1. Fair Allocation with Succinct Representation Azarakhsh Malekian (NWU) Joint Work with Saeed Alaei, Ravi Kumar, Erik Vee UMDYahoo! Research

2. 2 Online Advertising query=travel 4 slots Sponsored links Main Problem search engines face: Which ad to show for which query Subject to: Maximizing revenue Maximizing user Safisfaction

3. 3 Categories of Advertisers Non-Guaranteed Delivery (small advertiser)  Main purpose: Selling your item An action from user Allocation is not guaranteed Guaranteed Delivery (large advertiser)  Main purpose: Brand recognition  Contracts  Ask for a minimum # impressions: fixed price per item  Prepaid charge I want 10K impressions per day for august to users from california! Can we sign a contract? We focus on Guaranteed Delivery in this talk

4. 4 Introduction We have a set of advertisers and a set of impression types (buckets). Each advertiser is only  Interested in impressions of certain types.  Required minimum number of impression from its desired buckets For each impression:  There is only a limited number of impressions available Furthermore  Advertisers want the allocation to be representative of the supply as much as possible. Due to the online nature of the problem and the huge size of data, we are seeking a solution:  Can be represented by a compact plan  Can be reconstructed efficiently in real time

5. Justifying Representativeness Each bucket has some of user attributes explicitly  The unspecified ones are subject to interpretation  Most often, advertisers are equally interested in all the users who belong to the bucket  Example: It is undesirable to assign old men to a Sport car dealer interested in men There can be a large number of attributes at different level of granularity  It is not fully possible for the advertiser to specify the desired bucket to the finest conceivable detail  Example: Toy store

6. 6 Agenda Formal Problem Definition Our Main Results Compact plan Reconstructing the original solution in constant time

7. 7 Problem Definition J: set of contracts (advertisers) I: set of impression types (buckets) d j : Total demand of contract j w j : weight of contract j s i : Total supply of impression i 20 Fair: 10 Fair: 15  i j : 15 d j =30 w j = 1 Goal: finding an allocation that minimizes the distance from the ideal fair allocation We use L1 distance function We are interested in a method that: Can compute the allocation efficiently Can store the allocation succinctly

8. Main Results An efficient combinatorial solution for finding allocation that minimizes L1 distance using min cost flow A compact representation of the solution requiring only linear space in number of impression types and advertisers (as opposed to quadratic) Reconstructing the allocation in constant time Robustness Experimental Results Also: We compute the approximation ratio of greedy  Experimental results Combinatorial way of computing succint plan for L2 distance function (Based on the solution of Vee et al [VVS10]

9. 9 Formal Model (LP Formulation) J: set of contracts I: set of impression buckets d j : Total demand of contract j w j : weight of contract j s i : Total supply of impression i The allocation

10. 10 Idea Consider the perfectly fair allocation (possibly infeasible) To make it feasible  reassign the overfilled portions of the contracts to other buckets with available capacity. If we remove x ij for contract j it increases the objective by 2w j x j 510 9 12 36 66 Overfull: Should reassign 2 510 6-26+2 6 3

11. 11 Min Cost Flow Solution Theorem: The min cost solution to the flow network on left is the solution to the LP for L1 distance function. Capacity d j Cost 0 Capacity  ij Cost 0 Capacity  Cost 2W j Capacity  s i Cost 0

12. 12 Compact Plan? Min cost flow can be computed efficiently  We still need to store the whole allocation The space required to store the allocation plan should be linear in the number of vertices. We should be able to reconstruct the flow along each edge in constant time.

13. 13 Reconstruction (Primary Steps) Writing the dual of min cost flow P rimal (min cost flow) Dual allocation Dual variables

14. 14 Reconstruction Compute the dual variables of the min cost flow LP. We only need O(|I|+|J|) space to store the dual (Z i and Y j ). The allocation along any edge (primal) can be computed using dual and complementary slackness except  for a few slack edges. For the slack edges,  we show how to compute an extra variable for each vertex call it height which allows us: to reconstruct the flow along any slack edge.

15. 15 If there exists a feasible allocation We compute the most representative allocation using min cost flow We can reconstruct the solution in O(1) by only storing O(|I|+|J|) information.

16. 16 Reconstruction: Network Flow Solution Lemma:  A i j = max(0, Z i - Y j )  The value of x’ i j in primal is 0: if Z i - Y j < 0  i j : if Z i - Y j > 0 Z i - Y j =0 : slack edges  Make a new instance of max flow problem on this set of edges.  The cost of all max flow in the new network is the same. Find a height function for this network flow such that:  Flow(i,j) = min(capacity, (h(i)-h(j))capacity)

17. 17 Storing the Solution Extract the subgraph of edges for which:  Z i - Y j = 0 or  Z i – Y j = 2w j For all other edges the primal can be identified uniquely. We ignore the costs in the new graph as all the paths have the same cost. Any maximum flow in the new subgraph would work.

18. 18 Storing the Solution Height based Maximum Flow:  We find a height function h(v) that assigns height to each vertex such that:

19. 19 Storing the Solution We find a height function h(v) that assigns height to each vertex such that: We can approximate the above for any given  in time polynomial in 1/  The obtained solution is robust

20. Summary Compact Plan:  Write the primal/Dual min cost flow  Make a network flow instane on vertices with Z i - Y j = 0  Compute the height for vertices of the flow Reconstruction: For each edge if: Z i - Y j ≠ 0 then it is either full or empty based on the sign Z i – Y j =0 then use height function  Flow(i,j) = min(capacity, (h(i)-h(j))capacity)

21. 21 Experimental Results Data set:  Actual impression buckets and advertiser contracts from Yahoo! Display advertisement The results for the largest graph:  Min Cost Flow is much faster than solving LP 178 seconds versus 4000 seconds  More than 99% percent of the edges are either empty or saturated in practice, as a result: We only need to address this small proportion by height

22. Experimental Results Results on the rest of data sets: 22

23. 23 Idea It would be interesting to solve the following network flow problem: Set a height for each vertex so that in the maximum flow the flow on each edge is Min( capacity, height difference)

24. Related Works Vee et al  Strictly convex version of the problem  Given approximation of the online supply Find a reconstructible plan for other norms  Using KKT method  Focus on sampling aspects of the problem Gosh et al  Combined variant of guaranteed and non guaranteed  A randomized mechanism

25. 25 Future Directions Adapting our solution to highest degree norm and comparing the results Consider the fair allocation from the mechanism design point of view  When advertisers are strategic