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Strategy-Proof Classification
Reshef MeirSchool of Computer Science and Engineering, Hebrew University
A joint work with Ariel. D. Procaccia and Jeffrey S. Rosenschein
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Strategy-Proof Classification
• An Example of Strategic Labels in Classification• Motivation• Our Model• Previous work (positive results)
• An impossibility theoremAn impossibility theorem• More results (if there is time)More results (if there is time)
(~12 minutes)
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ERM
Motivation Model Results
Strategic labeling: an example
Introduction
5 errors
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There is a better classifier! (for me…)
Motivation Model ResultsIntroduction
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If I will only change the
labels…
Motivation Model ResultsIntroduction
2+4 = 6 errors
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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
R(c) ≡the expected number of errors w.r.t. the distribution D
Motivation ResultsIntroduction Model
E(x,y)~D[ c(x)≠y ]
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Classification (cont.)• A common approach is to return the ERMERM, i.e.
the concept in C that is the best w.r.t. the given samples (has the lowest number of errors)
• Generalizes well under some assumptions on the concept class C
With multiple experts, we can’t trust our ERM!
Motivation ResultsIntroduction Model
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Where do we find “experts” with incentives?
Example 1: A firm learning purchase patterns– Information gathered from local retailers– The resulting policy affects them – “the best policy, is the policy that fits my pattern”
Introduction Model ResultsMotivation
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Users Reported Dataset
Classification AlgorithmClassifier
Introduction Model Results
Example 2: Internet polls / expert systems
Motivation
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Related work• A study of SP mechanisms in Regression learning
– O. Dekel, F. Fischer and A. D. Procaccia, Incentive Compatible Regression Learning, SODA 2008
• No SP mechanisms for Clustering
– J. Perote-Peña and J. Perote. The impossibility of strategy-proof clustering, Economics Bulletin, 2003
Introduction Motivation Model Results
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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’
Introduction Motivation ResultsModel
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Input: Example
++–––– ++
––––
––––––
++++ ++++ ++++
––
X1 Xm1 X2 Xm2 X3 Xm3
Y1 -,+m1 Y2 -,+m2 Y3 -,+m3
S = S1, S2,…, Sn = (X1,Y1),…, (Xn,Yn)
Introduction Motivation ResultsModel
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Incentives and 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 ResultsModel
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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 ResultsModel
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Requirements
1. Good approximation: S R(M(S),S) ≤ β∙r*
2. Strategy-Proofness (SP): 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?
Introduction Motivation ResultsModel
MOST IMPORTANT
SLIDE
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Restricted settings• A very small concept class: |C| = 2
– There is a deterministic SP mechanism that obtains a 3-approximation ratio
– This bound is tight– Randomization can improve the bound to 2
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
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Restricted settings (cont.)• Agents with similar interests:
– There is a randomized SP 3-approximation mechanism (works for any class C)
Introduction Motivation Model Results
R. Meir, A. D. Procaccia and J. S. Rosenschein, Incentive Compatible Classification with Shared Inputs, IJCAI 2009.
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But not everything shines
• Without restrictions on the input, we cannot guarantee a constant approximation ratio
Our main result:Theorem: There is a concept class C, for which
there are no deterministic SP mechanisms with o(m)-approximation ratio
Introduction Motivation Model Results
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Deterministic lower bound
Proof idea: – First construct a classification problem that is
equivalent to a voting problem with 3 candidates
– Then use the Gibbard-Satterthwaite theorem to prove that there must be a dictator
– Finally, the dictator’s opinion might be very far from the optimal classification
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Proof (1)
Construction: We have X=a,b, and 3 classifiers as follows
The dataset contains two types of agents, with samples distributed unevenly over a and b
Introduction Motivation Model Results
We do not set the labels.
Instead, we denote by Y all the possible labelings of an agent’s dataset.
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Proof (2)Let P be the set of all 6 orders over C A voting rule is a function of the form f: Pn CBut our mechanism is a function M: Yn C !
(its input are labels and not orders)
Lemma 1: there is a valid mapping g: Pn Yn, s.t. (M*g) is a voting rule
Introduction Motivation Model Results
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Proof (3)Lemma 2: If M is SP, and guarantees any bounded
approximation ratio, then f=M*g is dictatorialProof: (f is onto) any profile that c classifies perfectly
must induce the selection of c
(f is SP) suppose there is a manipulationBy mapping this profile to labels with g, we find a
manipulation of M, in contradiction to its SP
From the G-S theorem, f must be dictatorial
Introduction Motivation Model Results
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Proof (4)Introduction Motivation Model Results
Finally, f (and thus M) can only be dictatorial. We assume w.l.o.g. that the dictator is agent 1 of
type Ia. We now label the data points as follows:
The optimal classifier is cab, which makes 2 errors
The dictator selects ca, which makes m/2 errors
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Real concept classesIntroduction Motivation Model Results
• We managed to show that there are no good (deterministic) SP mechanisms, but only for a synthetically constructed class.
• We are interested in more common classes, that are really used in machine learning. For example:
• Linear Classifiers• Boolean Conjunctions
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Linear classifiers
Only 2 errors
Introduction Motivation Model Results
“b”
cacb
cab
“a”
Ω(√m) errors
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A lower bound for randomized SP mechanisms
• A lottery over dictatorships is still bad– Ω(k) instead of Ω(m), where k is the size of the
largest dataset controlled by an agent ( m ≈ k*n )
• However, it is not clear how to eliminate other mechanisms – G-S works only for deterministic mechanisms– Another theorem by Gibbard [’79] can help
• But only under additional assumptions
Introduction Motivation Model Results
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Upper bounds
• So, our lower bounds do not leave much hope for good SP mechanisms
• We would still like to know if they are tight
A deterministic SP O(m)-approximation is easy:– break ties iteratively according to dictators
What about randomized SP O(k) mechanisms?
Introduction Motivation Model Results
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The iterative random dictator (IRD)
(example with linear classifiers on R1)
Introduction Motivation Model Results
v v
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The iterative random dictator (IRD)
(example with linear classifiers on R1)
Introduction Motivation Model Results
v v
Iteration 1: 2 errors
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The iterative random dictator (IRD)
(example with linear classifiers on R1)
Introduction Motivation Model Results
v v
Iteration 1: 2 errorsIteration 2: 5 errorsIteration 3: 0 errors
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The iterative random dictator (IRD)
(example with linear classifiers on R1)
Introduction Motivation Model Results
v v
Iteration 1: 2 errorsIteration 2: 5 errorsIteration 3: 0 errorsIteration 4: 0 errors
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The iterative random dictator (IRD)
(example with linear classifiers on R1)
Introduction Motivation Model Results
v v
Iteration 1: 2 errorsIteration 2: 5 errorsIteration 3: 0 errorsIteration 4: 0 errorsIteration 5: 1 error
Theorem: The IRD is O(k2) approximating for Linear Classifiers in R1
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Future work• Other concept classes
• Other loss functions
• Alternative assumptions on structure of data
• Other models of strategic behavior
• …
Introduction Motivation Model Results
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