Privacy and Fault-Tolerance PowerPoint Presentation, PPT - DocSlides

Privacy and Fault-Tolerance PowerPoint Presentation, PPT - DocSlides

2018-11-04 1K 1 0 0

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in Distributed . Optimization. Nitin Vaidya. University of Illinois at Urbana-Champaign. Shripad. . Gade. Lili Su. Acknowledgements. i. Applications. f. i. (x). . = cost for robot . i. to go to location . ID: 713035

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Presentations text content in Privacy and Fault-Tolerance

Slide1

Privacy and Fault-Tolerancein Distributed OptimizationNitin VaidyaUniversity of Illinois at Urbana-Champaign

Slide2

Shripad

Gade

Lili Su

Acknowledgements

Slide3

i

Slide4

Applicationsfi(x) = cost for robot i to go to location

x

Minimize total cost

of rendezvous

i

x

f

1

(x)

f

2

(x)

x

1

x

2

Slide5

Applications5

Minimize

cost

Σ

f

i(x)

i

f

1

(x)

f

2

(x)

f

4

(x)

f

3

(x)

Learning

Slide6

 

 

 

 

 

Distributed

Optimization

 

 

 

 

 

Fault-tolerance

 

 

 

 

 

Privacy

i

Outline

Slide7

Distributed Optimization7

 

 

 

 

 

 

Server

 

 

Slide8

Client-Server Architecture8f

1

(x)

f

2

(x)

f

4

(x)

f

3

(x)

 

Server

 

 

Slide9

Client-Server ArchitectureServer maintains estimate Client i knows

 

 

 

Server

 

 

Slide10

Client-Server ArchitectureServer maintains estimate

Client

i

knows

In iteration k+1

Client

i

Download

from server

U

pload

gradient

 

 

 

 

Server

 

 

Slide11

Client-Server Architecture

Server maintains estimate

Client

i

knows

In iteration k+1

Client

i

Download

from server

U

pload

gradient

Server

 

 

 

Server

 

 

Slide12

VariationsStochasticAsynchronous…12

Slide13

Peer-to-Peer Architecturef

1

(x)

f

2

(x)

f

4

(x)

f

3

(x)

 

 

 

 

 

Slide14

Peer-to-Peer ArchitectureEach agent maintains local estimate xConsensus step with neighborsApply own gradient to own estimate

 

 

 

 

 

 

Slide15

 

 

 

 

 

Distributed

Optimization

 

 

 

 

 

Fault-tolerance

 

 

 

 

 

Privacy

i

Outline

Slide16

 

Server

 

 

 

Slide17

Server observes gradients  privacy compromised

 

Server

 

 

 

Slide18

 

Server

 

 

Achieve privacy and yet collaboratively optimize

 

Server observes gradients

 privacy compromised

Slide19

Related WorkCryptographic methods (homomorphic encryption)Function transformationDifferential privacy19

Slide20

Differential Privacy20

 

Server

 

 

 

Slide21

Differential Privacy21

 

Server

 

 

 

Trade-off privacy with accuracy

Slide22

Proposed Approach22

Motivated by secret sharing

Exploit diversity

Multiple

servers

/

neighbors

Slide23

Proposed Approach23

Server 1

 

 

Server 2

 

Privacy if

subset of servers

adversarial

Slide24

Proposed Approach24

 

 

 

 

 

Privacy if

subset of neighbors

adversarial

Slide25

Proposed ApproachStructured noise that “cancels” over servers/neighbors25

Slide26

Intuition26

Server 1

 

 

Server 2

 

x

1

x

2

Slide27

Intuition27

Server 1

 

Server 2

 

 

 

 

 

x

1

x

2

Each client

simulates

multiple clients

Slide28

Intuition28

Server 1

 

Server 2

 

+

 

 

 

 

 

x

1

x

2

not necessarily convex

 

Slide29

AlgorithmEach server maintains an estimateIn each iterationClient iDownload estimates from corresponding server

U

pload gradient of

Each server updates estimate using received gradients

 

Slide30

AlgorithmEach server maintains an estimate

In each iteration

Client

i

Download estimates from

corresponding serverUpload gradient of

Each server updates estimate using received gradientsServers periodically exchange estimates to perform a consensus step

 

Slide31

ClaimUnder suitable assumptions, servers eventually reach consensus in31

i

Slide32

Privacy32

Server 1

 

Server 2

 

 

 

 

 

+

+

 

+

+

 

Slide33

PrivacyServer 1 may learn

,

+

+

Not sufficient to learn

 

33

Server 1

 

Server 2

 

 

 

 

 

+

+

 

+

+

 

Slide34

Function splitting not necessarily practicalStructured randomization as an alternative34

+

 

Slide35

Structured RandomizationMultiplicative or additive noise in gradientsNoise cancels over servers35

Slide36

Multiplicative Noise36

Server 1

 

 

Server 2

 

x

1

x

2

Slide37

Multiplicative Noise37

Server 1

 

 

Server 2

 

x

1

x

2

Slide38

Multiplicative Noise38

Server 1

 

 

Server 2

 

(x

1

)

 

 

x

1

x

2

+

=1

 

Slide39

Multiplicative NoiseServer 1

 

 

Server 2

 

(x

1

)

 

 

x

1

x

2

+

=1

 

Suffices for this invariant to hold

o

ver a larger number of iterations

Slide40

Multiplicative NoiseServer 1

 

 

Server 2

 

(x

1

)

 

 

x

1

x

2

+

=1

 

Noise from client

i

to server

j

not

zero-mean

Slide41

ClaimUnder suitable assumptions, servers eventually reach consensus in41

i

Slide42

Peer-to-Peer Architecture

 

 

 

 

 

Slide43

Reminder …Each agent maintains local estimate xConsensus step with neighborsApply own gradient to own estimate

 

 

 

 

 

 

Slide44

Proposed ApproachEach agent shares noisy estimate with neighborsScheme 1 – Noise cancels over neighborsScheme 2 – Noise cancels network-wide

 

 

 

 

 

Slide45

Proposed ApproachEach agent shares noisy estimate with neighbors

Scheme 1 – Noise cancels over neighbors

Scheme 2 – Noise cancels network-wide

 

 

 

 

 

x

+ ε

1

x

+ ε

2

ε

1

+ ε

2

= 0 (over iterations)

Slide46

Peer-to-Peer ArchitecturePoster today

Shripad

Gade

Slide47

 

 

 

 

 

Distributed

Optimization

 

 

 

 

 

Fault-tolerance

 

 

 

 

 

Privacy

i

Outline

Slide48

Fault-ToleranceSome agents may be faultyNeed to produce “correct” output despite the faults48

Slide49

Byzantine Fault ModelNo constraint on misbehavior of a faulty agentMay send bogus messagesFaulty agents can collude49

Slide50

Peer-to-Peer Architecturefi(x) = cost for robot i to go to location x

Faulty agent may

choose

arbitrary

cost function

x

f

1

(x)

f

2

(x)

x

1

x

2

Slide51

Peer-to-Peer Architecture51

 

 

 

 

 

Slide52

 

Server

 

 

Client-Server Architecture

 

Slide53

Fault-Tolerant OptimizationThe original problem is not meaningful53

i

Slide54

Fault-Tolerant OptimizationThe original problem is not meaningfulOptimize cost over only non-faulty agents

i

i

good

Slide55

Fault-Tolerant OptimizationThe original problem is not meaningfulOptimize cost over only non-faulty agents

i

i

good

Impossible!

Slide56

Fault-Tolerant OptimizationOptimize weighted cost over only non-faulty agentsWith 𝛂i

as close to

1/ good

as possible

i

good

𝛂

i

Slide57

Fault-Tolerant OptimizationOptimize weighted cost over only non-faulty agents

i

good

𝛂

i

With

t

Byzantine faulty agents:

t

weights may be

0

Slide58

Fault-Tolerant OptimizationOptimize weighted cost over only non-faulty agents

i

good

𝛂

i

t

Byzantine agents, n total agents

At least n-2t weights guaranteed to be > 1/2(n-t)

Slide59

Centralized AlgorithmOf the n agents, any t may be faultyHow to filter cost functions of faulty agents?

X

Slide60

Centralized Algorithm: Scalar argument xDefine a virtual function G(x) whose gradient isobtained as follows60

Slide61

Define a virtual function G(x) whose gradient isobtained as follows

At a given

x

Sort the gradients of the

n

local cost functions

61Centralized Algorithm: Scalar argument x

Slide62

Define a virtual function G(x) whose gradient isobtained as follows

At a given

x

Sort the gradients of the

n

local cost functions

Discard smallest t and largest t gradients62

Centralized Algorithm: Scalar argument x

Slide63

Define a virtual function G(x) whose gradient isobtained as follows

At a given

x

Sort the gradients of the

n

local cost functions

Discard smallest t and largest t gradientsMean of remaining gradients = Gradient of G at x

63Centralized Algorithm: Scalar argument x

Slide64

Define a virtual function G(

x

) whose gradient is

obtained as follows

At a given

x

Sort the gradients of the

n local cost functions Discard smallest t and largest t gradientsMean of remaining gradients =

Gradient of G at x

Virtual function G(x) is convex

Centralized Algorithm:

Scalar argument

x

Slide65

Define a virtual function G(

x

) whose gradient is

obtained as follows

At a given

x

Sort the gradients of the

n local cost functions Discard smallest t and largest t gradientsMean of remaining gradients = Gradient of G at

xVirtual function G(

x) is convex 

Can optimize easily

Centralized Algorithm:

Scalar argument

x

Slide66

Gradient filtering similar to centralized algorithm … require “rich enough” connectivity … correlation between functions helpsVector case harder … redundancy between functions helps

66

Peer-to-Peer Fault-Tolerant

Optimization

Slide67

 

 

 

 

 

Distributed

Optimization

 

 

 

 

 

Fault-tolerance

 

 

 

 

 

Privacy

i

Summary

Slide68

Thanks! disc.ece.illinois.edu

Slide69

69

Slide70

70

Slide71

Distributed Peer-to-Peer OptimizationEach agent maintains local estimate x

In each

iteration

Compute weighted average with neighbors’ estimates

 

 

 

 

 

Slide72

Distributed Peer-to-Peer OptimizationEach agent maintains local estimate x

In each

iteration

Compute weighted average with neighbors’ estimates

Apply

own gradient

to own estimate

 

 

 

 

 

 

Slide73

Distributed Peer-to-Peer Optimization

Each agent maintains local estimate

x

In each

iteration

Compute weighted average with neighbors’ estimates

Apply

own gradient to own estimateLocal estimates converge to

 

 

 

 

 

i

 

Slide74

74

RSS – Locally Balanced

Perturbations

Add to zero (locally per node)

Bounded (

)

Algorithm

Node j selects

such that

and

Share

with node

i

Consensus and (Stochastic) Gradient Descent

 

Slide75

75

RSS – Network Balanced

Perturbations

Add to zero (over network)

Bounded (

)

Algorithm

Node j computes perturbation

- sends

to

i

- add received

and subtract sent

Obfuscate state

shared with neighbors

Consensus and (Stochastic) Gradient Descent

 

Slide76

ConvergenceLet

and

Asymptotic convergence of iterates to optimum

Privacy-Convergence Trade-off

Stochastic gradient updates

work too

 

76

Slide77

Function SharingLet be bounded degree polynomials AlgorithmNode j shares

with node

i

Node j obfuscates using

Use

and use distributed gradient descent

 

77

Slide78

Function Sharing - ConvergenceFunction Sharing iterates converge to correct optimum (

)

Privacy:

If vertex connectivity of graph

f then no group of f nodes can estimate true functions

(or any good subset)

is also similar to

then it can hide

well

 

78

Slide79


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