Distributed Machine Learning: Communication, Efficiency, and

Privacy

Avrim Blum

[RaviKannan60]

Joint work with Maria-Florina Balcan, Shai Fine, and Yishay Mansour

Carnegie Mellon University

Happy birthday Ravi!

And thank you for many enjoyable years working

togetheron challenging problems where

machine learning meets high-dimensional geometry

This talkAlgorithms for machine learning in distributed,

cloud-computing context.

Related to interest of Ravi’s in algorithms for cloud-computing.

For full details see [Balcan-B-Fine-Mansour COLT’12]

Machine LearningWhat is Machine Learning about?

– Making useful, accurate generalizations or predictions from data.

– Given access to sample of some population, classified in some way, want to learn some rule that will have high accuracy over population as a whole.

Typical ML problems:

Given sample of images, classified as male or female, learn a rule to classify new images.

Machine LearningWhat is Machine Learning about?

– Making useful, accurate generalizations or predictions from data.

– Given access to sample of some population, classified in some way, want to learn some rule that will have high accuracy over population as a whole.

Typical ML problems:

Given set of protein sequences, labeled by function, learn rule to predict functions of new proteins.

Distributed Learning

Many ML problems today involve massive amounts of data distributed across multiple locations.

Distributed Learning

Many ML problems today involve massive amounts of data distributed across multiple locations.

Click data

Distributed Learning

Many ML problems today involve massive amounts of data distributed across multiple locations.

Customer data

Distributed Learning

Many ML problems today involve massive amounts of data distributed across multiple locations.

Scientific data

Distributed Learning

Many ML problems today involve massive amounts of data distributed across multiple locations.

Each has only a piece of the overall

data pie

Distributed Learning

Many ML problems today involve massive amounts of data distributed across multiple locations.

In order to learn over the combined D,

holders will need to communicate.

Distributed Learning

Many ML problems today involve massive amounts of data distributed across multiple locations.

Classic ML question: how much data is needed to learn a

given type of function well?

Distributed Learning

Many ML problems today involve massive amounts of data distributed across multiple locations.

These settings bring up a new question:

how much communication?Plus issues like privacy, etc.

That is the focus of this talk.

Distributed Learning: Scenarios

Two natural high-level scenarios:1. Each location has data from same distribution.

– So each could in principle learn on its own.– But want to use limited communication to speed up

– ideally to centralized learning rate. [Dekel, Giliad-Bachrach, Shamir, Xiao]

2. Overall distribution arbitrarily partitioned.– Learning without communication is impossible.– This will be our focus here.

The distributed PAC learning model• Goal is to learn unknown function f 2 C

given labeled data from some prob. distribution D.

• However, D is arbitrarily partitioned among k entities (players) 1,2,…,k. [k=2 is interesting]

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The distributed PAC learning model• Goal is to learn unknown function f 2 C

given labeled data from some prob. distribution D.

• However, D is arbitrarily partitioned among k entities (players) 1,2,…,k. [k=2 is interesting]

• Players can sample (x,f(x)) from their own Di.

1 2 … kD1 D2 … Dk

D = (D1 + D2 + … + Dk)/k

The distributed PAC learning model• Goal is to learn unknown function f 2 C

given labeled data from some prob. distribution D.

• However, D is arbitrarily partitioned among k entities (players) 1,2,…,k. [k=2 is interesting]

• Players can sample (x,f(x)) from their own Di.

1 2 … kD1 D2 … Dk

Goal: learn good rule over combined D.

The distributed PAC learning model

Interesting special case to think about:– k=2.– One has the positives and one has the

negatives.– How much communication to learn, e.g., a

good linear separator?

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– In general, view k as small compared to sample size needed for learning.

The distributed PAC learning model

Some simple baselines.• Baseline #1: based on fact that can learn any

class of VC-dim d to error ² from O(d/² log 1/²) samples– Each player sends 1/k fraction of this to

player 1– Player 1 finds good rule h over sample. Sends

h to others.– Total: 1 round, O(d/² log 1/²) examples sent.

D1 D2 … Dk

The distributed PAC learning model

Some simple baselines.• Baseline #2: Suppose function class has an

online algorithm A with mistake-bound M. E.g., Perceptron algorithm learns linear separators of margin ° with mistake-bound O(1/°2).

D1 D2 … Dk

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The distributed PAC learning model

Some simple baselines.• Baseline #2: Suppose function class has an

online algorithm A with mistake-bound M. – Player 1 runs A, broadcasts current

hypothesis.– If any player has a counterexample, sends to

player 1. Player 1 updates, re-broadcasts.– At most M examples and rules

communicated.

D1 D2 … Dk

Dependence on 1/²

Had linear dependence in d and 1/², or M and no dependence on 1/². [² = final error rate]• Can you get O(d log 1/²) examples of

communication?• Yes.

Distributed boosting

D1 D2 … Dk

Distributed Boosting

Idea: • Run baseline #1 for ² = ¼. [everyone sends a

small amount of data to player 1, enough to learn to error ¼]

• Get initial rule h1, send to others.

D1 D2 … Dk

Distributed Boosting

Idea: • Players then reweight their Di to focus on

regions h1 did poorly.• Repeat

D1 D2 … Dk

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• Distributed implementation of Adaboost Algorithm.

• Some additional low-order communication needed too (players send current performance level to #1, so can request more data from players where h doing badly).

• Key point: each round uses only O(d) samples and lowers error multiplicatively.

Distributed Boosting

Final result: • O(d) examples of communication per

round + low order extra bits.• O(log 1/²) rounds of communication.• So, O(d log 1/²) examples of

communication in total plus low order extra info.

D1 D2 … Dk

Agnostic learning (no perfect h)

[Balcan-Hanneke] give robust halving alg that can be implemented in distributed setting.• Based on analysis of a generalized active

learning model.• Algorithms especially suited to distributed

setting.

D1 D2 … Dk

Agnostic learning (no perfect h)

[Balcan-Hanneke] give robust halving alg that can be implemented in distributed setting.• Get error 2*OPT(C) + ² using total of only

O(k log|C| log(1/²)) examples.

• Not computationally efficient, but says logarithmic dependence possible in principle.

D1 D2 … Dk

Can we do better for specific classes of functions?

D1 D2 … Dk

Interesting class: parity functions

Examples x 2 {0,1}d. f(x) = x¢vf mod 2, for unknown vf.• Interesting for k=2.• Classic communication LB for

determining if two subspaces intersect. • Implies (d2) bits LB to output good v.• What if allow rules that “look different”?

D1 D2 … Dk

D1 D2

Interesting class: parity functions

Examples x 2 {0,1}d. f(x) = x¢vf mod 2, for unknown vf.• Parity has interesting property that:

(a) Can be learned using . [Given dataset S of size O(d/²), just solve the linear system]

(b) Can be learned using in reliable-useful model of Rivest-Sloan’88. [if x in subspace spanned by S, predict accordingly, else say “??”]

S vector vh

S

xf(x)

??

D1 D2

Interesting class: parity functions

Examples x 2 {0,1}d. f(x) = x¢vf mod 2, for unknown vf.• Algorithm:

– Each player i PAC-learns over Di to get parity function gi. Also R-U learns to get rule hi. Sends gi to other player.

– Uses rule: “if hi predicts, use it; else use g3-i.”– Can one extend to k=3?

g1

h1

g2

h2

Linear Separators

Can one do better?

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Linear separators thru origin. (can assume pts on sphere)• Say we have a near-uniform prob. distrib. D over Sd.• VC-bound, margin bound, Perceptron mistake-bound all

give O(d) examples needed to learn, so O(d) examples of communication using baselines (for constant k, ²).

Linear Separators

Idea: Use margin-version of Perceptron alg [update until f(x)(w ¢ x) ¸ 1 for all x] and run round-robin.

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Linear Separators

Idea: Use margin-version of Perceptron alg [update until f(x)(w ¢ x) ¸ 1 for all x] and run round-robin.• So long as examples xi of player i and xj of

player j are reasonably orthogonal, updates of player j don’t mess too much with data of player i.– Few updates ) no damage.– Many updates ) lots of progress!

Linear Separators

Idea: Use margin-version of Perceptron alg [update until f(x)(w ¢ x) ¸ 1 for all x] and run round-robin.• If overall distrib. D is near uniform [density

bounded by c¢unif], then total communication (for constant k, ²) is O((d log d)1/2) rather than O(d).

Get similar savings for general distributions?

Preserving Privacy of DataNatural also to consider privacy in this setting.• Data elements could be patient records,

customer records, click data.• Want to preserve privacy of individuals

involved.• Compelling notion of differential privacy: if

replace any one record with fake record, nobody else can tell. [Dwork, Nissim, …]S1 ~ D1 S2 ~ D2 … Sk ~ Dk

10110110111010111011001

Preserving Privacy of DataNatural also to consider privacy in this setting.

S1 ~ D1 S2 ~ D2 … Sk ~ Dk

10110110111010111011001

¼ 1-² ¼ 1+²

probability over randomness in A

For all sequences of interactions ¾, e-² · Pr(A(Si)=¾)/Pr(A(Si’)=¾) · e²

Preserving Privacy of DataNatural also to consider privacy in this setting.

S1 ~ D1 S2 ~ D2 … Sk ~ Dk

10110110111010111011001

• A number of algorithms have been developed for differentially-private learning in centralized setting.

• Can ask how to maintain without increasing communication overhead.

Preserving Privacy of DataAnother notion that is natural to consider in this setting.

S1 ~ D1 S2 ~ D2 … Sk ~ Dk

• A kind of privacy for data holder.• View distrib Di as non-sensitive (statistical info

about population of people who are sick in city i).

• But the sample Si » Di is sensitive (actual patients).

• Reveal no more about Si other than inherent in Di?

Preserving Privacy of DataAnother notion that is natural to consider in this setting.

S1 ~ D1 S2 ~ D2 … Sk ~ Dk

Di SiProtocol

• Want to reveal no more info about Si than is inherent in Di.

Di Si

S’i

Protocol

PrSi,S’i[8¾, Pr(A(Si)=¾)/Pr(A(S’i)=¾) 2 1 § ²] ¸ 1 -

±.

Preserving Privacy of DataAnother notion that is natural to consider in this setting.

Actual sample

“Ghost sample” sample

Can get algorithms with this guarantee

Conclusions

As we move to large distributed datasets, communication issues become important.• Rather than only ask “how much data is needed

to learn well”, also ask “how much communication do we need?”

• Also issues like privacy become more critical.

Quite a number of open questions.