Privacy and Fault-Tolerance PowerPoint Presentation, PPT - DocSlides

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

Shripad

Gade

Lili Su

Acknowledgements

i

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

Applications5

Minimize

cost

Σ

f

i(x)

i

f

1

(x)

f

2

(x)

f

4

(x)

f

3

(x)

Learning

Distributed

Optimization

Fault-tolerance

Privacy

i

Outline

Distributed 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

Peer-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

Server

Server observes gradients  privacy compromised

Server

Server

Achieve privacy and yet collaboratively optimize

Server observes gradients

 privacy compromised

Related WorkCryptographic methods (homomorphic encryption)Function transformationDifferential privacy19

Differential Privacy20

Server

Differential Privacy21

Server

Trade-off privacy with accuracy

Proposed Approach22

Motivated by secret sharing

Exploit diversity

…

Multiple

servers

/

neighbors

Proposed Approach23

Server 1

Server 2

Privacy if

subset of servers

adversarial

Proposed Approach24

Privacy if

subset of neighbors

adversarial

Proposed ApproachStructured noise that “cancels” over servers/neighbors25

Intuition26

Server 1

Server 2

x

1

x

2

Intuition27

Server 1

Server 2

x

1

x

2

Each client

simulates

multiple clients

Intuition28

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

Privacy32

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

Multiplicative Noise36

Server 1

Server 2

x

1

x

2

Multiplicative Noise37

Server 1

Server 2

x

1

x

2

Multiplicative 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

Multiplicative NoiseServer 1

Server 2

(x

1

)

x

1

x

2

+

=1

Noise from client

i

to server

j

not

zero-mean

ClaimUnder suitable assumptions, servers eventually reach consensus in41

i

Peer-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)

Peer-to-Peer ArchitecturePoster today

Shripad

Gade

Distributed

Optimization

Fault-tolerance

Privacy

i

Outline

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

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

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

Peer-to-Peer Architecture51

Server

Client-Server Architecture

Fault-Tolerant OptimizationThe original problem is not meaningful53

i

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

i

i

good

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

i

i

good

Impossible!

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

as close to

1/ good

as possible

i

good

𝛂

i

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

i

good

𝛂

i

With

t

Byzantine faulty agents:

t

weights may be

0

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)

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

X

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

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

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

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

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

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

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

Distributed

Optimization

Fault-tolerance

Privacy

i

Summary

Thanks! disc.ece.illinois.edu

69

70

Distributed 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

Function SharingLet be bounded degree polynomials AlgorithmNode j shares

with node

i

Node j obfuscates using

Use

and use distributed gradient descent

77

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