![BALANCE algorithm [MSVV]
[Mehta, Saberi, Vazirani, and Vazirani]
For each query, pick the advertiser with
the largest unspent budget
Break ties arbitrarily](https://image.slidesharecdn.com/advertising-160311123905/85/Advertising-25-320.jpg)
Advertising
- 1. CS 345 Data Mining Online algorithms Search advertising
- 2. Online algorithms Classic model of algorithms You get to see the entire input, then compute some function of it In this context, “offline algorithm” Online algorithm You get to see the input one piece at a time, and need to make irrevocable decisions along the way Similar to data stream models
- 4. Example: Bipartite matching 1 2 3 4 a b c d M = {(1,a),(2,b),(3,d)} is a matching Cardinality of matching = |M| = 3 Girls Boys
- 5. Example: Bipartite matching 1 2 3 4 a b c dGirls Boys M = {(1,c),(2,b),(3,d),(4,a)} is a perfect matching
- 6. Matching Algorithm Problem: Find a maximum-cardinality matching for a given bipartite graph A perfect one if it exists There is a polynomial-time offline algorithm (Hopcroft and Karp 1973) But what if we don’t have the entire graph upfront?
- 7. Online problem Initially, we are given the set Boys In each round, one girl’s choices are revealed At that time, we have to decide to either: Pair the girl with a boy Don’t pair the girl with any boy Example of application: assigning tasks to servers
- 9. Greedy algorithm Pair the new girl with any eligible boy If there is none, don’t pair girl How good is the algorithm?
- 10. Competitive Ratio For input I, suppose greedy produces matching Mgreedy while an optimal matching is Mopt Competitive ratio = minall possible inputs I (|Mgreedy|/|Mopt|)
- 11. Analyzing the greedy algorithm Consider the set G of girls matched in Mopt but not in Mgreedy Then it must be the case that every boy adjacent to girls in G is already matched in Mgreedy There must be at least |G| such boys Otherwise the optimal algorithm could not have matched all the G girls Therefore |Mgreedy| ¸ |G| = |Mopt - Mgreedy| |Mgreedy|/|Mopt| ¸ 1/2
- 13. History of web advertising Banner ads (1995-2001) Initial form of web advertising Popular websites charged X$ for every 1000 “impressions” of ad Called “CPM” rate Modeled similar to TV, magazine ads Untargeted to demographically tageted Low clickthrough rates low ROI for advertisers
- 14. Performance-based advertising Introduced by Overture around 2000 Advertisers “bid” on search keywords When someone searches for that keyword, the highest bidder’s ad is shown Advertiser is charged only if the ad is clicked on Similar model adopted by Google with some changes around 2002 Called “Adwords”
- 15. Ads vs. search results
- 16. Web 2.0 Performance-based advertising works! Multi-billion-dollar industry Interesting problems What ads to show for a search? If I’m an advertiser, which search terms should I bid on and how much to bid?
- 17. Adwords problem A stream of queries arrives at the search engine q1, q2,… Several advertisers bid on each query When query qi arrives, search engine must pick a subset of advertisers whose ads are shown Goal: maximize search engine’s revenues Clearly we need an online algorithm!
- 18. Greedy algorithm Simplest algorithm is greedy It’s easy to see that the greedy algorithm is actually optimal!
- 19. Complications (1) Each ad has a different likelihood of being clicked Advertiser 1 bids $2, click probability = 0.1 Advertiser 2 bids $1, click probability = 0.5 Clickthrough rate measured historically Simple solution Instead of raw bids, use the “expected revenue per click”
- 20. The Adwords Innovation Advertiser Bid CTR Bid * CTR A B C $1.00 $0.75 $0.50 1% 2% 2.5% 1 cent 1.5 cents 1.125 cents
- 21. The Adwords Innovation Advertiser Bid CTR Bid * CTR A B C $1.00 $0.75 $0.50 1% 2% 2.5% 1 cent 1.5 cents 1.125 cents
- 22. Complications (2) Each advertiser has a limited budget Search engine guarantees that the advertiser will not be charged more than their daily budget
- 23. Simplified model (for now) Assume all bids are 0 or 1 Each advertiser has the same budget B One advertiser per query Let’s try the greedy algorithm Arbitrarily pick an eligible advertiser for each keyword
- 24. Bad scenario for greedy Two advertisers A and B A bids on query x, B bids on x and y Both have budgets of $4 Query stream: xxxxyyyy Worst case greedy choice: BBBB____ Optimal: AAAABBBB Competitive ratio = ½ Simple analysis shows this is the worst case
- 25. BALANCE algorithm [MSVV] [Mehta, Saberi, Vazirani, and Vazirani] For each query, pick the advertiser with the largest unspent budget Break ties arbitrarily
- 26. Example: BALANCE Two advertisers A and B A bids on query x, B bids on x and y Both have budgets of $4 Query stream: xxxxyyyy BALANCE choice: ABABBB__ Optimal: AAAABBBB Competitive ratio = ¾
- 27. Analyzing BALANCE Consider simple case: two advertisers, A1 and A2, each with budget B (assume B À 1) Assume optimal solution exhausts both advertisers’ budgets BALANCE must exhaust at least one advertiser’s budget If not, we can allocate more queries Assume BALANCE exhausts A2’s budget
- 28. Analyzing Balance A1 A2 B xy B A1 A2 x Opt revenue = 2B Balance revenue = 2B-x = B+y We have y ¸ x Balance revenue is minimum for x=y=B/2 Minimum Balance revenue = 3B/2 Competitive Ratio = 3/4 Queries allocated to A1 in optimal solution Queries allocated to A2 in optimal solution
- 29. General Result In the general case, worst competitive ratio of BALANCE is 1–1/e = approx. 0.63 Interestingly, no online algorithm has a better competitive ratio Won’t go through the details here, but let’s see the worst case that gives this ratio
- 30. Worst case for BALANCE N advertisers, each with budget B À N À 1 NB queries appear in N rounds of B queries each Round 1 queries: bidders A1, A2, …, AN Round 2 queries: bidders A2, A3, …, AN Round i queries: bidders Ai, …, AN Optimum allocation: allocate round i queries to Ai Optimum revenue NB
- 31. BALANCE allocation … A1 A2 A3 AN-1 AN B/N B/(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 rounds we cannot allocate any queries to any advertiser
- 32. BALANCE analysis B/1 B/2 B/3 … B/(N-k+1) … B/(N-1) B/N S1 S2 Sk = B 1/1 1/2 1/3 … 1/(N-k+1) … 1/(N-1) 1/N S1 S2 Sk = 1
- 33. BALANCE analysis Fact: Hn = ∑1· i· n1/i = approx. log(n) for large n Result due to Euler 1/1 1/2 1/3 … 1/(N-k+1) … 1/(N-1) 1/N Sk = 1 log(N) log(N)-1 Sk = 1 implies HN-k = log(N)-1 = log(N/e) N-k = N/e k = N(1-1/e)
- 34. BALANCE analysis So after the first N(1-1/e) rounds, we cannot allocate a query to any advertiser Revenue = BN(1-1/e) Competitive ratio = 1-1/e
- 35. General version of problem Arbitrary bids, budgets Consider query q, advertiser i Bid = xi Budget = bi BALANCE can be terrible Consider two advertisers A1 and A2 A1: x1 = 1, b1 = 110 A2: x2 = 10, b2 = 100
- 36. Generalized BALANCE Arbitrary bids; consider query q, bidder i Bid = xi Budget = bi Amount spent so far = mi Fraction of budget left over fi = 1-mi/bi Define ψi(q) = xi(1-e-fi) Allocate query q to bidder i with largest value of ψi(q) Same competitive ratio (1-1/e)