Adwords, An Algorithmic Perspective - Brooklyn Collegeshoshana/pub/CS514_adwords_slides.pdf ·...

OverviewGreedy AlgorithmMSVV Algorithm

Random Input Models

Adwords, An Algorithmic Perspective

Shoshana Neuburger

April 20, 2009

Shoshana Neuburger Adwords, An Algorithmic Perspective


Random Input Models

What is the Adwords problem?Online Algorithm



Random Input Models


What are Adwords?

◮ Search engine displays search results.



Random Input Models


What are Adwords?


◮ For each query, relevant ads are also returned.



Random Input Models


What are Adwords?


◮ For each query, relevant ads are also returned.

◮ The search results and ads are displayed separately.



Random Input Models


Some searches reveal many sponsored ads...



Random Input Models

What is the Adwords problem?Online Algorithm��

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Random Input Models


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Random Input Models

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Random Input Models


Advertiser

◮ An advertiser is charged for his ads.



Random Input Models


Advertiser


◮ Each advertiser can value each keyword differently.



Random Input Models


Advertiser



◮ An advertiser bids for a keyword that should display his ad.



Random Input Models


Advertiser



◮ An advertiser bids for a keyword that should display his ad.

◮ Some keywords are more popular among advertisers.



Random Input Models


Search engine’s perspective:

◮ Maximize revenue each day.

◮ Limitations:◮ must respect each advertiser’s daily budget

and◮ the profit depends on an advertiser’s bid for a keyword he wins.

The bids and daily budgets are specified in advance.



Random Input Models


Search engine’s perspective:

◮ Maximize revenue each day.

◮ Limitations:◮ must respect each advertiser’s daily budget

and◮ the profit depends on an advertiser’s bid for a keyword he wins.

The bids and daily budgets are specified in advance.

Decision: which ads to display for a query?



Random Input Models


Online Algorithm

The Adwords problem is an online allocation problem.



Random Input Models


Online Algorithm


The effectiveness of an online algorithm is measured bycompetitive analysis.



Random Input Models


Online Algorithm


The effectiveness of an online algorithm is measured bycompetitive analysis.

ALG is α-competitive if the ratio between its performance and theoptimal offline performance is bounded by α.

ALG(I )OPT (I ) ≥ α for all instances I.



Random Input Models

Greedy Algorithm

Greedy Criteria

Maximize the profit for each query.

As a keyword arrives, choose the ad that offers the highest bid.

Until the advertiser’s budget is depleted.



Random Input Models

Greedy Algorithm

Greedy Algorithm

ExampleBidder1 Bidder2

Bicycle $1 $0.99Flowers $1 $0

Budget $100 $100



Random Input Models

Greedy Algorithm

Greedy Algorithm



Budget $100 $100

Queries:

100 Bicyclesthen100 Flowers.



Random Input Models

Greedy Algorithm

Greedy Algorithm



Budget $100 $100

Queries:


Greedy allocates 100 Bicycles to Bidder1.



Random Input Models

Greedy Algorithm

Greedy Algorithm



Budget $100 $100

Queries:


OPT (offline) allocates 100 Bicycles to Bidder2 and 100 Flowers toBidder1.



Random Input Models

Greedy Algorithm

Greedy Algorithm



Budget $100 $100

Queries:


Algorithm Allocation Revenue

Greedy Bidder1: 100 Bicycles $100

OPT Bidder1: 100 FlowersBidder2: 100 Bicycles $199



Random Input Models

Greedy Algorithm

Competitive Analysis

ALG is α-competitive if the ratio between its performance and theoptimal offline performance is bounded by α.

ALG(I )OPT (I ) ≥ α for all instances I.

Algorithm α

Greedy 12

Mehta, Saberi, Vazirani, Vazirani (’05) 1 − 1e≈ .63 optimal



Random Input Models

PerspectiveRelated WorkMSVVMore Realistic Models

AdWords and Generalized On-line Matching

Aranyak Mehta, Amin Saberi, Umesh Vazirani, Vijay VaziraniFOCS, 2005



Random Input Models


In search of a better algorithm

As a query arrives, the algorithm

◮ Should favor advertisers with high bids

◮ Should not exhaust the budget of any advertiser too quickly



Random Input Models


In search of a better algorithm

As a query arrives, the algorithm

◮ Should favor advertisers with high bids

◮ Should not exhaust the budget of any advertiser too quickly

The algorithm weighs the remaining fraction of each advertiser’sbudget against the amount of its bid.



Random Input Models


Previous Results

Karp, Vazirani, Vazirani (1990)

◮ Online bipartite matching

◮ Randomized algorithmranking

◮ Fixes random permutation ofbidders in advance.

◮ Budgets = 1, Bids = 0/1

◮ Factor: 1 − 1e



Random Input Models


Previous Results

Karp, Vazirani, Vazirani (1990)

◮ Online bipartite matching

◮ Randomized algorithmranking

◮ Fixes random permutation ofbidders in advance.

◮ Budgets = 1, Bids = 0/1


Kalyanasundaram, Pruhs (2000)

◮ Online b-matching

◮ Deterministic algorithmbalance

◮ Matches query to bidder withhighest remaining budget.

◮ Budgets = 1, Bids = 0/ǫ




Random Input Models


KVV ’90

“We saw online matching as a beautiful research

problem with purely theoretical appeal. At the time, we

had no idea it would turn out to have practical value.”

- Umesh Vazirani, SIAM News, April 2005



Random Input Models


Applying balance

The new algorithm generalizes b-matching to arbitrary bids.

Natural Algorithm:

◮ Assign query to highest bidder

◮ Break ties with largest remaining budget

Achieves competitive ratio < 1 − 1e.



Random Input Models


Applying balance

The new algorithm generalizes b-matching to arbitrary bids.

Natural Algorithm:

◮ Assign query to highest bidder

◮ Break ties with largest remaining budget

Achieves competitive ratio < 1 − 1e.

We would like to do better!



Random Input Models


Adwords problem:

◮ N bidders

◮ Each bidder i has daily budget bi

◮ Each bidder i specifies a bid ciq for query word q ∈ Q

◮ A sequence of query words q1q2 · · · qM , qj ∈ Q, arrive onlineduring the day.

◮ Each query qj must be assigned to some bidder i as it arrives;the revenue is ciqj

Objective: maximize total daily revenue while respecting dailybudget of bidders.



Random Input Models


Simplified version of Adwords problem:

◮ Bidder pays as soon as ad is displayed

◮ Bidder pays his own bid

◮ One ad displayed per search page

Assumption: bids are small compared to budgets.



Random Input Models


New Algorithm

Algorithm: Give query to bidder that maximizes

bid × ψ(fraction of budget spent)

ψ is tradeoff function between bid and unspent budget.

ψ(x) = 1 − e−(1−x)



Random Input Models


Where does ψ come from?

New Proof forBALANCE {0,ε} Factor Revealing LP

Modify LP for arbitrary bids [0,ε]

Use dual to get

1-1/e

1-1/eUse dual to get Tradeoff function

Tradeoff Revealing LP1-1/e



Random Input Models


Factor-Revealing LP

Choose large k .

Discretize the budget of each bidder into k equal slabs.

Define the type of a bidder by the fraction of budget spent at endof balance.

Define x1, x2, . . . , xk :xi = number of bidders of type i .



Random Input Models


Factor-Revealing LP

w.l.o.g. OPT = $N.

Revenue =k

∑

i=1

xii

k



Random Input Models


Factor-Revealing LP

w.l.o.g. OPT = $N.

Revenue =k

∑

i=1

xii

k

Constraint 1: x1 ≤ Nk

Constraint 2: x1 + x2 ≤ 2Nk−

x1k

∀i , 1 ≤ i ≤ k − 1:i

∑

j=1

(1 +i − j

k)xj ≤

i

kN



Random Input Models


Factor-Revealing LP

Minimizek

∑

i=1

xii

k

such thati

∑

j=1

(1 +i − j

k)xj ≤

i

kN

∑

j

xj = N



Random Input Models


Factor-Revealing LP

Minimizek

∑

i=1

xii

k

such thati

∑

j=1

(1 +i − j

k)xj ≤

i

kN

∑

j

xj = N

Solve LP by finding the opimum primal and dual.Optimal solution is xi = N

k(1 − 1

k)i−1

which tends to N(1 − 1e) as k → ∞.

Thus, balance achieves a factor of 1 − 1e.



Random Input Models


Where does ψ come from?

New Proof forBALANCE {0,ε} Factor Revealing LP

Modify LP for arbitrary bids [0,ε]

Use dual to get

1-1/e

1-1/eUse dual to get Tradeoff function

Tradeoff Revealing LP1-1/e



Random Input Models


Modify the LP for arbitrary bids

Subtle tradeoff between bid and unspent budget

We generalize LP L and its dual D to the case with arbitrary bidsusing LPs L(π, ψ).

L: D:Max c · x Min b · y

Ax ≤ b yTA ≥ c

L(π, ψ) D(π, ψ)Max c · x Min b · y + ∆(π, ψ) · y

Ax ≤ b + ∆(π, ψ) yTA ≥ c



Random Input Models




Ax ≤ b + ∆(π, ψ) yTA ≥ c

Observation: For every ψ, dual achieves optimal value at samevertex.



Random Input Models




Ax ≤ b + ∆(π, ψ) yTA ≥ c


There is a way to choose ψ so that the objective function does notdecrease.Thus, the competitive factor remains 1 − 1

e.



Random Input Models




Ax ≤ b + ∆(π, ψ) yTA ≥ c


There is a way to choose ψ so that the objective function does notdecrease.Thus, the competitive factor remains 1 − 1

e.

This competitive factor is optimal.



Random Input Models


More Realistic Models

The algorithm introduced by MSVV generalizes to

◮ Advertisers with different daily budgets.

◮ The optimal allocation does not exhaust all budgets.

◮ Several ads per search query.

◮ Cost-per-Click

◮ Second Price



Random Input Models

Goel and Mehta

Online budgeted matching in random input models

with applications to Adwords

Gagan Goel and Aranyak MehtaSODA, 2008



Random Input Models

Goel and Mehta

Outline of article

◮ Distributional assumption about query sequence: although theset of queries is arbitrary, the order of queries is random.

◮ Main Result: Greedy has competitive ratio 1 − 1e

in therandom permutation input model

◮ This result applies to the i.i.d. model as well.

◮ Approach: modify KVV (fix hole in proof) and then applyresults to Adwords problem



Random Input Models

Goel and Mehta

Permutation Classes

For each item p, classify permutations

◮ Into those in which p remains unmatched, miss.

◮ Into those in which p gets matched.Subclasses depending on the structure of the match:

◮ good : the correct match is available when p arrives◮ bad : the correct match is allocated prior to the arrival of p



Random Input Models

Goel and Mehta

GM Algorithm

◮ Properties of Greedy:◮ Monotonicity◮ Prefix◮ Partition

◮ These properties are simple observations in bipartite matching.

◮ There can be several different bids for the same query in theAdwords problem.

◮ Not every mismatch can be reversed easily; a taggingprocedure is used to generalize the results.

◮ The tagging method works on permutations of the input.



Random Input Models

Goel and Mehta

GM Algorithm

Revenue =m

∑

i=1

min

Bi ,∑

q∈Q

bidiq

In the last bid the algorithm allocates to a bidder, his budget maybe exceeded.

Inequalities bound the sizes of miss, bad, good.

A linear program maximizes the loss of revenue over theseinequalities.

Factor revealing LP proves competitive ratio of 1 − 1e

in randominput model.



Random Input Models

Goel and Mehta

GM Results

Competitive factor of Greedy in

◮ random-permutation input model

◮ independent distribution input model (i.i.d.)

is exactly 1 − 1e.



Random Input Models

Goel and Mehta

Thank you!


Adwords, An Algorithmic Perspective - Brooklyn Collegeshoshana/pub/CS514_adwords_slides.pdf ·...

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