Adwords, An Algorithmic Perspective

Transcription

1 April 20, 2009

2 What is the Adwords problem? Online Algorithm

3 What is the Adwords problem? Online Algorithm What are Adwords? Search engine displays search results.

4 What is the Adwords problem? Online Algorithm What are Adwords? Search engine displays search results. For each query, relevant ads are also returned.

5 What is the Adwords problem? Online Algorithm What are Adwords? Search engine displays search results. For each query, relevant ads are also returned. The search results and ads are displayed separately.

6 What is the Adwords problem? Online Algorithm Some searches reveal many sponsored ads...

7 What is the Adwords problem? Online Algorithm lmnmoppq rsttuvuwxyz{ xyu}~ssvtuu yr DE1DEF"F"G3HF#2 I J1HK,LK College and Dorm room Furniture and Rug Boston Massachusetts )+ ;c, !"#$! 7!"#$! d +!"#$ %J122 3! %!"#$! % e\ff\5 \ \ College Furniture & Dorm Room Furniture - Free Shipping !"#$! ( & Z 4!"#$! C!"#$! % * +, J122 3 III %,!"#$! % ej122 3 \!"#$! % \=g\ 5 \ \ Dorm Essentials : College : Home : Target & A 0122 B B C %III !"#$! !"#$! & A % III % % e4!"#$! \& \A \ \@ eh i/c 6 j( i;gcf<kcff\fjj\5 \ \ Dorm Furniture - College Bunk Beds and Computer Desks & Z J122 3!"#$! ' % %& +, % % % e\=<\5 \ \ MNOPQRSNTSPURV WUSXSUSQRSY Z [\[] ^_`]]]`]]]J122 3!"#$! %a ] I _[ b School Furniture for Less!"#$! %& ' ( % % ) % * +,,- Student Travel Deals %. + % % /+ % Dorm & + 4!"#$ Furniture! '5 '6 ' (7 8 %&, % Buy 9 :;<= Dorm Room Furniture % Dorm Furniture 6 *4!"#$! 7 '> & +,8 %? 5 % , Furniture & A B,7 C 8 JCPenney: Beanbags, Shoshana Dorm Chairs, Neuburger Dorm Room

8 What is the Adwords problem? Online Algorithm And others result in none...

9 What is the Adwords problem? Online Algorithm åæçèæçèèé êëìêëíîíîïðñðòóïòôõììóöï ïíòøë ƒ ˆ Š ŒŠ ŽŠ ˆ Šš Šš ˆ š œ žÿ œ ž Ÿ ˆ ƒ Š ªŠ««ª ««± ²³ ²³µ µ ƒ ¹ƒ º» ¼ ½½Š Š¾ FACULTY: Department of CIS: Brooklyn College CUNY ³ ²³µ µàƒ ¹ƒ Á²³ ²³µ µâš º Ã Ä º Ä º ºÅ Šª ª Š Ã Œ Š Á Šª ŠÂŠ º Ã Ä º Ä º ºÆ ÇÈˆ Á Èˆ ÂŠ º Ã Ä º Ä º ¼¼¼ ŽŽŽ º Š º Ã Ä º Ä º É ŠÉ É ªÄ º šª Ê ËÃ Ì š Œ ˆ Š Discrete Algorithms Seminar ªš Œ ªª ªš ª Íªªš Š º ª ªÎšªªŒÎÉÉŒ ª º º ˆÉ ª ª º Ï ÐÑ Ê ÒÓÔÕÖ º Ñ Ê ÒÓÖÔÑ º Œ Ã Î ³ ²³µ µàƒ ¹ƒ º¼¼¼ ŽŽŽ º Š º Ã Ä º Ä º É ªÈÉÕÔÖÉ ÔÕ Šª ªŠ º Œ Œ ˆ Š Š ªŠ ŽŽŽ º Š º Ã Ä º Ä º Ø Advanced Algorithms : Topics in Game Theory Ù ÑÚ Á š Û ˆš Á Ã ª Ì ˆ Æ ŒŒ ª Š Á ³ ²³µ µàƒ ¹ƒ ºÒ º Ù Ö Ê ÁÆ Á ª ª ˆÎ ª Œ ªšŠŒ ŠŒ ª Žš Š ¼¼¼ Œ º ªˆ Š º É ˆ ŠšÉ ŠÜÑÒ º šª ÑÚÃ Ì š Œ ˆ Š Stringology Bar Ilan University ¼¼¼ Š Ž Ý ÙÝ Á Ý ¾ÞÝ ª š Áß à Š ªá Þâ ˆÌ š Á ã ˆ ß à Š ªÄÞ ³ ²³µ µàƒ ¹ƒ ÁÌ ª Ä ß à Š ªÄ Žä Ã ¼¼¼ º Š º º º É ÄÉŠª ˆ ˆÄÉŒ ª Œ ªŠ º ŒšŒ ÕÃ Ì š Œ ˆ Š

10 Advertiser Overview What is the Adwords problem? Online Algorithm An advertiser is charged for his ads.

11 What is the Adwords problem? Online Algorithm Advertiser An advertiser is charged for his ads. Each advertiser can value each keyword differently.

12 What is the Adwords problem? Online Algorithm Advertiser An advertiser is charged for his ads. Each advertiser can value each keyword differently. An advertiser bids for a keyword that should display his ad.

13 What is the Adwords problem? Online Algorithm Advertiser An advertiser is charged for his ads. Each advertiser can value each keyword differently. An advertiser bids for a keyword that should display his ad. Some keywords are more popular among advertisers.

14 Search engine s perspective: What is the Adwords problem? Online Algorithm 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.

15 Search engine s perspective: What is the Adwords problem? Online Algorithm 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?

16 Online Algorithm Overview What is the Adwords problem? Online Algorithm The Adwords problem is an online allocation problem.

17 Online Algorithm Overview What is the Adwords problem? Online Algorithm The Adwords problem is an online allocation problem. The effectiveness of an online algorithm is measured by competitive analysis.

18 Online Algorithm Overview What is the Adwords problem? Online Algorithm The Adwords problem is an online allocation problem. The effectiveness of an online algorithm is measured by competitive analysis. ALG is α-competitive if the ratio between its performance and the optimal offline performance is bounded by α. ALG(I) OPT(I) α for all instances I.

19 Greedy Criteria Overview 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.

20 Overview Example Bidder 1 Bidder 2 Bicycle $1 $0.99 Flowers $1 $0 Budget $100 $100

21 Overview Example Bidder 1 Bidder 2 Bicycle $1 $0.99 Flowers $1 $0 Budget $100 $100 Queries: 100 Bicycles then 100 Flowers.

22 Overview Example Bidder 1 Bidder 2 Bicycle $1 $0.99 Flowers $1 $0 Budget $100 $100 Queries: 100 Bicycles then 100 Flowers. Greedy allocates 100 Bicycles to Bidder 1.

23 Overview Example Bidder 1 Bidder 2 Bicycle $1 $0.99 Flowers $1 $0 Budget $100 $100 Queries: 100 Bicycles then 100 Flowers. OPT (offline) allocates 100 Bicycles to Bidder 2 and 100 Flowers to Bidder 1.

24 Overview Example Bidder 1 Bidder 2 Bicycle $1 $0.99 Flowers $1 $0 Budget $100 $100 Queries: 100 Bicycles then 100 Flowers. Algorithm Allocation Revenue Greedy Bidder 1 : 100 Bicycles $100 OPT Bidder 1 : 100 Flowers Bidder 2 : 100 Bicycles $199

25 Competitive Analysis Overview ALG is α-competitive if the ratio between its performance and the optimal offline performance is bounded by α. ALG(I) OPT(I) α for all instances I. Algorithm α Greedy 1 2 Mehta, Saberi, Vazirani, Vazirani ( 05) 1 1 e.63 optimal

26 Perspective Related Work MSVV More Realistic Models AdWords and Generalized On-line Matching Aranyak Mehta, Amin Saberi, Umesh Vazirani, Vijay Vazirani FOCS, 2005

27 Perspective Related Work MSVV More Realistic 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

28 Perspective Related Work MSVV More Realistic 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 s budget against the amount of its bid.

29 Perspective Related Work MSVV More Realistic Models Previous Results Karp, Vazirani, Vazirani (1990) Online bipartite matching Randomized algorithm ranking Fixes random permutation of bidders in advance. Budgets = 1, Bids = 0/1 Factor: 1 1 e

30 Perspective Related Work MSVV More Realistic Models Previous Results Karp, Vazirani, Vazirani (1990) Online bipartite matching Randomized algorithm ranking Fixes random permutation of bidders in advance. Budgets = 1, Bids = 0/1 Factor: 1 1 e Kalyanasundaram, Pruhs (2000) Online b-matching Deterministic algorithm balance Matches query to bidder with highest remaining budget. Budgets = 1, Bids = 0/ǫ Factor: 1 1 e

31 Perspective Related Work MSVV More Realistic 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

32 Perspective Related Work MSVV More Realistic 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 1 e.

33 Perspective Related Work MSVV More Realistic 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 1 e. We would like to do better!

34 Perspective Related Work MSVV More Realistic Models Adwords problem: N bidders Each bidder i has daily budget b i Each bidder i specifies a bid c iq for query word q Q A sequence of query words q 1 q 2 q M, q j Q, arrive online during the day. Each query q j must be assigned to some bidder i as it arrives; the revenue is c iqj Objective: maximize total daily revenue while respecting daily budget of bidders.

35 Perspective Related Work MSVV More Realistic 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.

36 Perspective Related Work MSVV More Realistic 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)

37 Perspective Related Work MSVV More Realistic Models Where does ψ come from? 1-1/e New Proof for BALANCE {0,ε} Factor Revealing LP Modify LP for arbitrary bids [0,ε] 1-1/e Use dual to get Tradeoff function Tradeoff Revealing LP

38 Perspective Related Work MSVV More Realistic 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 end of balance. Define x 1, x 2,...,x k : x i = number of bidders of type i.

39 Perspective Related Work MSVV More Realistic Models Factor-Revealing LP w.l.o.g. OPT = $N. k i Revenue = x i k i=1

40 Perspective Related Work MSVV More Realistic Models Factor-Revealing LP Constraint 1: x 1 N k Constraint 2: x 1 + x 2 2 N k x 1 k i i, 1 i k 1: (1 + i j k )x j i k N j=1 w.l.o.g. OPT = $N. k i Revenue = x i k i=1

41 Perspective Related Work MSVV More Realistic Models Factor-Revealing LP Minimize such that k i x i k i=1 i (1 + i j k )x j i k N j=1 x j = N j

42 Perspective Related Work MSVV More Realistic Models Factor-Revealing LP Minimize such that k i x i k i=1 i (1 + i j k )x j i k N j=1 x j = N j Solve LP by finding the opimum primal and dual. Optimal solution is x i = N k (1 1 k )i 1 which tends to N(1 1 e ) as k. Thus, balance achieves a factor of 1 1 e.

43 Perspective Related Work MSVV More Realistic Models Where does ψ come from? 1-1/e New Proof for BALANCE {0,ε} Factor Revealing LP Modify LP for arbitrary bids [0,ε] 1-1/e Use dual to get Tradeoff function Tradeoff Revealing LP

44 Perspective Related Work MSVV More Realistic 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 bids using LPs L(π, ψ). L: D: Max c x Ax b L(π, ψ) D(π, ψ) Max c x Ax b + (π, ψ) Min b y y T A c Min b y + (π, ψ) y y T A c

45 Perspective Related Work MSVV More Realistic Models Modify the LP for arbitrary bids L(π, ψ) D(π, ψ) Max c x Ax b + (π, ψ) Min b y + (π, ψ) y y T A c Observation: For every ψ, dual achieves optimal value at same vertex.

46 Perspective Related Work MSVV More Realistic Models Modify the LP for arbitrary bids L(π, ψ) D(π, ψ) Max c x Ax b + (π, ψ) Min b y + (π, ψ) y y T A c Observation: For every ψ, dual achieves optimal value at same vertex. There is a way to choose ψ so that the objective function does not decrease. Thus, the competitive factor remains 1 1 e.

47 Perspective Related Work MSVV More Realistic Models Modify the LP for arbitrary bids L(π, ψ) D(π, ψ) Max c x Ax b + (π, ψ) Min b y + (π, ψ) y y T A c Observation: For every ψ, dual achieves optimal value at same vertex. There is a way to choose ψ so that the objective function does not decrease. Thus, the competitive factor remains 1 1 e. This competitive factor is optimal.

48 Perspective Related Work MSVV More Realistic 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

49 Goel and Mehta Online budgeted matching in random input models with applications to Adwords Gagan Goel and Aranyak Mehta SODA, 2008

50 Outline of article Overview Goel and Mehta Distributional assumption about query sequence: although the set of queries is arbitrary, the order of queries is random. Main Result: Greedy has competitive ratio 1 1 e in the random permutation input model This result applies to the i.i.d. model as well. Approach: modify KVV (fix hole in proof) and then apply results to Adwords problem

51 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

52 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 the Adwords problem. Not every mismatch can be reversed easily; a tagging procedure is used to generalize the results. The tagging method works on permutations of the input.

53 GM Algorithm Overview Goel and Mehta m Revenue = min B i, bid iq i=1 q Q In the last bid the algorithm allocates to a bidder, his budget may be exceeded. Inequalities bound the sizes of miss, bad, good. A linear program maximizes the loss of revenue over these inequalities. Factor revealing LP proves competitive ratio of 1 1 e input model. in random

54 Goel and Mehta GM Results Competitive factor of Greedy in random-permutation input model independent distribution input model (i.i.d.) is exactly 1 1 e.

55 Goel and Mehta Thank you!