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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

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

Slide2Shripad

Gade

Lili Su

Acknowledgements

Slide3i

Slide4Applicationsfi(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

Slide5Applications5

Minimize

cost

Σ

f

i(x)

i

f

1

(x)

f

2

(x)

f

4

(x)

f

3

(x)

Learning

Slide6Distributed

Optimization

Fault-tolerance

Privacy

i

Outline

Slide7Distributed Optimization7

Server

Client-Server Architecture8f

1

(x)

f

2

(x)

f

4

(x)

f

3

(x)

Server

Client-Server ArchitectureServer maintains estimate Client i knows

Server

Client-Server ArchitectureServer maintains estimate

Client

i

knows

In iteration k+1

Client

i

Download

from server

U

pload

gradient

Server

Client-Server Architecture

Server maintains estimate

Client

i

knows

In iteration k+1

Client

i

Download

from server

U

pload

gradient

Server

Server

VariationsStochasticAsynchronous…12

Slide13Peer-to-Peer Architecturef

1

(x)

f

2

(x)

f

4

(x)

f

3

(x)

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

Distributed

Optimization

Fault-tolerance

Privacy

i

Outline

Slide16Server

Server observes gradients privacy compromised

Server

Server

Achieve privacy and yet collaboratively optimize

Server observes gradients

privacy compromised

Slide19Related WorkCryptographic methods (homomorphic encryption)Function transformationDifferential privacy19

Slide20Differential Privacy20

Server

Differential Privacy21

Server

Trade-off privacy with accuracy

Slide22Proposed Approach22

Motivated by secret sharing

Exploit diversity

…

Multiple

servers

/

neighbors

Slide23Proposed Approach23

Server 1

Server 2

Privacy if

subset of servers

adversarial

Slide24Proposed Approach24

Privacy if

subset of neighbors

adversarial

Slide25Proposed ApproachStructured noise that “cancels” over servers/neighbors25

Slide26Intuition26

Server 1

Server 2

x

1

x

2

Slide27Intuition27

Server 1

Server 2

x

1

x

2

Each client

simulates

multiple clients

Slide28Intuition28

Server 1

Server 2

+

x

1

x

2

not necessarily convex

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

U

pload gradient of

Each server updates estimate using received gradients

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

ClaimUnder suitable assumptions, servers eventually reach consensus in31

i

Slide32Privacy32

Server 1

Server 2

+

+

+

+

PrivacyServer 1 may learn

,

+

+

Not sufficient to learn

33

Server 1

Server 2

+

+

+

+

Function splitting not necessarily practicalStructured randomization as an alternative34

+

Structured RandomizationMultiplicative or additive noise in gradientsNoise cancels over servers35

Slide36Multiplicative Noise36

Server 1

Server 2

x

1

x

2

Slide37Multiplicative Noise37

Server 1

Server 2

x

1

x

2

Slide38Multiplicative Noise38

Server 1

Server 2

(x

1

)

x

1

x

2

+

=1

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

Slide40Multiplicative NoiseServer 1

Server 2

(x

1

)

x

1

x

2

+

=1

Noise from client

i

to server

j

not

zero-mean

Slide41ClaimUnder suitable assumptions, servers eventually reach consensus in41

i

Slide42Peer-to-Peer Architecture

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

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

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)

Slide46Peer-to-Peer ArchitecturePoster today

Shripad

Gade

Slide47Distributed

Optimization

Fault-tolerance

Privacy

i

Outline

Slide48Fault-ToleranceSome agents may be faultyNeed to produce “correct” output despite the faults48

Slide49Byzantine Fault ModelNo constraint on misbehavior of a faulty agentMay send bogus messagesFaulty agents can collude49

Slide50Peer-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

Slide51Peer-to-Peer Architecture51

Server

Client-Server Architecture

Fault-Tolerant OptimizationThe original problem is not meaningful53

i

Slide54Fault-Tolerant OptimizationThe original problem is not meaningfulOptimize cost over only non-faulty agents

i

i

good

Slide55Fault-Tolerant OptimizationThe original problem is not meaningfulOptimize cost over only non-faulty agents

i

i

good

Impossible!

Slide56Fault-Tolerant OptimizationOptimize weighted cost over only non-faulty agentsWith 𝛂i

as close to

1/ good

as possible

i

good

𝛂

i

Slide57Fault-Tolerant OptimizationOptimize weighted cost over only non-faulty agents

i

good

𝛂

i

With

t

Byzantine faulty agents:

t

weights may be

0

Slide58Fault-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)

Slide59Centralized AlgorithmOf the n agents, any t may be faultyHow to filter cost functions of faulty agents?

X

Slide60Centralized Algorithm: Scalar argument xDefine a virtual function G(x) whose gradient isobtained as follows60

Slide61Define 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

Slide62Define 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

Slide63Define 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

Slide64Define 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

Slide65Define 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

Slide66Gradient 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

Slide67Distributed

Optimization

Fault-tolerance

Privacy

i

Summary

Slide68Thanks! disc.ece.illinois.edu

Slide6969

Slide7070

Slide71Distributed Peer-to-Peer OptimizationEach agent maintains local estimate x

In each

iteration

Compute weighted average with neighbors’ estimates

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

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

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

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

ConvergenceLet

and

Asymptotic convergence of iterates to optimum

Privacy-Convergence Trade-off

Stochastic gradient updates

work too

76

Slide77Function SharingLet be bounded degree polynomials AlgorithmNode j shares

with node

i

Node j obfuscates using

Use

and use distributed gradient descent

77

Slide78Function 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