f : 2S → Rが劣モジュラ関数:

f (X ∪ s) − f (X) ≥ f (Y ∪ s) − f (Y ) (X ⊆ Y , s < Y )“限界効用逓減 (diminishing return)”

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劣モジュラ関数最大化

max f (X) sub. to X ∈ C

解の品質制約 (コスト)

既存研究• 一般にはNP困難• 多くの制約に対して効率的な定数近似アルゴリズム

Applications• Influence Maximization• Data Summarization, etc

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Multiple Criteria

1. coverage

2. diversity

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Multiple Criteria

1. coverage

2. diversity

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Multiple Criteria

1. coverage

2. diversity

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Multiple Criteria

1. coverage

2. diversity

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多目的最適化の出番?

× Pareto最適解は指数個ありうる!

Pareto解の “良い”代表集合を選ぶ既存手法• 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]

既存手法の問題点すべての実行可能解が陽に与えられている必要...

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Tao et al. 09]

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Tao et al. 09]

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概要Regret minimizing databaseを劣モジュラ最大化に拡張

アルゴリズム単目的最大化に対してα近似アルゴリズムがあるとき

• regret ratio 1 − α/d (d:任意)• regret ratio 1 − α + O(1/k) (k:任意, d = 2)• regret ratio 1 − α + O(1/k2) (k:任意, d = O(1))

d = # objectives, k = # feasible solutions

情報理論的下界• α = 1 (厳密アルゴリズム), d = 2であっても，

regret ratio o(1/k2)は達成不可能.

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Regret RatioSingle ObjectiveThe regret ratio for S ⊆ C and f is

rr(S) = maxX∈C f (X) − maxX∈S f (X)maxX∈C f (X) .

Multi ObjectiveThe regret ratio for S ⊆ C and f1, ... , fd is

rrf1,...,fd ,C(S) = maxa∈Rd

rrfa,C(S),

where fa := a1f1 + · · · + ad fd . (linear weighting)

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Regret RatioSingle ObjectiveThe regret ratio for S ⊆ C and f is

rr(S) = maxX∈C f (X) − maxX∈S f (X)maxX∈C f (X) .

Multi ObjectiveThe regret ratio for S ⊆ C and f1, ... , fd is

rrf1,...,fd ,C(S) = maxa∈Rd

rrfa,C(S),

where fa := a1f1 + · · · + ad fd . (linear weighting)

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Geometry of Regret Ratio

f2Pareto opt point

point in S

ε−1P(S)

rr(S) ≤ 1 − ε ⇐⇒ f(X) ∈ ε−1P(S) (X ∈ C).

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f2Pareto opt point

point in S

ε−1P(S)

rr(S) ≤ 1 − ε ⇐⇒ f(X) ∈ ε−1P(S) (X ∈ C).

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f2Pareto opt point

point in S

ε−1P(S)

rr(S) ≤ 1 − ε ⇐⇒ f(X) ∈ ε−1P(S) (X ∈ C).9 / 20

Regret Ratio Minimization

Given: f1, ... , fd : submodular, C ⊆ 2E , k > 0

minimize rr(S)subject to S ⊆ C, |S| ≤ k.

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Algorithm 1: Coordinate

approx solution tomaxX∈C f1(X)

approx solution tomaxX∈C f2(X)

Scoord: α-approx. solutions to maxX∈C fi(X) (i = 1, ... , d)=⇒ rr(Scoord) ≤ 1 − α/d.

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Algorithm 2: Polytope

a: normal vector

S: output of Polytope and d = 2,=⇒ rr(S) ≤ 1 − α + O(1/k).

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a: normal vector

approx solution tomaxX∈C fa(X)

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a: normal vector

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a: normal vector

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a: normal vector

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Algorithm 3: Reduction Method

Fix ε > 0.

Q ⊆ Rd is called an (1 − ε)-convex Pareto set⇐⇒ (1 − ε)f(X) is dominated by conv(Q) (X ∈ C).

Theorem [Diakonikolas ’11]For any ε > 0, we can find (1 − ε)-convex Pareto set Q inpolynomial time, by polymany calls to weighted singleobjective optimization oracle. |Q | = O

( (dε

)d−1 log∆),

where ∆ is the maximum possible objective value.

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What if we only have α-approx?

Theorem [S.–Yoshida, ’17]For any ε > 0, we can find (1 − ε)α-convex Pareto set Q inpolynomial time, by polymany calls to weighted singleobjective α-approx oracle. |Q | = O

( (dε

)d−1 log∆), where

∆ is the maximum possible objective value.

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Reduction Method1: Find (1 − ε)α-convex Pareto set Q.2: Find regret minimizing set P for point set Q, by e.g.

[Agarwal et al. ’17].3: return corresponding set S of feasible X ∈ C.

Theorem [S.–Yoshida, ’17]This alg finds S with regret ratio 1 − α + O(ε) + O(1/k2).

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Lower Bound

f1(X) = cosπ |X |2n

, f2(X) = sinπ |X |2n

> π2k

distance = O(1/k2)

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数値実験

アルゴリズム• Coordinate

• Polytope

• Random: k 個のランダムベクトル a1, ... , ak を生成，{X1, ... , Xk} (Xi : max

X∈Cfai (X)の近似解)を出力

Machine• Intel Xeon E5-2690 (2.90 GHz) CPU, 256 GB RAM• implemented in C#

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Data Summarization

Dataset: MovieLensE: 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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Result

0 5 10 15 20k

10−3

10−2

10−1

PolytopeRandomCoordinate

regret ratiodecreases dramatically

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Result

0 5 10 15 20k

10−3

10−2

10−1

PolytopeRandomCoordinate

regret ratiodecreases dramatically

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regret ratio minimization in multi-objective submodular ... · 自己紹介...

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