Non-linear objectives - PowerPoint Presentation

Slide1

Non-linear objectivesin mechanism design

Shuchi Chawla

University of Wisconsin – Madison

FOCS 2012Slide2

So far today…

Revenue & Social Welfare

This talk:

Non-linear functions of type & allocationQuestion: how well can we optimize in strategic settings? Do Bayesian assumptions help?

Shuchi Chawla: Non-linear objectivesSlide3

Algorithmic mechanism design

Three desiderata:

Computational efficiency

Incentive compatibility

Optimize/approximate objective

Main theme in AMD: all three not always achievable together

What should we give up?

Shuchi Chawla: Non-linear objectivesSlide4

AMD tradeoffs

Shuchi Chawla: Non-linear objectives

Overall OPT

OPT-IC

OPT-IC+E

OPT-E

Black-box

Social welfare has gap=1

Bayesian social welfare has small gap

Standard approximation question

Social welfare can have large gap, e.g. comb. auctionsSlide5

Social welfare has gap=1

Bayesian social welfare has small gap

AMD tradeoffs

Shuchi Chawla: Non-linear objectives

Overall OPT

OPT-IC

OPT-IC+E

OPT-E

Black-box

Question 1: OPT

vs

OPT-IC gap for multi-parameter non-linear objectives

Question 2: Black-box reductions for single-parameter monotone objectives

Single-parameter

: each agent has a single value

Monotone objectives

: unilateral increase in an agent’s value causes OPT to allocate more to the agent

IC condition

: unilateral increase in

an agent’s

value results in larger

allocation

All single-parameter “monotone” objectives have gap=1

Prior-free

Bayesian

(sometimes)Slide6

Rest of this talk

Part I

The

makespan objectiveImpossibility of black-box reductions for makespanPart IIBayesian truthful approximations for makespan

Other non-linear objectives; Open problems

Shuchi Chawla: Non-linear objectivesSlide7

Part I.1: Minimizing makespan

Shuchi Chawla: Non-linear objectivesSlide8

Scheduling to minimize makespan

n jobs, m machines

Jobs have different runtimes on different machines

Makespan = completion time of last job

Shuchi Chawla: Non-linear objectives

J1

J2

J3

J4

M1

M2

M3

Makespan

“Unrelated instance”Slide9

Scheduling to minimize makespan

Strategic setting [Nisan Ronen’99]:

Machines are “selfish workers”; jobs’ runtimes are private

Mechanism = (schedule, payments to machines)Machines’ objective: maximize payment – work doneWant assignment+payments to induce

truthtelling

Shuchi Chawla: Non-linear objectivesSlide10

Why makespan?

Important CS problem

Captures the difficulty with non-linear objectives

A single agent can disproportionately affect objectiveHas received the most attention in AGT

Shuchi Chawla: Non-linear objectivesSlide11

J1

J2

J3

J4

M1

M2

M3

Single-parameter

makespan

Each machine has a speed; each job has a size

Runtime of job j on machine

i

= (size of j)/(speed of

i

)

Monotone objective

Shuchi Chawla: Non-linear objectives

Makespan

“

R

elated instance”Slide12

A history of prior-free scheduling

Truthful approximations for related machines

Archer-Tardos’01

: constant approxDhangwatnotai

et al.’08

:

PTAS

Unrelated machines: upper & lower boundsNisan-Ronen’99: m approximationNisan-Ronen’99: lower bound of 2Christodoulou et al.’07:

2.41

; Koutsoupias-Vidali’07

:

2.61

Mu’alem-Shapira’07

: randomized, fractional mechanisms

Ashlagi-Dobzinski-Lavi’09: lower bound of m for anonymous mechanismsShuchi Chawla: Non-linear objectives

Overall OPT

OPT-IC

OPT-IC+E

OPT-ESlide13

Bayesian model for schedulingUnrelated setting: Running time of every job on every machine drawn independently from known distribution

Related setting: Speed of every machine drawn independently from known distribution; jobs sizes fixed

Objective: Expected min

makespanShuchi Chawla: Non-linear objectivesSlide14

Part I.2: Black-box transformations

Shuchi Chawla: Non-linear objectivesSlide15

Black-box transformationsShuchi Chawla: Non-linear objectives

Transformation

Algorithm

Input v

Allocation x

Payment p

GOAL: for

every

algorithm, transformation

preserves quality of solution

and

satisfies

incentive compatibility

.

(cf. Nicole’s talk)Slide16

Black-box transformations

Social welfare: can transform any approx. algorithm into BIC mechanism with “no” loss in expected performance.

[

Hartline-Lucier’10, Hartline-Kleinberg-Malekian’11, Bei-Huang’11]

Is this possible for other objectives?

Makespan

: For any

polytime BIC transformation, there is a makespan problem and algorithm such that mech.’s expected makespan is polynomially larger than alg.’s.[C.-Immorlica-Lucier’12]

Shuchi Chawla: Non-linear objectives

NO!Slide17

Single-parameter makespan

v

1

v

2

v

3

v

4

v

5

v

6

v

7

v

8

x

1

m

machines,

machine i has speed vi ~ Fi

n jobs,size of job j is xjx2

x3

x4collection F of feasible assignments

Shuchi Chawla: Non-linear objectivesSlide18

Proof outline

Define

makespan

instance (feasibility constraints, value distribution).Find algorithm with low expected makespan.Use monotonicity condition to show any BIC transformation has high expected makespan.

Key

issue: Transformation must rely on algorithm to

understand/satisfy

feasibility constraintThat is, transformation must return an allocation that it observes the algorithm returnShuchi Chawla: Non-linear objectives

Higher speed ⇒ higher expected loadSlide19

Makespan Instance

f

easibility set

F = {at most one job per machine}

v

1

v

2

v

3

v

4

v

5

v

6

v

7

v

8

x

m/2m

machines, speeds vi ~ Uniform{low = 1, high = α}x2

x1

x

k

x

2

x

1

m

/2 jobs, small size

x

j

= 1

m

1/2

jobs, large size

x

j

=

α

Shuchi Chawla: Non-linear objectivesSlide20

Approximation Algorithm

If (m/2 ± m

3

/4) machines report high speed,assign large jobs to fast machines (at random)assign small jobs to slow machines (at random)assign NO job to all remaining machinesElse

assign all jobs randomly

Shuchi Chawla: Non-linear objectivesSlide21

Approximation Algorithm

h

igh speeds

l

ow speeds

Note 1: By

Chernoff

, expected

makespan

is low.

Note 2:

E

xpected allocation is not monotone.

Shuchi Chawla: Non-linear objectivesSlide22

Transformation

To fix non-monotonicity, must more often:

allocate nothing to low speed machines, or

allocate something to high speed machines.Shuchi Chawla: Non-linear objectivesSlide23

Transformation

1

1

1

1

α

α

α

α

Query v’: pretend some low machines are high and vice versa...

Input v:

α

α

α

α

1

1

1

1

Each “fast” machine gets large job with probability m

-1/2

t

hen with high probability,

makespan

is high.

Shuchi Chawla: Non-linear objectivesSlide24

Transformation

1

1

1

1

α

α

α

α

Query v’:

pretend number of high machines deviates from expectation..

Input v:

1

1

1

1

1

1

1

1

Each machine gets large job with

probability m

-

1/

2

t

hen with high probability,

makespan

is high.

Shuchi Chawla: Non-linear objectivesSlide25

Recap and other resultsFor any BIC transformation, there is an alg. such that the transformation’s

makespan

is

polynomially larger than the algorithm’seven when the algorithm is a constant approximationWhat about other non-linear functions?Ironing doesn’t workGap increases with non-linearity

Shuchi Chawla: Non-linear objectives

[C.-Immorlica-Lucier’12]Slide26

Non-linear objectivesin mechanism design

Shuchi Chawla

University of Wisconsin – Madison

Part IISlide27

Recap of part I

A representative non-linear objective:

makespan

Black-box transformations are essentially impossible for makespan: objective function increases by polynomial factorShuchi Chawla: Non-linear objectives

Overall OPT

OPT-IC

OPT-IC+E

OPT-E

Black-boxSlide28

Part II.1: Bayesian approximation for makespan

Shuchi Chawla: Non-linear objectivesSlide29

Recall: scheduling to minimize makespan

n jobs, m machines

Jobs’ runtimes drawn from known

indep. distributionsMakespan = completion time of last jobPrior-free setting: any anonymous truthful mechanism is at best an m approximation.

Shuchi Chawla: Non-linear objectives

J1

J2

J3

J4

M1

M2

M3

MakespanSlide30

A truthful mechanism: M

inWork

For every job:

Assign the job to the machine that reports the lowest runtimePay the machine the job’s running time on its “second best” machine m’

“Second-price” payments: induce

truthtelling

Makespan

≤ sum of best runtimes of all jobs ≤ total work done in optimal schedule ≤ m x optimal makespan⇒ m-approximation to makespanShuchi Chawla: Non-linear objectivesSlide31

Overcoming the lower bound

Ashlagi

et al.’s lower bound of m for

makespanOrdered instance: machine i is better than machine i+1 for all jobsRunning times within 1+eps of each otherAny truthful mechanism must allocate all jobs to machine 1How do Bayesian assumptions help?

Knowledge of distribution => we can penalize allocations that are always bad for the given instance

A priori identical machines: bad instances have extremely low probability

Shuchi Chawla: Bayesian scheduling

31Slide32

Prior-independent approximation

Unknown Bayesian prior, but belongs to some “nice” family

In particular, the runtime of a job j is identically distributed on every machine.

That is, machines are a priori identicalHowever, any instantiation of runtimes is an unrelated instanceResult: There exists a truthful prior-independent mechanism that achieves an O(n/m) approximation to expected makespan

(*)

[C.-Hartline-Malec-Sivan’12]

Shuchi Chawla: Non-linear objectives

(cf. Tim’s talk)Slide33

Benchmark

Hindsight OPT

For any instantiation, finds the optimal

makespanOPT1/2 Discards m/2 machines randomlyFor any instantiation, finds optimal makespan

over remaining machines

For many distributions, OPT

1/2

~ constant. OPTKey property: min over 2 draws ~ 2 times a single drawIncludes all “MHR” distributions, e.g. uniform, exponential, normal,…Shuchi Chawla: Non-linear objectivesSlide34

How to design a truthful multi-parameter mechanism?

A simple powerful class: affine

maximizers

Maximize an appropriate linear a.k.a. affine functionEssentially, an extension of VCGFor example:Can assign “costs” to some outcomes, and, minimize total (work – cost) Can forbid certain outcomes by setting cost = ∞

Can assign more weight to the work of some agents than that of others

Shuchi Chawla: Non-linear objectivesSlide35

The MinWork

mechanism again

Essentially VCG: schedule every job on its best machineObserve: job j’s runtime in MinWork ≤

job j’s runtime in

OPT Furthermore,

every job goes to a random

machineIf jobs were to be distributed uniformly across machines, we would get good makespanHowever, balls-in-bins analysis ⇒ some machine has O(log m/log log m) jobsShuchi Chawla: Non-linear objectivesSlide36

The MinWork(k) mechanism

Find a min-size matching between jobs and machines that assigns at most k jobs to each machine.

Claim:

MinWork(k) is truthfulProof: It is VCG over a restricted domain.

Shuchi Chawla: Non-linear objectivesSlide37

The MinWork(k) mechanism

Find a min-size matching between jobs and machines that assigns at most k jobs to each machine.

Claim:

MinWork(k) is truthfulClaim: MinWork(10) gets a constant approximation

Obs1: The schedule is almost balanced

Obs2: Every job still goes to roughly its best machine

Shuchi Chawla: Non-linear objectivesSlide38

Obs2: the last entry procedure

Fix job j and imagine adding it last in a greedy fashion.

Shuchi Chawla: Non-linear objectives

1

2

3

4

5

6

MinWork

(3) schedule for all but job j

1

2

3

4

5

6

MinWork

(3) schedule sorted by j’s preferences

Machine full

Space available,

so j goes hereSlide39

Obs2: The last entry procedureFix job j and imagine adding it last in a greedy fashion.

The probability that j goes to one of its top

i

machines is at least 1-(1/k)iMinWork(k) places j in an even better positionKey claim: Placing j on its i

th

best machine is no worse than placing 5

i

independent copies of j on their best (of n/2) machinesShuchi Chawla: Non-linear objectivesSlide40

MinWork(10) analysis

Job j’s runtime in

MinWork

(k) ≤ max5^i independent copies j’s runtime in OPT1/2

≤ 5

i

times j’s runtime in OPT1/2 Here i is an exponential random variable; Note: E[5i] = constant.∴ MinWork(10)’s makespan ≤ 10 E[maxj (j’s

runtime in

MW)]

≤ constant times OPT

1/2

Shuchi Chawla: Non-linear objectives

Stochastic dominanceSlide41

Key technical claim

Placing j on its

i

th best machine is no worse than placing 5i independent copies of j on their best (of n/2) machines

Shuchi Chawla: Non-linear objectives

Expt. 1

n copies of j’s runtime

Expt. 2

5

i

/2 blocks

n/2 copies of j’s runtime

i

th

min over n copies

m

ax over 5

i

mins

over n/2 copiesSlide42

Recap and other results

Machines a priori identical, “few” jobs

O(1) prior-independent approximation:

MinWork(k) ≤ O(1) OPT1/2Compare to Bulow-Klemperer’s result for revenue with k items:

VCG ≥ O(1)

OPT

less

k agentsJobs are also a priori identical: multi-stage mechanismsPrior-ind. O(√log m) approximation to OPT1/2Prior-

ind.

O((log log m)

2

)

approx to OPT for MHR distributions

Shuchi Chawla: Non-linear objectives[C.-Hartline-Malec-Sivan’12]

Hindsight-OPT1/2(needs regularity)Slide43

Part II.2: Other objectives & open problems

Shuchi Chawla: Non-linear objectivesSlide44

Open problems for makespan

O(1) prior-

ind.

approximation for non-identical jobsBayesian approximation for non-identical machinesWill need to use the knowledge of priorEven logarithmic approx is non-trivialA potential approach: charge a prior-dependent amount for placing each additional job on a machine

Approximation for small-support priors

LP based?

Shuchi Chawla: Non-linear objectivesSlide45

Other non-linear objectives

Max-min fairness in scheduling a.k.a. load balancing

Prior-free PTAS for related setting

[Epstein-van Stee’10]Unrelated approximation?Max-min fairness in welfare a.k.a. the Santa Claus problem

Not monotone!

Single-parameter Bayesian

approx

?Shuchi Chawla: Non-linear objectives

Min

m

akespan

10

3

3

2Slide46

ConclusionsNon-linear objectives in general much harder than social welfare

Mild stochastic assumptions can help us circumvent strong impossibility results

Multi-parameter mechanisms are difficult to understand, but “affine

maximizers” is a powerful subclass.Lots of nice open problems!Shuchi Chawla: Non-linear objectives