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

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

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

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

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