stanford大学大数据挖掘

1、CS 345Data MiningOnline algorithmsSearch advertisingOnline algorithmsClassic model of algorithmsYou get to see the entire input, then compute some function of itIn this context, “offline algorithm”Online algorithmYou get to see the input one piece at a time, and need to make irrevocable decisions al

2、ong the waySimilar to data stream modelsExample: Bipartite matching1234abcdGirlsBoysExample: Bipartite matching1234abcdM = (1,a),(2,b),(3,d) is a matchingCardinality of matching = |M| = 3GirlsBoysExample: Bipartite matching1234abcdGirlsBoysM = (1,c),(2,b),(3,d),(4,a) is a perfect matchingMatching Al

3、gorithmProblem: Find a maximum-cardinality matching for a given bipartite graphA perfect one if it existsThere is a polynomial-time offline algorithm (Hopcroft and Karp 1973)But what if we dont have the entire graph upfront?Online problemInitially, we are given the set BoysIn each round, one girls c

4、hoices are revealedAt that time, we have to decide to either:Pair the girl with a boyDont pair the girl with any boyExample of application: assigning tasks to serversOnline problem1234abcd(1,a)(2,b)(3,d)Greedy algorithmPair the new girl with any eligible boyIf there is none, dont pair girlHow good i

5、s the algorithm?Competitive RatioFor input I, suppose greedy produces matching Mgreedy while an optimal matching is MoptCompetitive ratio = minall possible inputs I (|Mgreedy|/|Mopt|)Analyzing the greedy algorithmConsider the set G of girls matched in Mopt but not in MgreedyThen it must be the case

6、that every boy adjacent to girls in G is already matched in MgreedyThere must be at least |G| such boysOtherwise the optimal algorithm could not have matched all the G girlsTherefore |Mgreedy| |G| = |Mopt - Mgreedy|Mgreedy|/|Mopt| 1/2Worst-case scenario1234abc(1,a)(2,b)dHistory of web advertisingBan

7、ner ads (1995-2001)Initial form of web advertisingPopular websites charged X$ for every 1000 “impressions” of adCalled “CPM” rateModeled similar to TV, magazine adsUntargeted to demographically tagetedLow clickthrough rateslow ROI for advertisersPerformance-based advertisingIntroduced by Overture ar

8、ound 2000Advertisers “bid” on search keywordsWhen someone searches for that keyword, the highest bidders ad is shownAdvertiser is charged only if the ad is clicked onSimilar model adopted by Google with some changes around 2002Called “Adwords”Ads vs. search resultsWeb 2.0Performance-based advertisin

9、g works!Multi-billion-dollar industryInteresting problemsWhat ads to show for a search?If Im an advertiser, which search terms should I bid on and how much to bid?Adwords problemA stream of queries arrives at the search engineq1, q2,Several advertisers bid on each queryWhen query qi arrives, search

10、engine must pick a subset of advertisers whose ads are shownGoal: maximize search engines revenuesClearly we need an online algorithm!Greedy algorithmSimplest algorithm is greedyIts easy to see that the greedy algorithm is actually optimal!Complications (1)Each ad has a different likelihood of being

11、 clickedAdvertiser 1 bids $2, click probability = 0.1Advertiser 2 bids $1, click probability = 0.5Clickthrough rate measured historicallySimple solutionInstead of raw bids, use the “expected revenue per click”The Adwords InnovationAdvertiserBidCTRBid * CTRABC$1.00$0.75$0.501%2%2.5%1 cent1.5 cents1.1

12、25 centsThe Adwords InnovationAdvertiserBidCTRBid * CTRABC$1.00$0.75$0.501%2%2.5%1 cent1.5 cents1.125 centsComplications (2)Each advertiser has a limited budgetSearch engine guarantees that the advertiser will not be charged more than their daily budgetSimplified model (for now)Assume all bids are 0

13、 or 1Each advertiser has the same budget BOne advertiser per queryLets try the greedy algorithmArbitrarily pick an eligible advertiser for each keywordBad scenario for greedyTwo advertisers A and BA bids on query x, B bids on x and yBoth have budgets of $4Query stream: xxxxyyyyWorst case greedy choi

14、ce: BBBB_Optimal: AAAABBBBCompetitive ratio = Simple analysis shows this is the worst caseBALANCE algorithm MSVVMehta, Saberi, Vazirani, and VaziraniFor each query, pick the advertiser with the largest unspent budgetBreak ties arbitrarilyExample: BALANCETwo advertisers A and BA bids on query x, B bi

15、ds on x and yBoth have budgets of $4Query stream: xxxxyyyyBALANCE choice: ABABBB_Optimal: AAAABBBBCompetitive ratio = Analyzing BALANCEConsider simple case: two advertisers, A1 and A2, each with budget B (assume B 1)Assume optimal solution exhausts both advertisers budgetsBALANCE must exhaust at lea

16、st one advertisers budgetIf not, we can allocate more queriesAssume BALANCE exhausts A2s budgetAnalyzing BalanceA1A2BxyBA1A2xOpt revenue = 2BBalance revenue = 2B-x = B+yWe have y xBalance revenue is minimum for x=y=B/2Minimum Balance revenue = 3B/2Competitive Ratio = 3/4Queries allocated to A1 in op

17、timal solutionQueries allocated to A2 in optimal solutionGeneral ResultIn the general case, worst competitive ratio of BALANCE is 11/e = approx. 0.63Interestingly, no online algorithm has a better competitive ratioWont go through the details here, but lets see the worst case that gives this ratioWor

18、st case for BALANCEN advertisers, each with budget B N 1NB queries appear in N rounds of B queries eachRound 1 queries: bidders A1, A2, , ANRound 2 queries: bidders A2, A3, , ANRound i queries: bidders Ai, , ANOptimum allocation: allocate round i queries to AiOptimum revenue NBBALANCE allocationA1A2

19、A3AN-1ANB/NB/(N-1)B/(N-2)After k rounds, sum of allocations to each of bins Ak,AN is Sk = Sk+1 = = SN = 1 1 kB/(N-i+1)If we find the smallest k such that Sk B, then after k roundswe cannot allocate any queries to any advertiserBALANCE analysisB/1 B/2 B/3 B/(N-k+1) B/(N-1) B/NS1S2Sk = B 1/1 1/2 1/3 1/(N-k+1) 1/(N-1) 1/NS1S2Sk = 1 BALANCE analysisFact: Hn = 1 i n1/i = approx. log(n) for large nResult due to Euler1/1 1/2 1/3 1/(N-k+1) 1/(N-1) 1/NSk = 1 log(N)log(N)-1S