strategy-proof classification

Strategy-Proof Classification

Reshef MeirSchool of Computer Science and Engineering, Hebrew University

A joint work with Ariel. D. Procaccia and Jeffrey S. Rosenschein

Strategy-Proof Classification

• Introduction– Learning and Classification– An Example of Strategic Behavior

• Motivation:– Decision Making– Machine Learning

• Our Model• Some Results

ClassificationThe Supervised Classification problem:

– Input: a set of labeled data points {(xi,yi)}i=1..m

– output: a classifier c from some predefined concept class C ( functions of the form f : X{-,+} )

– We usually want c to classify correctly not just the sample, but to generalize well, i.e .to minimize

Risk(c) ≡ E(x,y)~D[ L(c(x)≠y) ] ,Where D is the distribution from which we sampled the

training data, L is some loss function.

Motivation Model ResultsIntroduction

Classification (cont.)• A common approach is to return the ERM, i.e.

the concept in C that is the best w.r.t. the given samples (a.k.a. training data)– Try to approximate it if finding it is hard

• Works well under some assumptions on the concept class C

Should we do the same with many experts?


ERM

Motivation Model Results

Strategic labeling: an exampleIntroduction

5 errors

There is a better classifier! (for me…)


If I will only change the

labels…


2+4 = 6 errors

-

Decision making

• ECB makes decisions based on reports from national banks

• National bankers gather positive/negative data from local institutions

• Each country reports to ECB• Yes/no decision taken at

European level

• Bankers might misreport their data in order to sway the central decision

Introduction Model ResultsMotivation

Labels

Managers Reported Dataset

Classification AlgorithmClassifier (Spam filter)

Outlook

Introduction Model Results

Machine Learning (spam filter)

Motivation

Learning (cont.)

• Some e-mails may be considered spam by certain managers, and relevant by others

• A manager might misreport labels to bias the final classifier towards her point-of-view

Introduction Model ResultsMotivation

A Problem is characterized by• An input space X• A set of classifiers (concept class) C Every classifier c C is a function c: X{+,-}• Optional assumptions and restrictions

• Example 1: All Linear Separators in Rn

• Example 2: All subsets of a finite set Q

Introduction Motivation ResultsModel

A problem instance is defined by

• Set of agents I = {1,...,n}• A partial dataset for each agent i I,

Xi = {xi1,...,xi,m(i)} X• For each xikXi agent i has a label yik{,}

– Each pair sik=xik,yik is an example– All examples of a single agent compose the labeled dataset

Si = {si1,...,si,m(i)} • The joint dataset S= S1 , S2 ,…, Sn is our input

– m=|S|• We denote the dataset with the reported labels by S’


Input: Example

+–– +

––

–– –

++ ++ ++

–

X1 Xm1 X2 Xm2 X3 Xm3

Y1 {-,+}m1 Y2 {-,+}m2 Y3 {-,+}m3

S = S1, S2,…, Sn = (X1,Y1),…, (Xn,Yn)


Mechanisms

• A Mechanism M receives a labeled dataset S’ and outputs c C

• Private risk of i: Ri(c,S) = |{k: c(xik) yik}| / mi

• Global risk: R(c,S) = |{i,k: c(xik) yik}| / m

• We allow non-deterministic mechanisms– The outcome is a random variable– Measure the expected risk


ERM

We compare the outcome of M to the ERM:c* = ERM(S) = argmin(R(c),S)r* = R(c*,S)

c C

Can our mechanism simply compute and return the ERM?


Requirements

1. Good approximation: S R(M(S),S) ≤ β∙r*

2. Strategy-Proofness: i,S,Si‘ Ri(M(S-i , Si‘),S) ≤ Ri(M(S),S)

• ERM(S) is 1-approximating but not SP• ERM(S1) is SP but gives bad approximation

Are there any mechanisms

that guarantee both SP and

good approximation?


Suppose |C|=2

• Like in the ECB example• There is a trivial deterministic SP 3-

approximation mechanism• Theorem:

There are no deterministic SP α-approximation mechanisms, for any α<3

R. Meir, A. D. Procaccia and J. S. Rosenschein, Incentive Compatible Classification under Constant Hypotheses: A Tale of Two Functions, AAAI 2008

Introduction Motivation Model Results

ProofC = {“all positive”, “all negative”}



Randomization comes to the rescue

• There is a randomized SP 2-approximation mechanism (when |C|=2)– Randomization is non-trivial

• Once again, there is no better SP mechanism



Negative results

• Theorem: There are concept classes (including linear separators), for which there are no SP mechanisms with constant approximation

• Proof idea: – we first construct a classification problem that is

equivalent to a voting problem– Then use impossibility results from Social-Choice

theory to prove that there must be a dictator


R. Meir, A. D. Procaccia and J. S. Rosenschein, On the Power of Dictatorial Classification, in submission.

More positive results • Suppose all agents control the same data points,

i.e. X1 = X2 =…= Xn

• Theorem: Selecting a dictator at random is SP and guarantees 3-approximation– True for any concept class C– 2-approximation when each Si is separable


R. Meir, A. D. Procaccia and J. S. Rosenschein, Incentive Compatible Classification with Shared Inputs, in submission.

Proof ideaIntroduction Motivation Model Results

The average pair-wise distance between green dots, cannot be much higher than the average distance to the star

Generalization

• So far, we only compared our results to the ERM, i.e. to the data at hand

• We want learning algorithms that can generalize well from sampled data– with minimal strategic bias– Can we ask for SP algorithms?


Generalization (cont.)• There is a fixed distribution DX on X• Each agent holds a private function Yi : X {+,-}

– Possibly non-deterministic• The algorithm is allowed to sample from DX and ask

agents for their labels• We evaluate the result vs. the optimal risk, averaging over

all agents, i.e.

Introduction Motivation Model ResultsResultsModel

n

iiDxCcopt xxYxcr

X1

~ |)()(Prinf: E

Generalization (cont.)Introduction Motivation Model ResultsResultsModel

DX Y1Y3

Y2

Generalization Mechanisms

Our mechanism is used as follows:1. Sample m data points i.i.d2. Ask agents for their labels3. Use the SP mechanism on the labeled data, and

return the result

• Does it work? – Depends on our game-theoretic and learning-

theoretic assumptions


The “truthful approach”

• Assumption A: Agents do not lie unless they gain at least ε

• Theorem: W.h.p. the following occurs– There is no ε-beneficial lie– Approximation ratio (if no one lies) is close to 3

• Corollary: with enough samples, the expected approximation ratio is close to 3

• The number of required samples is polynomial in n and 1/ε



The “Rational approach”

• Assumption B: Agents always pick a dominant strategy, if one exists.

• Theorem: with enough samples, the expected approximation ratio is close to 3

• The number of required samples is polynomial in 1/ε (and not on n)



Previous and future work• A study of SP mechanisms in Regression learning 1

• No SP mechanisms for Clustering 2

Future directions• Other concept classes• Other loss functions• Alternative assumptions on structure of data

1 O. Dekel, F. Fischer and A. D. Procaccia, Incentive Compatible Regression Learning, SODA 20082 J. Perote-Peña and J. Perote. The impossibility of strategy-proof clustering, EconomicsBulletin, 2003.


strategy-proof classification

Documents

classifier c c

function c

outputs c c private

classifiers concept

xikxi agent i

joint dataset s

labeled dataset si

partial dataset