ブランディング広告のための最適⼊札戦略

Takanori Maehara1,2, Atsuhiro Narita3, Jun Baba3, Takayuki Kawabata31) Shizuoka University, 2) Riken Center for Advanced Intelligence Project3) Cyberagent, Inc.

February 21, 2017 @ Kiban (S) Seminar, Hokkaido University

Introduction

Real-Time Bidding [8]

• When a user opens a website, the user’s view of a website(“ impression”) is instantly sold in an ad-auction.

• Each advertiser bids for the impression to display hisadvertisement.

Advantages: Cost Efficiency and Personalizability... It can optimize the advertisement for each impression

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Branding Advertising

• Objective: increase the awareness(cf: direct response advertising to sell specific products)

• Branding advertising increases future rewards [4]

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Branding Advertising in Real-Time Bidding

No existing study for branding advertising in RTB because

• (Bad) The advertising effect of branding advertising isdifficult to observe

• Questionnaire survey• Access/search log analysis

and is difficult to quantify. Thus it cannot be directly usedin RTB algorithms.

However,

• (Good) Branding advertising is highly compatible with RTB• If a user is already exposed to an advertisement with ahigh frequency, we should limit the level of exposurebecause he may already remember the advertisement.

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Bid Optimization Problem for Branding Advertising

Our GoalMaximize the advertising effect of branding advertising:

maximize f(X)subject to c(X) ≤ B

where f(X) is the advertising effect when we bid theimpressions X, and c(X) ≤ B is the budget constraint.The problem must be solved in online.

Main IssueThe advertising effect f(X) cannot be observed. Therefore wehave to use a reasonable and tractable mathematical model.

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Problem Formulation

How to define f(X) ?

f(X) should be the number of individuals who recall theadvertisement, summed over time:

f(X) =∫ ∞

0

∑u:user

g(Xu; t)dt

where Xu is the impressions related with user u and g(X; t) isthe probability of recalling the advertisement at time t whena user is exposed the advertisement at the impressions X(we ignore the individual differences)

We are required to model a human memory retention!

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Memory Retention: Ebbinghaus Forgetting Curve

Ebbinghaus forgetting curve[5]

• A forgetting curve decreasesrapidly

• The decline of forgettingcurve slows down due torepetition

These are well-established in cognitive psychology [2]

However, there is no theoretical model that is unanimouslyagreed upon by several researchers [1]

Moreover, existing models in cognitive psychology is not suitedfor mathematical analysis and algorithms

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Our Model of Forgetting Curve

Let p(t) be the forgetting curve when a user is exposed theadvertisement. We use the Wickelgren’s power lawp(t) = λ(1+ αt)β . Then our forgetting curve is given by

g(X; t) = 1−∏j∈X

(1− p(t− tj))

Psychological background: Study-Phase Retrieval TheoryPrevious presentations are retrieved at a later time.

Our definition means the probability of recalling theadvertisement at time t is the probability of recalling at leastone presentation shown for impression j ∈ X

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Psychological Justification

X = {1, 2, 3}, t1 = 0, t1 = 1/5, t2 = 1

0 0.2 0.4 0.6 0.8 1 1.2 1.40.4

0.6

0.8

1

t

g(X;t)

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Algorithmic Property

TheoremOur objective function f(X) is monotone submodular.- Monotone: f(X) ≤ f(Y) for X ⊆ Y- Submodular: f(i|X) ≥ f(i|Y) for X ⊆ Y; f(i|X) = f(X+ i)− f(X)

Proof.g(X; t) is the objective function of the budget allocationproblem. It is known to be monotone submodular for each t.

The sum of monotone submodular functions is alsomonotone submodular.

This shows our problem is an online knapsack constrainedsubmodular maximization problem!

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Algorithm

Problem Formulation

We construct an online algorithm for the following problemwith monotone submodular f

maximize f(X)subject to c(X) ≤ B

We assume that ci/B ≤ ϵ.

Offline Algorithm: GreedySelect item i that has the largest marginal efficiency f(i|X)/ciand repeat the procedure.

Basic Idea for Online AlgorithmSelect item i if marginal efficiency f(i|X)/ci is greater thansome threshold Ψ

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Proposed Algorithm (Submodular version of [9])

Select i-th item if

f(i|X)ci

≥ Ψ(c(X)B ).

The threshold function is defined by

Ψ(z) = Le

(UeL

)z/(1−ϵ)

=Le exp ((1+ log(U/L))z/(1− ϵ))

where U = max f(i|X)/ci and L = min f(i|X)/ci

TheoremThis has competitive ratio of (2+ O(ϵ))(1+ log(U/L))

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Proof 1/2

Let X be the obtained solution and X∗ be the optimal solution.

f(X∗) ≤ f(X ∪ X∗) =∑i∈X∪X∗

f(i|(X ∪ X∗)<i) ≤∑i∈X∗∪X

f(i|X<i)

=∑i∈X

f(i|X<i) +∑i∈X∗\X

f(i|X<i),

f(X) =∑i∈X

f(i|X<i).

Thereforef(X∗)f(X) ≤

∑i∈X f(i|X<i) +

∑i∈X∗\X f(i|X<i)∑

i∈X f(i|X<i)

≤∑

i∈X ciΨ(zi) +∑

i∈X∗\X f(i|X<i)∑i∈X ciΨ(zi)

≤∑

i∈X ciΨ(zi) +∑

i∈X∗\X ciΨ(zi)∑i∈X ciΨ(zi)

=

∑i∈X∗∪X ciΨ(zi)∑i∈X ciΨ(zi)

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Proof 2/2

f(X∗)f(X) ≤

∑i∈X∗∪X ciΨ(zi)∑i∈X ciΨ(zi)

≤∑

i∈X∗∪X ciΨ(Z)∑i∈X ciΨ(zi)

≤ 2BΨ(Z)∑i∈X ciΨ(zi)

The summation is approximated by the integral∑i∈X

ciBΨ(zi) =

∑i∈X

∆ziΨ(zi) ≥∫ Z−ϵ

0Ψ(z)dz = (1− O(ϵ))

∫ Z

0Ψ(z)dz

Since Ψ(z) ≤ L if z ≤ c = (1− ϵ)/(1+ log(U/L)), we can redefineΨ(z) = L for z ≤ c.∫ Z

0Ψ(z)dz = cL+

∫ Z

cΨ(z)dz = cL+ cΨ(Z)− cL = (1− ϵ)ψ(Z)

1+ log(U/L)Thus

f(X∗)f(X) ≤ (2+ O(ϵ))(1+ log(U/L)).

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Comment

• This is the first algorithm for this setting• If the order of items is random, it is a submodularsecretary problem and is O(1) competitive [3, 6]

• If we can maintain multiple solutions, it is a streamingknapsack constrained submoular maximization problem,which is O(1) competitive [7]

... both are impossible• If f is modular, the same algorithm gives a solution with(1+ O(ϵ))(1+ log(U/l)) competitive [9]⇒ submodularity makes the competitive ratio twice worse(Open: Is this tight?)

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Experiments

Dataset

We collected datasets of a branding advertising campaignfrom a real-world running RTB system

• The campaign was conducted during November, 2016• After the campaign, a questionnaire survey wasconducted on 2000 individuals. This asked “Do youremember this advertisement?”.

• We used the impressions in 12th–13th, November, 2016 fora bid optimization experiment, which contains 2,895,643impressions for 1,216,702 users

• Maximum impressions for a user is 171 (heavy tailed).

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Forgetting Curve in real dataset

λ = 1.0, α = 1.0, β = 1.4

0 5 10 15 20 25 30

0.15

0.2

0.25

0.3

time t [day]

awareness

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Submodularity is observed in real dataset

0 2 4 6 8 10 12 140

5 · 10−2

0.1

0.15

0.2

#impressions

awareness

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Bid Optimization Result 1/2

Algorithm Objective Unspent MaxImpProposed 806982.86 4.656% 11Random 707455.82 0.011% 20AllBid 662126.52 0.000% 23Offline 1715348.28 0.000% 19

The proposed algorithm outperformed basic algorithm, but isfar from the offline algorithm

The proposed algorithm left a lot of unspent budget ...Reason: the proof of the correctness relies the property thatthe algorithm never rejects the impressions by the shortage ofbudget.

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Bid Optimization Result 2/2

0 1 2 3 4 50

1

2

3·105

time [day]

g(X,t)

ProposedRandomAllBidOffline

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Conclusion / Future Work

Conclusion / Future Work

• The bid optimization problem for branding advertising isformulated as an online knapsack constrainedsubmodular maximization problem

• Here, a new submodular objective function is introduced.It is psychologically reasonable and computationallytractable.

• An online algorithm with a provable competitive ratio isproposed.

• Some applications of the proposed function?• Better competitive ratio? / Matching lower bound?• Practical algorithm?

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Bibliograpy

Lee Averell and Andrew Heathcote.The form of the forgetting curve and the fate ofmemories.Journal of Mathematical Psychology, 55(1):25–35, 2011.

Alan D Baddeley.Essentials of human memory.Psychology Press, 1999.

MohammadHossein Bateni, MohammadTaghi Hajiaghayi,and Morteza Zadimoghaddam.Submodular secretary problem and extensions.In Approximation, Randomization, and CombinatorialOptimization. Algorithms and Techniques, pages 39–52.Springer, 2010.

Rong Chen and Feng He.

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Examination of brand knowledge, perceived risk andconsumers’ intention to adopt an online retailer.Total Quality Management & Business Excellence,14(6):677–693, 2003.Hermann Ebbinghaus.Memory: A contribution to experimental psychology.Number 3. University Microfilms, 1913 (Original workpublished 1885).

Thomas Kesselheim and Andreas Tönnis.Submodular secretary problems: Cardinality, matching,and linear constraints.arXiv preprint arXiv:1607.08805, 2016.

Ravi Kumar, Benjamin Moseley, Sergei Vassilvitskii, andAndrea Vattani.

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Fast greedy algorithms in mapreduce and streaming.ACM Transactions on Parallel Computing, 2(3):14, 2015.

Eric K Lee and Charles U Martel.When to use splay trees.Software-Practice and Experience, 37(15):1559–1576, 2007.

Yunhong Zhou, Deeparnab Chakrabarty, and Rajan Lukose.Budget constrained bidding in keyword auctions andonline knapsack problems.In Internet and Network Economics, pages 566–576.Springer, 2008.

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