Download Machine Learning Game Theory

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
Introduction
Motivation
Model
Results
Classification
The 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.
Introduction
Motivation
Model
Results
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?
Introduction
Motivation
Model
Results
Strategic labeling: an example
ERM
5 errors
Introduction
Motivation
Model
Results
There is a better
classifier!
(for me…)
Introduction
Motivation
Model
Results
If I will only
change the
labels…
2+4 = 6 errors
Introduction
Motivation
Model
Results
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
Motivation
Model
Results
Machine Learning (spam filter)
Managers
Outlook
Reported Dataset
Labels
Classifier (Spam filter)
Classification
Algorithm
Introduction
Motivation
Model
Results
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
Motivation
Model
Results
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
Model
Results
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 xikXi 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’
Introduction
Motivation
Model
Results
Input: Example
–
–
–
–
+
+ –
–
+
–
+
+ +
+ +
–
Agent 1
Agent 2
Agent 3
X1  Xm1
Y1  {-,+}m1
X2  Xm2
Y2  {-,+}m2
X3  Xm3
Y3  {-,+}m3
S = S1, S2,…, Sn = (X1,Y1),…, (Xn,Yn)
Introduction
Motivation
Model
Results
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
Introduction
Motivation
Model
Results
ERM
We compare the outcome of M to the ERM:
c* = ERM(S) = argmin(R(c),S)
cC
r* = R(c*,S)
Can our mechanism
simply compute and
return the ERM?
Introduction
Motivation
Model
Results
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
Introduction
Motivation
Model
Results
Suppose |C|=2
• Like in the ECB example
• There is a trivial deterministic SP 3approximation 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
R. Meir, A. D. Procaccia and J. S. Rosenschein, On the Power of Dictatorial Classification, in submission.
Introduction
Motivation
Model
Results
More positive results 
• Suppose all agents control the same data
points, i.e. X1 = X2 =…= Xn
Agent 1
Agent 2
Agent 3
• 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.
Introduction
Motivation
Model
Results
Proof idea
The average pair-wise distance between green dots, cannot be
much higher than the average distance to the star
Introduction
Motivation
Model
Results
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
Model
Results
Results
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. n


ropt : inf E x ~ DX  Pr c( x)  Yi ( x) | x 
cC
 i 1

Introduction
Motivation
Model
Model
Results
Results
Generalization (cont.)
Y3
Y1
DX
Y2
Introduction
Motivation
Model
Results
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 learningtheoretic 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
R. Meir, A. D. Procaccia and J. S. Rosenschein, Incentive Compatible Classification with Shared
Inputs, in submission.
Introduction
Motivation
Model
Results
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)
R. Meir, A. D. Procaccia and J. S. Rosenschein, Incentive Compatible Classification with Shared
Inputs, in submission.
Introduction
Motivation
Model
Results
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.