Case study I: Advertising on the Web

Transcription

1 1 Case study I: Advertisig o the Web Geoveva Vargas-Solar Frech Coucil of Scietific Research, LIG & LAFMIA Labs Motevideo, 22 d November 4 th December, 2015 INFORMATIQUE

2 + Ifiite data / App 2 High dim. data Graph data Ifiite data Machie learig Apps Locality sesitive hashig PageRak, SimRak Filterig data streams SVM Recommeder systems Clusterig Commuity Detectio Web advertisig Decisio Trees Associatio Rules Dimesioality reductio Spam Detectio Queries o streams Perceptro, knn Duplicate documet detectio

3 + Olie Algorithms 3 Classic model of algorithms You get to see the etire iput, the compute some fuctio of it I this cotext, offlie algorithm Olie Algorithms You get to see the iput oe piece at a time, ad eed to make irrevocable decisios alog the way Similar to the data stream model

4 Olie Bipartite Matchig

5 + Example: Bipartite Matchig 5 1 a 2 b 3 c Boys 4 d Girls Nodes: Boys ad Girls; Edges: Prefereces Goal: Match boys to girls so that maximum umber of prefereces is satisfied

6 + Example: Bipartite Matchig 6 1 a 2 b 3 c Boys 4 d Girls M = {(1,a),(2,b),(3,d)} is a matchig Cardiality of matchig = M = 3

7 + Example: Bipartite Matchig 7 1 a 2 b 3 c Boys 4 d Girls M = {(1,c),(2,b),(3,d),(4,a)} is a perfect matchig Perfect matchig all vertices of the graph are matched Maximum matchig a matchig that cotais the largest possible umber of matches

8 + Matchig Algorithm 8 Problem: Fid a maximum matchig for a give bipartite graph A perfect oe if it exists There is a polyomial-time offlie algorithm based o augmetig paths (Hopcroft & Karp 1973, see But what if we do ot kow the etire graph upfrot?

9 + Olie Graph Matchig Problem 9 Iitially, we are give the set boys I each roud, oe girl s choices are revealed That is, girl s edges are revealed At that time, we have to decide to either: Pair the girl with a boy Do ot pair the girl with ay boy Example of applicatio: Assigig tasks to servers

10 + Olie Graph Matchig: Example a b c d (1,a) (2,b) (3,d)

11 + Greedy Algorithm 11 Greedy algorithm for the olie graph matchig problem: Pair the ew girl with ay eligible boy If there is oe, do ot pair girl How good is the algorithm?

12 + Competitive Ratio 12 For iput I, suppose greedy produces matchig M greedy while a optimal matchig is M opt Competitive ratio = mi all possible iputs I ( M greedy / M opt ) (what is greedy s worst performace over all possible iputs I)

13 + Aalyzig the Greedy Algorithm 13 Cosider a case: M greedy M opt Cosider the set G of girls matched i M opt but ot i M greedy 1 M opt a The every boy B adjacet to girls i G is already matched i M greedy : If there would exist such o-matched (by M greedy ) boy adjacet to a o-matched girl the greedy would have matched them Sice boys B are already matched i M greedy the (1) M greedy B B={ } b c d G={ }

14 Web advertisemet

15 + History of Web Advertisig 15 Baer ads ( ) Iitial form of web advertisig Popular websites charged X$ for every 1,000 impressios of the ad Called CPM rate (Cost per thousad impressios) Modeled similar to TV, magazie ads From utargeted to demographically targeted Low click-through rates Low ROI for advertisers CPM cost per mille Mille thousad i Lati

16 + Performace-based Advertisig 16 Itroduced by Overture aroud 2000 Advertisers bid o search keywords Whe someoe searches for that keyword, the highest bidder s ad is show Advertiser is charged oly if the ad is clicked o Similar model adapted by Google with some chages aroud 2002 Called Adwords

17 + Ads vs. Search Results 17

18 + Web Performace-based advertisig works! Multi-billio-dollar idustry Iterestig problem: What ads to show for a give query? If I am a advertiser, which search terms should I bid o ad how much should I bid?

19 Adwords Problem 19 Give: 1. A set of bids by advertisers for search queries 2. A click-through rate for each advertiser-query pair 3. A budget for each advertiser (say for 1 moth) 4. A limit o the umber of ads to be displayed with each search query Respod to each search query with a set of advertisers such that: 1. The size of the set is o larger tha the limit o the umber of ads per query 2. Each advertiser has bid o the search query 3. Each advertiser has eough budget left to pay for the ad if it is clicked upo

20 + Adwords Problem 20 A stream of queries arrives at the search egie: q 1, q 2, Several advertisers bid o each query Whe query q i arrives, search egie must pick a subset of advertisers whose ads are show Goal: Maximize search egie s reveues Simple solutio: Istead of raw bids, use the expected reveue per click (i.e., Bid*CTR Click Through Rate) Clearly we eed a olie algorithm!

21 + The Adwords Iovatio 21 Advertiser Bid CTR Bid * CTR A $1.00 1% 1 cet B $0.75 2% 1.5 cets C $ % cets Click through rate Expected reveue

22 + The Adwords Iovatio 22 Advertiser Bid CTR Bid * CTR B $0.75 2% 1.5 cets C $ % cets A $1.00 1% 1 cet

23 + Complicatios: Budget 23 Two complicatios: Budget CTR of a ad is ukow Each advertiser has a limited budget Search egie guaratees that the advertiser will ot be charged more tha their daily budget

24 + Complicatios: CTR 24 CTR: Each ad has a differet likelihood of beig clicked Advertiser 1 bids $2, click probability = 0.1 Advertiser 2 bids $1, click probability = 0.5 Clickthrough rate (CTR) is measured historically Very hard problem: Exploratio vs. exploitatio Exploit: Should we keep showig a ad for which we have good estimates of click-through rate or Explore: Shall we show a brad ew ad to get a better sese of its click-through rate

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26 + Greedy Algorithm 26 Our settig: Simplified eviromet There is 1 ad show for each query All advertisers have the same budget B All ads are equally likely to be clicked Value of each ad is the same (=1) Simplest algorithm is greedy: For a query pick ay advertiser who has bid 1 for that query Competitive ratio of greedy is 1/2

27 + Bad Sceario for Greedy 27 Two advertisers A ad B A bids o query x, B bids o x ad y Both have budgets of $4 Query stream: x x x x y y y y Worst case greedy choice: B B B B Optimal: A A A A B B B B Competitive ratio = ½ This is the worst case! Note: Greedy algorithm is determiistic it always resolves draws i the same way

28 + BALANCE Algorithm [MSVV] 28 BALANCE Algorithm by Mehta, Saberi, Vazirai, ad Vazirai For each query, pick the advertiser with the largest uspet budget Break ties arbitrarily (but i a determiistic way)

29 + Example: BALANCE 29 Two advertisers A ad B A bids o query x, B bids o x ad y Both have budgets of $4 Query stream: x x x x y y y y BALANCE choice: A B A B B B Optimal: A A A A B B B B I geeral: For BALANCE o 2 advertisers Competitive ratio = ¾

30 + Aalyzig BALANCE 30 Cosider simple case (w.l.o.g.): 2 advertisers, A 1 ad A 2, each with budget B ( 1) Optimal solutio exhausts both advertisers budgets BALANCE must exhaust at least oe advertiser s budget: If ot, we ca allocate more queries Wheever BALANCE makes a mistake (both advertisers bid o the query), advertiser s uspet budget oly decreases Sice optimal exhausts both budgets, oe will for sure get exhausted Assume BALANCE exhausts A 2 s budget, but allocates x queries fewer tha the optimal Reveue: BAL = 2B - x

31 + Aalyzig Balace 31 Queries allocated to A 1 i the optimal solutio B Queries allocated to A 2 i the optimal solutio x y x y A 1 A 2 A 1 A 2 A 1 A 2 Not used Not used x x B B Optimal reveue = 2B Assume Balace gives reveue = 2B-x = B+y Uassiged queries should be assiged to A 2 (if we could assig to A 1 we would sice we still have the budget) Goal: Show we have y x Case 1) ½ of A 1 s queries got assiged to A 2 the y ³ B/2 Case 2) > ½ of A 1 s queries got assiged to A 2 the x B/2 ad x + y = B Balace reveue is miimum for x = y = B/2 Miimum Balace reveue = 3B/2 Competitive Ratio = 3/4 BALANCE exhausts A 2 s budget

32 + BALANCE: Geeral Result 32 I the geeral case, worst competitive ratio of BALANCE is 1 1/e = approx Iterestigly, o olie algorithm has a better competitive ratio! Let s see the worst case example that gives this ratio

33 + Worst case for BALANCE 33 N advertisers: A 1, A 2, A N Each with budget B > N Queries: N B queries appear i N rouds of B queries each Biddig: Roud 1 queries: bidders A 1, A 2,, A N Roud 2 queries: bidders A 2, A 3,, A N Roud i queries: bidders A i,, A N Optimum allocatio: Allocate roud i queries to A i Optimum reveue N B

34 + BALANCE Allocatio 34 A 1 A 2 A 3 A N-1 A N B/N B/(N-2) B/(N-1) BALANCE assigs each of the queries i roud 1 to N advertisers. After k rouds, sum of allocatios to each of advertisers A k,,a N k21 B is S k = S k-1 = = S N = i61 N2(i21) If we fid the smallest k such that S k B, the after k rouds we caot allocate ay queries to ay advertiser

35 + BALANCE: Aalysis 35 B/1 B/2 B/3 B/(N-(k-1)) B/(N-1) B/N S 1 S 2 S k = B 1/1 1/2 1/3 1/(N-(k-1)) 1/(N-1) 1/N S 1 S 2 S k = 1

36 BALANCE: Aalysis Fact: H = i61 1/i Result due to Euler l for large 1/1 1/2 1/3 1/(N-(k-1)) 1/(N-1) 1/N l(n)-1 l(n) S k = 1 implies: H N2k = l (N) 1 = l ( N e ) We also kow: H N2k = l (N k) S k = 1 So: N k = N e The: k = N(1 1 e ) N terms sum to l(n). Last k terms sum to 1. First N-k terms sum to l(n-k) but also to l(n)-1 36

37 + BALANCE: Aalysis 37 So after the first k=n(1-1/e) rouds, we caot allocate a query to ay advertiser Reveue = B N (1-1/e) Competitive ratio = 1-1/e

38 + Geeral Versio of the Problem 38 Arbitrary bids ad arbitrary budgets! Cosider we have 1 query q, advertiser i Bid = x i Budget = b i I a geeral settig BALANCE ca be terrible Cosider two advertisers A 1 ad A 2 A 1 : x 1 = 1, b 1 = 110 A 2 : x 2 = 10, b 2 = 100 Cosider we see 10 istaces of q BALANCE always selects A 1 ad ears 10 Optimal ears 100

39 Geeralized BALANCE Arbitrary bids: cosider query q, bidder i Bid = x i Budget = b i Amout spet so far = m i Fractio of budget left over f i = 1-m i /b i Defie ψ i (q) = x i (1-e -f i) Allocate query q to bidder i with largest value of ψ i (q) Same competitive ratio (1-1/e) 39