Pareto Optimal Solutionsfor Smoothed Analysts

Ankur Moitra, IAS

joint work with Ryan O’Donnell, CMU

September 26, 2011

Ankur Moitra (IAS) Pareto September 26, 2011

1

vi+1

wi+1

vi

wi

vn

wn

values

weights

...

...

...

...

v W1

w

1

vi+1

wi+1

vi

wi

vn

wn

values

weights

...

...

...

...

v W1

w

1

vi+1

wi+1

vi

wi

vn

wn

values

weights

...

...

...

...val

ue

W − weight

v W1

w

1

vi+1

wi+1

vi

wi

vn

wnW1

w

W − weight

vvalues

weights

...

...

...

...val

ue

1

vi+1

wi+1

vi

wi

vn

wnW1

w

W − weight

vvalues

weights

...

...

...

...val

ue

1

vi+1

wi+1

vi

wi

vn

wnW1

w

W − weight

vvalues

weights

...

...

...

...

Not Pareto Optimal

val

ue

1

vi+1

wi+1

vi

wi

vn

wnW1

w

W − weight

vvalues

weights

...

...

...

...

Pareto Optimal!

val

ue

1

vi+1

wi+1

vi

wi

vn

wnW1

w

W − weight

vvalues

weights

...

...

...

...val

ue

1

vi+1

wi+1

vi

wi

vn

wn

PO(i)

1

w Wvvalues

weights

...

...

...

...val

ue

W − weight

1

vi+1

wi+1

vi

wi

vn

wnW1

w

W − weight

vvalues

weights

...

...

...

...val

ue

1

vi+1

wi+1

vi

wi

vn

wn

v

vi+1

wi+1

1

w ...

...

...val

ue

W − weight

Wvalues

weights

...

1

vi+1

wi+1

vi

wi

vn

wnW1

w ...

...

...val

ue

W − weight

vvalues

weights

...

1

vi+1

wi+1

vi

wi

vn

wnW1

w ...

...

...val

ue

W − weight

vvalues

weights

...

1

vi+1

wi+1

vi

wi

vn

wnW1

w ...

...

...val

ue

W − weight

vvalues

weights

...

1

vi+1

wi+1

vi

wi

vn

wnW1

w ...

...

...val

ue

W − weight

vvalues

weights

...

1

vi+1

wi+1

vi

wi

vn

wnW1

w ...

...

...val

ue

W − weight

vvalues

weights

...

1

vi+1

wi+1

vi

wi

vn

wnW1

w ...

...

...val

ue

W − weight

vvalues

weights

...

1

vi+1

wi+1

vi

wi

vn

wn

PO(i+1)

1

w

...

...val

ue

W − weight

Wvvalues

weights

...

...

Dynamic Programming with Lists (Nemhauser, Ullmann):

PO(i + 1) can be computed from PO(i)...

... in linear (in |PO(i)|) time

an optimal solution can be computed from PO(n)

Bottleneck in many algorithms: enumerate the set ofPareto optimal solutions

Dynamic Programming with Lists (Nemhauser, Ullmann):

PO(i + 1) can be computed from PO(i)...

... in linear (in |PO(i)|) time

an optimal solution can be computed from PO(n)

Bottleneck in many algorithms: enumerate the set ofPareto optimal solutions

Dynamic Programming with Lists (Nemhauser, Ullmann):

PO(i + 1) can be computed from PO(i)...

... in linear (in |PO(i)|) time

an optimal solution can be computed from PO(n)

Bottleneck in many algorithms: enumerate the set ofPareto optimal solutions

Dynamic Programming with Lists (Nemhauser, Ullmann):

PO(i + 1) can be computed from PO(i)...

... in linear (in |PO(i)|) time

an optimal solution can be computed from PO(n)

Bottleneck in many algorithms: enumerate the set ofPareto optimal solutions

Dynamic Programming with Lists (Nemhauser, Ullmann):

PO(i + 1) can be computed from PO(i)...

... in linear (in |PO(i)|) time

an optimal solution can be computed from PO(n)

Bottleneck in many algorithms: enumerate the set ofPareto optimal solutions

Results of Beier and Vocking (STOC 2003, STOC 2004)

Smoothed Analysis of Pareto Curves:

Polynomial bound on |PO(i)|s (in two dimensions) inthe framework of Smoothed Analysis (Spielman, Teng)

Knapsack has polynomial smoothed complexity:

first NP-hard problem that is (smoothed) easy

generalizes long line of results on random instances

Results of Beier and Vocking (STOC 2003, STOC 2004)

Smoothed Analysis of Pareto Curves:

Polynomial bound on |PO(i)|s (in two dimensions) inthe framework of Smoothed Analysis (Spielman, Teng)

Knapsack has polynomial smoothed complexity:

first NP-hard problem that is (smoothed) easy

generalizes long line of results on random instances

Results of Beier and Vocking (STOC 2003, STOC 2004)

Smoothed Analysis of Pareto Curves:

Polynomial bound on |PO(i)|s (in two dimensions) inthe framework of Smoothed Analysis (Spielman, Teng)

Knapsack has polynomial smoothed complexity:

first NP-hard problem that is (smoothed) easy

generalizes long line of results on random instances

Results of Beier and Vocking (STOC 2003, STOC 2004)

Smoothed Analysis of Pareto Curves:

Polynomial bound on |PO(i)|s (in two dimensions) inthe framework of Smoothed Analysis (Spielman, Teng)

Knapsack has polynomial smoothed complexity:

first NP-hard problem that is (smoothed) easy

generalizes long line of results on random instances

Results of Beier and Vocking (STOC 2003, STOC 2004)

Smoothed Analysis of Pareto Curves:

Polynomial bound on |PO(i)|s (in two dimensions) inthe framework of Smoothed Analysis (Spielman, Teng)

Knapsack has polynomial smoothed complexity:

first NP-hard problem that is (smoothed) easy

generalizes long line of results on random instances

Capturing Tradeoffs

Question

What if the precise objective function (of a decisionmaker) is unknown?

e.g. travel planning: prefer lower fare, shorter trip, fewer transfers

Question

Can we algorithmically help a decision maker?

Pareto curves capture tradeoffs among competingobjectives

Capturing Tradeoffs

Question

What if the precise objective function (of a decisionmaker) is unknown?

e.g. travel planning: prefer lower fare, shorter trip, fewer transfers

Question

Can we algorithmically help a decision maker?

Pareto curves capture tradeoffs among competingobjectives

Capturing Tradeoffs

Question

What if the precise objective function (of a decisionmaker) is unknown?

e.g. travel planning: prefer lower fare, shorter trip, fewer transfers

Question

Can we algorithmically help a decision maker?

Pareto curves capture tradeoffs among competingobjectives

Capturing Tradeoffs

Question

What if the precise objective function (of a decisionmaker) is unknown?

e.g. travel planning: prefer lower fare, shorter trip, fewer transfers

Question

Can we algorithmically help a decision maker?

Pareto curves capture tradeoffs among competingobjectives

Capturing Tradeoffs

Question

What if the precise objective function (of a decisionmaker) is unknown?

e.g. travel planning: prefer lower fare, shorter trip, fewer transfers

Question

Can we algorithmically help a decision maker?

Pareto curves capture tradeoffs among competingobjectives

Proposition

Only useful approach if Pareto curves are small

Confirmed empirically e.g. Muller-Hannemann, Weihe: German

train system

Question

Why should we expect Pareto curves to be small?

Caveat: Smoothed Analysis is not a complete explanation

Proposition

Only useful approach if Pareto curves are small

Confirmed empirically e.g. Muller-Hannemann, Weihe: German

train system

Question

Why should we expect Pareto curves to be small?

Caveat: Smoothed Analysis is not a complete explanation

Proposition

Only useful approach if Pareto curves are small

Confirmed empirically e.g. Muller-Hannemann, Weihe: German

train system

Question

Why should we expect Pareto curves to be small?

Caveat: Smoothed Analysis is not a complete explanation

Proposition

Only useful approach if Pareto curves are small

Confirmed empirically e.g. Muller-Hannemann, Weihe: German

train system

Question

Why should we expect Pareto curves to be small?

Caveat: Smoothed Analysis is not a complete explanation

The Model

Adversary chooses:

Z ⊂ {0, 1}n (e.g. spanning trees, Hamiltonian cycles)

an adversarial objective function OBJ1 : Z → <+

d − 1 linear objective functions

OBJi(~x) = [w i1,w

i2, ...w

in] · ~x

... each w ij is a random variable on [−1,+1] – density

is bounded by φ

The Model

Adversary chooses:

Z ⊂ {0, 1}n (e.g. spanning trees, Hamiltonian cycles)

an adversarial objective function OBJ1 : Z → <+

d − 1 linear objective functions

OBJi(~x) = [w i1,w

i2, ...w

in] · ~x

... each w ij is a random variable on [−1,+1] – density

is bounded by φ

The Model

Adversary chooses:

Z ⊂ {0, 1}n (e.g. spanning trees, Hamiltonian cycles)

an adversarial objective function OBJ1 : Z → <+

d − 1 linear objective functions

OBJi(~x) = [w i1,w

i2, ...w

in] · ~x

... each w ij is a random variable on [−1,+1] – density

is bounded by φ

The Model

Adversary chooses:

Z ⊂ {0, 1}n (e.g. spanning trees, Hamiltonian cycles)

an adversarial objective function OBJ1 : Z → <+

d − 1 linear objective functions

OBJi(~x) = [w i1,w

i2, ...w

in] · ~x

... each w ij is a random variable on [−1,+1] – density

is bounded by φ

The Model

Adversary chooses:

Z ⊂ {0, 1}n (e.g. spanning trees, Hamiltonian cycles)

an adversarial objective function OBJ1 : Z → <+

d − 1 linear objective functions

OBJi(~x) = [w i1,w

i2, ...w

in] · ~x

... each w ij is a random variable on [−1,+1] – density

is bounded by φ

History

Let PO be the set of Pareto optimal solutions...

[Beier, Vocking STOC 2003] (d = 2), E [|PO|] = O(n5φ)

[Beier, Vocking STOC 2004] (d = 2), E [|PO|] = O(n4φ)

[Beier, Roglin, Vocking IPCO 2007] (d = 2), E [|PO|] = O(n2φ)(tight)

[Roglin, Teng, FOCS 2009] E [|PO|] = O((n√φ)f (d))

... where f (d) = 2d−1(d + 1)!

[Dughmi, Roughgarden, FOCS 2010] any FPTAS can betransformed to a truthful in expectation FPTAS

History

Let PO be the set of Pareto optimal solutions...

[Beier, Vocking STOC 2003] (d = 2), E [|PO|] = O(n5φ)

[Beier, Vocking STOC 2004] (d = 2), E [|PO|] = O(n4φ)

[Beier, Roglin, Vocking IPCO 2007] (d = 2), E [|PO|] = O(n2φ)(tight)

[Roglin, Teng, FOCS 2009] E [|PO|] = O((n√φ)f (d))

... where f (d) = 2d−1(d + 1)!

[Dughmi, Roughgarden, FOCS 2010] any FPTAS can betransformed to a truthful in expectation FPTAS

History

Let PO be the set of Pareto optimal solutions...

[Beier, Vocking STOC 2003] (d = 2), E [|PO|] = O(n5φ)

[Beier, Vocking STOC 2004] (d = 2), E [|PO|] = O(n4φ)

[Beier, Roglin, Vocking IPCO 2007] (d = 2), E [|PO|] = O(n2φ)(tight)

[Roglin, Teng, FOCS 2009] E [|PO|] = O((n√φ)f (d))

... where f (d) = 2d−1(d + 1)!

[Dughmi, Roughgarden, FOCS 2010] any FPTAS can betransformed to a truthful in expectation FPTAS

History

Let PO be the set of Pareto optimal solutions...

[Beier, Vocking STOC 2003] (d = 2), E [|PO|] = O(n5φ)

[Beier, Vocking STOC 2004] (d = 2), E [|PO|] = O(n4φ)

[Beier, Roglin, Vocking IPCO 2007] (d = 2), E [|PO|] = O(n2φ)(tight)

[Roglin, Teng, FOCS 2009] E [|PO|] = O((n√φ)f (d))

... where f (d) = 2d−1(d + 1)!

[Dughmi, Roughgarden, FOCS 2010] any FPTAS can betransformed to a truthful in expectation FPTAS

History

Let PO be the set of Pareto optimal solutions...

[Beier, Vocking STOC 2003] (d = 2), E [|PO|] = O(n5φ)

[Beier, Vocking STOC 2004] (d = 2), E [|PO|] = O(n4φ)

[Beier, Roglin, Vocking IPCO 2007] (d = 2), E [|PO|] = O(n2φ)(tight)

[Roglin, Teng, FOCS 2009] E [|PO|] = O((n√φ)f (d))

... where f (d) = 2d−1(d + 1)!

[Dughmi, Roughgarden, FOCS 2010] any FPTAS can betransformed to a truthful in expectation FPTAS

History

Let PO be the set of Pareto optimal solutions...

[Beier, Vocking STOC 2003] (d = 2), E [|PO|] = O(n5φ)

[Beier, Vocking STOC 2004] (d = 2), E [|PO|] = O(n4φ)

[Beier, Roglin, Vocking IPCO 2007] (d = 2), E [|PO|] = O(n2φ)(tight)

[Roglin, Teng, FOCS 2009] E [|PO|] = O((n√φ)f (d))

... where f (d) = 2d−1(d + 1)!

[Dughmi, Roughgarden, FOCS 2010] any FPTAS can betransformed to a truthful in expectation FPTAS

Our Results

Theorem

E [|PO|] ≤ 2 · (4φd)d(d−1)/2n2d−2

...answers a conjecture of Teng

[Bently et al, JACM 1978]: 2n points sampled from a d-dimensionalGaussian,

E [|PO|] = Θ(nd−1)

square factor difference necessary for d = 2

Our Results

Theorem

E [|PO|] ≤ 2 · (4φd)d(d−1)/2n2d−2

...answers a conjecture of Teng

[Bently et al, JACM 1978]: 2n points sampled from a d-dimensionalGaussian,

E [|PO|] = Θ(nd−1)

square factor difference necessary for d = 2

Our Results

Theorem

E [|PO|] ≤ 2 · (4φd)d(d−1)/2n2d−2

...answers a conjecture of Teng

[Bently et al, JACM 1978]: 2n points sampled from a d-dimensionalGaussian,

E [|PO|] = Θ(nd−1)

square factor difference necessary for d = 2

On Smoothed Analysis

Bounds are notoriously pessimistic (e.g. Simplex)

Method of Analysis:

Define a ”bad” event

...that you can blame if your algorithm runs slowly

Prove this event is rare

Proposition

Randomness is your friend!

On Smoothed Analysis

Bounds are notoriously pessimistic (e.g. Simplex)

Method of Analysis:

Define a ”bad” event

...that you can blame if your algorithm runs slowly

Prove this event is rare

Proposition

Randomness is your friend!

On Smoothed Analysis

Bounds are notoriously pessimistic (e.g. Simplex)

Method of Analysis:

Define a ”bad” event

...that you can blame if your algorithm runs slowly

Prove this event is rare

Proposition

Randomness is your friend!

On Smoothed Analysis

Bounds are notoriously pessimistic (e.g. Simplex)

Method of Analysis:

Define a ”bad” event

...that you can blame if your algorithm runs slowly

Prove this event is rare

Proposition

Randomness is your friend!

On Smoothed Analysis

Bounds are notoriously pessimistic (e.g. Simplex)

Method of Analysis:

Define a ”bad” event

...that you can blame if your algorithm runs slowly

Prove this event is rare

Proposition

Randomness is your friend!

On Smoothed Analysis

Bounds are notoriously pessimistic (e.g. Simplex)

Method of Analysis:

Define a ”bad” event

...that you can blame if your algorithm runs slowly

Prove this event is rare

Proposition

Randomness is your friend!

Our Approach

Goal”Define” bad events using as little randomness as possible

... ”conserve” randomness, save for analysis

Challenge

Events become convoluted (to say the least!)

We give an algorithm to define these events

Our Approach

Goal”Define” bad events using as little randomness as possible

... ”conserve” randomness, save for analysis

Challenge

Events become convoluted (to say the least!)

We give an algorithm to define these events

Our Approach

Goal”Define” bad events using as little randomness as possible

... ”conserve” randomness, save for analysis

Challenge

Events become convoluted (to say the least!)

We give an algorithm to define these events

Our Approach

Goal”Define” bad events using as little randomness as possible

... ”conserve” randomness, save for analysis

Challenge

Events become convoluted (to say the least!)

We give an algorithm to define these events

Our Approach

Goal”Define” bad events using as little randomness as possible

... ”conserve” randomness, save for analysis

Challenge

Events become convoluted (to say the least!)

We give an algorithm to define these events

An Alternative Condition

time OBJ

OBJ2

1

An Alternative Condition

time OBJ

OBJ2

1

An Alternative Condition

time OBJ

OBJ2

1

An Alternative Condition

time OBJ

OBJ2

1

An Alternative Condition

time1OBJ

OBJ2

An Alternative Condition

time1OBJ

OBJ2

An Alternative Condition

time1OBJ

OBJ2

Method of Proof

Goal

Count the (expected) number of Pareto optimal solutions

Define a complete family of events... for each Pareto optimal solution at least one (unique) eventoccurs

Then bound the expected number of events

The key is in the definition

Method of Proof

Goal

Count the (expected) number of Pareto optimal solutions

Define a complete family of events... for each Pareto optimal solution at least one (unique) eventoccurs

Then bound the expected number of events

The key is in the definition

Method of Proof

Goal

Count the (expected) number of Pareto optimal solutions

Define a complete family of events... for each Pareto optimal solution at least one (unique) eventoccurs

Then bound the expected number of events

The key is in the definition

Method of Proof

Goal

Count the (expected) number of Pareto optimal solutions

Define a complete family of events... for each Pareto optimal solution at least one (unique) eventoccurs

Then bound the expected number of events

The key is in the definition

The Common Case (Beier, Roglin, Vocking)

time

x

OBJ

OBJ2

1

The Common Case (Beier, Roglin, Vocking)

time

x

y

1OBJ

OBJ2

The Common Case (Beier, Roglin, Vocking)

x

yEMPTY

time OBJ

OBJ2

1

The Common Case (Beier, Roglin, Vocking)

x

yEMPTY

i

OBJ

OBJ2

1time

The Common Case (Beier, Roglin, Vocking)

x

y = 0i

yEMPTY

i

OBJ

OBJ2

1time

i

x = 1

The Common Case (Beier, Roglin, Vocking)

x

y = 0i

yEMPTY

i

OBJ

OBJ2

1time

EMPTY

i

x = 1

Complete Family of Events

Let I = (a, b) be an interval of [−n,+n] of width ε

Let E be the event that there is an ~x ∈ PO:

OBJ2(~x) ∈ I

~y have the smallest OBJ1(~y) among all points for whichOBJ2(~y) > b

~yi = 0, ~xi = 1

Claim

Pr [E ] ≤ εφ

Complete Family of Events

Let I = (a, b) be an interval of [−n,+n] of width ε

Let E be the event that there is an ~x ∈ PO:

OBJ2(~x) ∈ I

~y have the smallest OBJ1(~y) among all points for whichOBJ2(~y) > b

~yi = 0, ~xi = 1

Claim

Pr [E ] ≤ εφ

Complete Family of Events

Let I = (a, b) be an interval of [−n,+n] of width ε

Let E be the event that there is an ~x ∈ PO:

OBJ2(~x) ∈ I

~y have the smallest OBJ1(~y) among all points for whichOBJ2(~y) > b

~yi = 0, ~xi = 1

Claim

Pr [E ] ≤ εφ

Complete Family of Events

Let I = (a, b) be an interval of [−n,+n] of width ε

Let E be the event that there is an ~x ∈ PO:

OBJ2(~x) ∈ I

~y have the smallest OBJ1(~y) among all points for whichOBJ2(~y) > b

~yi = 0, ~xi = 1

Claim

Pr [E ] ≤ εφ

Complete Family of Events

Let I = (a, b) be an interval of [−n,+n] of width ε

Let E be the event that there is an ~x ∈ PO:

OBJ2(~x) ∈ I

~y have the smallest OBJ1(~y) among all points for whichOBJ2(~y) > b

~yi = 0, ~xi = 1

Claim

Pr [E ] ≤ εφ

Complete Family of Events

Let I = (a, b) be an interval of [−n,+n] of width ε

Let E be the event that there is an ~x ∈ PO:

OBJ2(~x) ∈ I

~y have the smallest OBJ1(~y) among all points for whichOBJ2(~y) > b

~yi = 0, ~xi = 1

Claim

Pr [E ] ≤ εφ

Analysis

[ , , ... .... ] EOBJ2 x

OBJ

OBJ2

1

: : arbitrary

time

??

Z

?? ??

OBJ

??1

Analysis

[ , , ... .... ] EOBJ2 x

OBJ

OBJ2

1

: : arbitrary

time

??

I

Z

?? ??

OBJ

??1

Analysis

[ , , ... .... ] E2 xw1 w2

OBJ

OBJ

OBJ2

1

: : arbitrary

time

index i

??

I

wn

Z

1

OBJ

Analysis

[ , , ... .... ] E2 xw1 w2

z

1

OBJ

index i

??

I

wn

Z

x

OBJ

OBJ

OBJ2

1

: : arbitrary

time

Analysis

[ , , ... .... ] E2 xw1 w2

x = 0ix = 1i

z

1

OBJ n

i −iZ

Z

Z

x

OBJ

OBJ

OBJ2

1

: : arbitrary

time

index i

??

I

w

Analysis

[ , , ... .... ] E2 xw1 w2

x = 0ix = 1iy

z

1

OBJ n

i −iZ

Z

Z

x

OBJ

OBJ

OBJ2

1

: : arbitrary

time

index i

??

I

w

Analysis

[ , , ... .... ] E2 xw1 w2

x = 0ix = 1i

OBJ (x)>OBJ (y)1 1

y

w1

OBJ

OBJ

n

i −iZ

Z

Z

Tx

z

OBJ

OBJ2

1

: : arbitrary

time

index i

??

I

Analysis

[ , , ... .... ] E2 xw1 w2

x = 0ix = 1i

OBJ (x)>OBJ (y)1 1

y

w1

OBJ

OBJ

n

i −iZ

Z

Z

T

x

z

OBJ

OBJ2

1

: : arbitrary

time

index i

??

I

Analysis

[ , , ... .... ] E2 xw1 w2

x = 0ix = 1i

OBJ (x)>OBJ (y)1 1

y same on index i

1

OBJ n

i −iZ

Z

Z

T

OBJ

x

z

OBJ

OBJ2

1

: : arbitrary

time

index i

??

I

w

Analysis

[ , , ... .... ] E2 xw1 w2

x = 0ix = 1i

OBJ (x)>OBJ (y)1 1

y

w1

OBJ

OBJ

n

i −iZ

Z

Z

Tx

z

OBJ

OBJ2

1

: : arbitrary

time

index i

??

I

Analysis

[ , , ... .... ] E2 xw1 w2

i

1

OBJ

OBJ

−iZ

Z

Z

T

x

z

x = 0ix = 1i

OBJ (x)>OBJ (y)1 1

y

OBJ

OBJ2

1

: : arbitrary

time

index i

??

I

wn

Analysis

[ , , ... .... ] E2 xw1 w2

i

1

OBJ

Z

Z

T

x

z

and y = 0

OBJ

x = 0ix = 1i

OBJ (x)>OBJ (y)1 1

x = 1i

y

OBJ

OBJ2

1

: : arbitrary

time

index i

??

I

wn

i −iZ

A Re-interpretation

Implicitly defined a Transcription Algorithm:

Question

Given a Pareto optimal solution x, which event should weblame?

Blame event E : interval I , index i , xi and yi

Transcription Algorithm

Input: values of r.v.s (and Pareto optimal x)

Output: interval I , index i , xi and yi

A Re-interpretation

Implicitly defined a Transcription Algorithm:

Question

Given a Pareto optimal solution x, which event should weblame?

Blame event E : interval I , index i , xi and yi

Transcription Algorithm

Input: values of r.v.s (and Pareto optimal x)

Output: interval I , index i , xi and yi

A Re-interpretation

Implicitly defined a Transcription Algorithm:

Question

Given a Pareto optimal solution x, which event should weblame?

Blame event E : interval I , index i , xi and yi

Transcription Algorithm

Input: values of r.v.s (and Pareto optimal x)

Output: interval I , index i , xi and yi

A Re-interpretation

Implicitly defined a Transcription Algorithm:

Question

Given a Pareto optimal solution x, which event should weblame?

Blame event E : interval I , index i , xi and yi

Transcription Algorithm

Input: values of r.v.s (and Pareto optimal x)

Output: interval I , index i , xi and yi

A Re-interpretation

Implicitly defined a Transcription Algorithm:

Question

Given a Pareto optimal solution x, which event should weblame?

Blame event E : interval I , index i , xi and yi

Transcription Algorithm

Input: values of r.v.s (and Pareto optimal x)

Output: interval I , index i , xi and yi

A Re-interpretation

Suppose output is event E : interval I , index i , xi and yi

Question

Which solution x caused this event to occur?

We do not need to know wi to determine the identity of x!

Reverse Algorithm

Find y

Then find x

A Re-interpretation

Suppose output is event E : interval I , index i , xi and yi

Question

Which solution x caused this event to occur?

We do not need to know wi to determine the identity of x!

Reverse Algorithm

Find y

Then find x

A Re-interpretation

Suppose output is event E : interval I , index i , xi and yi

Question

Which solution x caused this event to occur?

We do not need to know wi to determine the identity of x!

Reverse Algorithm

Find y

Then find x

A Re-interpretation

Suppose output is event E : interval I , index i , xi and yi

Question

Which solution x caused this event to occur?

We do not need to know wi to determine the identity of x!

Reverse Algorithm

Find y

Then find x

A Re-interpretation

Suppose output is event E : interval I , index i , xi and yi

Question

Which solution x caused this event to occur?

We do not need to know wi to determine the identity of x!

Reverse Algorithm

Find y

Then find x

A Re-interpretation

Reverse Algorithm: Find x (without looking at wi)

Question

Does x fall into the interval I ?

Hidden random variable wi must fall into some small range

Proposition

We can deduce a missing input to TranscriptionAlgorithm, from just some of the inputs and outputs

Hence each output is unlikely

A Re-interpretation

Reverse Algorithm: Find x (without looking at wi)

Question

Does x fall into the interval I ?

Hidden random variable wi must fall into some small range

Proposition

We can deduce a missing input to TranscriptionAlgorithm, from just some of the inputs and outputs

Hence each output is unlikely

A Re-interpretation

Reverse Algorithm: Find x (without looking at wi)

Question

Does x fall into the interval I ?

Hidden random variable wi must fall into some small range

Proposition

We can deduce a missing input to TranscriptionAlgorithm, from just some of the inputs and outputs

Hence each output is unlikely

A Re-interpretation

Reverse Algorithm: Find x (without looking at wi)

Question

Does x fall into the interval I ?

Hidden random variable wi must fall into some small range

Proposition

We can deduce a missing input to TranscriptionAlgorithm, from just some of the inputs and outputs

Hence each output is unlikely

A Re-interpretation

Reverse Algorithm: Find x (without looking at wi)

Question

Does x fall into the interval I ?

Hidden random variable wi must fall into some small range

Proposition

We can deduce a missing input to TranscriptionAlgorithm, from just some of the inputs and outputs

Hence each output is unlikely

Transcription for d = 3

2

OBJ2

( , )

OBJ

OBJ3

: adversarial

: linear,

OBJ1

OBJ3

,Output: ,

Z

Transcription for d = 3

2

OBJ2

( , )

Z

OBJ

OBJ3

: adversarial

: linear,

OBJ1

OBJ3

,Output: ,

Transcription for d = 3

2

OBJ2

( , ),Z

OBJ

OBJ3

: adversarial

: linear,

OBJ1

OBJ3

,Output:

Transcription for d = 3

2

OBJ2

( , )Output: ,Z

OBJ

OBJ3

: adversarial

: linear,

OBJ1

OBJ3

,

Transcription for d = 3

x

OBJ2

OBJ2

x

( , )

,Output: ,

Z

OBJ3

: adversarial

: linear,

OBJ1

OBJ3

Transcription for d = 3

y

x

OBJ2

OBJ2

y x

( , )

3

,Output: ,

Z

OBJ3

: adversarial

: linear,

OBJ1

OBJ

Transcription for d = 3

y

x

OBJ2

OBJ2

y x

( , )

3

,Output: ,

Z

OBJ3

: adversarial

: linear,

OBJ1

OBJ

Transcription for d = 3

y

x

OBJ2

OBJ2

y x

Z3

,Output: ,(i, )

OBJ3

index i

: adversarial

: linear,

OBJ1

OBJ

Transcription for d = 3

y

x

OBJ2

OBJ2

y x

Z

10

Output: ,(i, )OBJ

3

index i

: adversarial

: linear,

OBJ1

OBJ3

,index i

Transcription for d = 3

y

x

OBJ2

OBJ2

y x

Z

10

Output: ,(i, )OBJ

3

index i

: adversarial

: linear,

OBJ1

OBJ3

,index i

Transcription for d = 3

y

x

OBJ2

OBJ2

y x

Z

10

Output: ,(i, )OBJ

3

index i

: adversarial

: linear,

OBJ1

OBJ3

,index i

Transcription for d = 3

y

x

OBJ2

OBJ2

y x

x = 0i

x = 1i

−i

(i, )

ZZ

Z i

OBJ3

index i

: adversarial

: linear,

OBJ1

OBJ3

,index i 10

Output: ,

Transcription for d = 3

x

OBJ2

OBJ2

y x

x = 0i

x = 1i

−i

y

time

ZZ

Z i

OBJ3

index i

: adversarial

: linear,

OBJ1

OBJ3

,index i 10

Output: ,(i, )

Transcription for d = 3

y

x

OBJ2

OBJ2

y x

x = 0i

x = 1i

−i

z

(i, )

ZZ

Z i

OBJ

time

3

index i

: adversarial

: linear,

OBJ1

OBJ3

,index i 10

Output: ,

Transcription for d = 3

y

x

OBJ2

OBJ2

zy x

x = 0i

x = 1i

−i

z

(i, )

ZZ

Z i

OBJ

time

3

index i

: adversarial

: linear,

OBJ1

OBJ3

,index i 1 10

Output: ,

Transcription for d = 3

z y

x

OBJ2

OBJ2

zy x

x = 0i

x = 1i

−i

(i, )

ZZ

Z i

OBJ3

index i

: adversarial

: linear,

OBJ1

OBJ3

,index i 1 10

Output: ,

Transcription for d = 3

z y

x

OBJ2

OBJ2

zy x

x = 0i

x = 1i

−i

(i, )

ZZ

Z i

OBJ3

index j index i

: adversarial

: linear,

OBJ1

OBJ3

,index i 1 10

Output: ,

Transcription for d = 3

z y

x

OBJ2

OBJ2

zy x

x = 0i

x = 1i

−i

,Z

Z

Z i

OBJ3

index j index i

: adversarial

: linear,

OBJ1

OBJ3

,index i

index j

1 1

1

0

0 1

Output: (i,j)

Transcription for d = 3

z y

x

OBJ2

OBJ2

zy x

x = 0i

x = 1i

−i

,Z

Z

Z i

OBJ3

index j index i

: adversarial

: linear,

OBJ1

OBJ3

,index i

index j

1 1

1

0

0 1

Output: (i,j)

Transcription for d = 3

z y

x

OBJ2

OBJ2

zy x

x = 0i

x = 1i

−i

,Z

Z

Z i

OBJ3

index j index i

: adversarial

: linear,

OBJ1

OBJ3

,index i

index j

1 1

1

0

0 1

Output: (i,j)

Transcription for d = 3

z y

x

zy x

index i

index j

1 1

1

0

0 1

Output: (i,j),

,,

OBJ2

OBJ2

x = 0i

x = 1i

−i3 Z

Z

Z i

OBJ3

index j index i

: adversarial

: linear,

OBJ1

OBJ

Future Directions?

Open Question

Are there additional applications of randomness”conservation” in Smoothed Analysis?

e.g. simplex algorithm

Open Question

Are there perturbation models that make sense fornon-linear objectives?

e.g. submodular functions

Future Directions?

Open Question

Are there additional applications of randomness”conservation” in Smoothed Analysis?

e.g. simplex algorithm

Open Question

Are there perturbation models that make sense fornon-linear objectives?

e.g. submodular functions

Future Directions?

Open Question

Are there additional applications of randomness”conservation” in Smoothed Analysis?

e.g. simplex algorithm

Open Question

Are there perturbation models that make sense fornon-linear objectives?

e.g. submodular functions

Future Directions?

Open Question

Are there additional applications of randomness”conservation” in Smoothed Analysis?

e.g. simplex algorithm

Open Question

Are there perturbation models that make sense fornon-linear objectives?

e.g. submodular functions

Future Directions?

Open Question

Are there additional applications of randomness”conservation” in Smoothed Analysis?

e.g. simplex algorithm

Open Question

Are there perturbation models that make sense fornon-linear objectives?

e.g. submodular functions

Questions?

Thanks!