Fair Allocation with Succinct Representation Azarakhsh Malekian (NWU) Joint Work with Saeed Alaei,...
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Transcript of Fair Allocation with Succinct Representation Azarakhsh Malekian (NWU) Joint Work with Saeed Alaei,...
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Fair Allocation with Succinct Representation
Azarakhsh Malekian (NWU)Joint Work with Saeed Alaei, Ravi Kumar, Erik VeeUMD Yahoo! Research
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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
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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
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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
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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
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Agenda
Formal Problem DefinitionOur Main ResultsCompact planReconstructing the original solution in
constant time
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Problem Definition
J: set of contracts (advertisers) I: set of impression types (buckets) dj: Total demand of contract j wj: weight of contract j si: Total supply of impression i
2020
20Fai
r: 1
0F
air:
10
Fai
r: 1
5 i
j : 15
dj=30wj= 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
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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]
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Formal Model (LP Formulation) J: set of contracts I: set of impression buckets dj: Total demand of contract j wj: weight of contract j si: Total supply of impression i
The allocation
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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 xij for contract j it increases the objective by 2wj xj
5 10 10
9 12
3 6 6 6
Overfull: Should reassign 2
5 10 10
6-2 6+263
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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 dj
Cost 0
Capacity ij
Cost 0
Capacity Cost 2Wj
Capacity si
Cost 0
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Compact Plan?
Min cost flow can be computed efficientlyWe 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.
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Reconstruction (Primary Steps) Writing the dual of min cost flow
Primal (min cost flow) Dual allocation
Dual variables
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Reconstruction
Compute the dual variables of the min cost flow LP.
We only need O(|I|+|J|) space to store the dual (Zi and Yj).
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.
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Reconstruction: Network Flow Solution Lemma:
Aij= max(0, Zi - Yj)
The value of x’ij in primal is
0: if Zi - Yj < 0
ij: if Zi - Yj > 0
Zi - Yj =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)
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Storing the Solution
Height based Maximum Flow: We find a height function h(v) that assigns
height to each vertex such that:
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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
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Summary
Compact Plan: Write the primal/Dual min cost flow Make a network flow instane on vertices with Z i - Yj = 0 Compute the height for vertices of the flow
Reconstruction: For each edge if: Zi - Yj ≠ 0 then it is either full or empty based on
the sign Zi – Yj =0 then use height function
Flow(i,j) = min(capacity, (h(i)-h(j))capacity)
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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
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Experimental Results
Results on the rest of data sets:
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Related Works
Vee et al Strictly convex version of the problemGiven approximation of the online supply
Find a reconstructible plan for other norms Using KKT method Focus on sampling aspects of the problem
Gosh et alCombined variant of guaranteed and non
guaranteedA randomized mechanism
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Future Directions
Adapting our solution to highest degree norm and comparing the results
Consider the fair allocation from the mechanism design point of viewWhen advertisers are strategic