Incentive Compatible Regression Learning Ofer Dekel, Felix A. Fischer and Ariel D. Procaccia
Dec 20, 2015
Incentive Compatible Regression LearningOfer Dekel, Felix A. Fischer and Ariel D. Procaccia
Lecture Outline
• Until now: applications of learning to game theory. Now: merge.
• The model:– Motivation– The learning game
• Three levels of generality:– Distributions which are degenerate at one
point– Uniform distributions– The general setting
Model Degenerate Uniform General
Motivation
• Internet search company: improve performance by learning ranking function from examples.
• Ranking function assigns real value to every (query,answer).
• Employ experts to evaluate examples. • Different experts may have diff.
interests and diff. ideas of good output. • Conflict Manipulation Bias in
training set.
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Jaguar vs. Panthera Onca
(“Ja
gu
ar”
, ja
gu
ar.
com
)
Model Degenerate Uniform General
Regression Learning
• Input space X=Rk ((query,answer) pairs).• Function class F:XR (ranking functions). • Target function o:XR.• Distribution over X.• Loss function l (a,b).
– Abs. loss: l (a,b)=|a-b|.– Squared loss: l (a,b)=(a-b)2.
• Learning process: – Given: Training set S={(xi,o(xi))}, i=1,...,m, xi
sampled from .
– R(h)=Ex[l (h(x),o(x))].
– Find: hF to minimize R(h).
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Our Setting
• Input space X=Rk ((query,answer) pairs).
• Function class F (ranking functions). • Set of players N={1,...,n} (experts).
• Target functions oi:XR.
• Distributions i over X.
• Training set?
Model Degenerate Uniform General
The Learning Game
i: controls xij, j=1,...,m, sampled w.r.t. i (common knowledge).
• Private info of i: oi(xij)=yij, j=1,...,m.
• Strategies of i: y’ij, j=1,...,m.
• h is obtained by learning S={(xij,y’ij)}
• Cost of i: Ri(h)=Exi [l (h(x),oi(x))].
• Goal: Social Welfare (please avg. player).
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Example: The learning game with ERM• Parameters: X=R, F=Constant Functions, l (a,b)=|a-
b|, N={1,2}, o1(x)=1, o2(x)=2, 1=2=uniform dist on [0,1000].
• Learning algorithm: Empirical Risk Minimization (ERM)
– Minimize R’(h,S)=1/|S| (x,y)Sl (h(x),y).
1
2
Model Degenerate Uniform General
Degenerate Distributions: ERM with abs. loss
• The Game:– Players: N={1,...n} i: degenerate at xi.
i: controls xi.
– Private info of i: oi(xi)=yi.
– Strategies of i: y’i.
– Cost of i: Ri(h)= l (h(xi),yi).
• Theorem: If l = absolute loss and F is convex. Then ERM is group incentive compatible.
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ERM with superlinear loss
• Theorem: l is “superlinear”, F is convex, |F|2, F is not “full” on x1,...,xn. Then y1,...,yn such that there is incentive to lie.
• Example: X=R, F=Constant Functions, l (a,b)=(a-b)2, N={1,2}.
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Uniform dist. over samples
• The Game:– Players: N={1,...n} i: Discrete uniform on {xi1,...,xim}
i: controls xij, j=1,...,m
– Private info of i: oi(xij)=yij.
– Strategies of i: y’ij, j=1,...,m.
– Cost of i: Ri(h)= R’i(h,S)= 1/mjl (h(xij),yij).
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ERM with abs. loss is not ICModel Degenerate Uniform General
1
0
VCG to the Rescue
• Use ERM.
• Each player pays jiR’j(h,S).
• Each player’s total cost is R’i(h,S)+jiRj’(h,S) = jR’j(h,S).
• Truthful for any loss function.• VCG has many faults:
– Not group incentive compatible. – Payments problematic in practice.
• Would like (group) IC mechanisms w/o payments.
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Mechanisms w/o Payments
• Absolute loss. -approximation mechanism: gives an -
approximation of the social welfare.• Theorem (upper bound): There exists a
group IC 3-approx mechanism for constant functions over Rk and homogeneous linear functions over R.
• Theorem (lower bound): There is no IC (3-)-approx mechanism for constant/hom. lin. functions over Rk.
• Conjecture: There is no IC mechanism with bounded approx. ratio for hom. lin. functions over Rk, k2.
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Generalization
• Theorem: If f,– (1) i, |R’i(f,S)-Ri(f)| /2 – (2) |R’(f,S)-1/ni Ri(f)| /2
Then:– (Group) IC in uniform -(group) IC in general. -approx in uniform -approx up to additive
in general. • If F has bounded complexity,
m=(log(1/)/), then cond. (1) holds with prob. 1-.
• Cond. (2) is obtained if (1) occurs for all i. Taking /n adds factor of logn.
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Discussion
• Given m large enough, with prob. 1- VCG is -truthful. This holds for any loss function.
• Given m large enough, abs loss, mechanism w/o payments which is -group IC and 3-approx for constant functions and hom. lin. functions.
• Most important direction for future work: extending to other models of learning, such as classification.
Model Degenerate Uniform General