Machine Learning  Game Theory

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16 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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Strategy
-
Proof Classification

Reshef Meir

School 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

Classification

The
Supervised Classification

problem:


Input
: a set of labeled data points {(x
i
,y
i
)}
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

Results

Introduction

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?

Motivation

Model

Results

Introduction

ERM

Motivation

Model

Results

Strategic labeling: an example

Introduction

5 errors

There is a better
classifier!

(for me…)

Motivation

Model

Results

Introduction

If I will only
change the
labels…

Motivation

Model

Results

Introduction

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

Results

Motivation

Labels

Managers

Reported Dataset

Classification


Algorithm

Classifier (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

Results

Motivation

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
R
n


Example
2
: All subsets of a finite set Q





Introduction

Motivation

Results

Model

A problem
instance

is defined by



Set of
agents

I

= {
1
,...,n}


A partial dataset for each agent i




I
,



X
i

= {x
i
1
,...,x
i,m(i)
}


X


For each x
ik

X
i

agent i has a label y
ik

{

,

}


Each pair s
ik=

x
ik
,y
ik


is an
example


All examples of a single agent compose the labeled dataset
S
i

= {s
i
1
,...,s
i,m(i)
}


The joint dataset
S
=

S
1

,
S
2

,…,
S
n


is our
input


m=|
S
|


We denote the dataset with the
reported
labels by
S’



Introduction

Motivation

Results

Model

Agent
1

Agent
2

Agent
3

Input: Example

+





+











+

+

+

+

+

+



X
1



X
m
1

X
2



X
m
2

X
3



X
m
3

Y
1



{
-
,+}
m
1

Y
2



{
-
,+}
m
2

Y
3



{
-
,+}
m
3

S

=

S
1
,
S
2
,…,
S
n


=

(X
1
,Y
1
),…,
(X
n
,Y
n
)



Introduction

Motivation

Results

Model

Mechanisms


A

Mechanism
M

receives a labeled dataset
S’

and outputs
c



C



Private risk of
i:
R
i
(
c
,S
) = |{k:
c
(x
ik
)


y
ik
}| / m
i


Global risk:
R
(
c
,S
) = |{i,k:
c
(x
ik
)


y
ik
}| / m



We allow non
-
deterministic mechanisms


The outcome is a random variable


Measure the
expected risk


Introduction

Motivation

Results

Model

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?

Introduction

Motivation

Results

Model

Requirements

1.
Good approximation:



S

R
(
M
(
S
),
S
) ≤
β

r*

2.
Strategy
-
Proofness:




i
,S,
S
i


R
i
(
M
(
S
-
i
, S
i

),
S
)


R
i
(
M
(
S
),
S
)




ERM(
S
) is
1
-
approximating but not SP


ERM(S
1
) is SP but gives bad approximation

Introduction

Motivation

Results

Model

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

Proof

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




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

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

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

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


Introduction

Motivation

Model

Results

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. X
1
= X
2
=…=

X
n






Theorem
: Selecting a dictator at random is SP
and guarantees
3
-
approximation


True for
any

concept class
C


2
-
approximation when each
S
i

is separable


Agent
1

Agent
2

Agent
3

Introduction

Motivation

Model

Results

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

Proof idea

Introduction

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?

Introduction

Motivation

Model

Results

Generalization (cont.)


There is a fixed distribution
D
X
on
X


Each agent holds a private function


Y
i
:
X


{+,
-
}


Possibly non
-
deterministic


The algorithm is allowed to sample from
D
X
and ask agents for their labels


We evaluate the result vs. the optimal risk,
averaging over all agents, i.e.

Introduction

Motivation

Model

Results

Results

Model

Generalization (cont.)

Introduction

Motivation

Model

Results

Results

Model

D
X

Y
1

Y
3

Y
2

Generalization Mechanisms

Our mechanism is used as follows:

1.
Sample
m

data points i.i.d

2.
Ask agents for their labels

3.
Use the SP mechanism on the labeled data, and
return the result



Does it work?


Depends on our game
-
theoretic and learning
-
theoretic assumptions

Introduction

Motivation

Model

Results

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

Introduction

Motivation

Model

Results

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

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
)

Introduction

Motivation

Model

Results

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

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
2008

2

J. Perote
-
Peña and J. Perote.
The impossibility of strategy
-
proof clustering,

Economics

Bulletin,
2003
.

Introduction

Motivation

Model

Results