Incentive Compatible Regression
Learning
Ofer Dekel, Felix A. Fischer and Ariel D. Procaccia
Lecture Outline
Model
Degenerate
Uniform
General
• 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
Motivation
Model
Degenerate
Uniform
General
• 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.
Jaguar vs. Panthera Onca
Degenerate
Uniform
(“Jaguar”, jaguar.com)
Model
General
Regression Learning
Model
•
•
•
•
•
Degenerate
Uniform
General
Input space X=Rk ((query,answer) pairs).
Function class F:XR (ranking functions).
Target function o:XR.
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: hF to minimize R(h).
Our Setting
Model
•
•
•
•
•
•
Degenerate
Uniform
General
Input space X=Rk ((query,answer) pairs).
Function class F (ranking functions).
Set of players N={1,...,n} (experts).
Target functions oi:XR.
Distributions i over X.
Training set?
The Learning Game
Model
Degenerate
Uniform
General
• 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)=Exi [l (h(x),oi(x))].
• Goal: Social Welfare (please avg. player).
Example: The learning game with ERM
Model
Degenerate
Uniform
General
• 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).
2
1
Degenerate Distributions: ERM with abs. loss
Model
Degenerate
Uniform
General
• 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.
ERM with superlinear loss
Model
Degenerate
Uniform
General
• 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}.
Uniform dist. over samples
Model
Degenerate
Uniform
• 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/mjl (h(xij),yij).
General
ERM with abs. loss is not IC
Model
1
0
Degenerate
Uniform
General
VCG to the Rescue
Model
Degenerate
Uniform
General
• Use ERM.
• Each player pays jiR’j(h,S).
• Each player’s total cost is
R’i(h,S)+jiRj’(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.
Mechanisms w/o Payments
Model
Degenerate
Uniform
General
• 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, k2.
Generalization
Model
Degenerate
Uniform
General
• Theorem: If f,
– (1) i, |R’i(f,S)-Ri(f)|  /2
– (2) |R’(f,S)-1/ni 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.
Discussion
Model
Degenerate
Uniform
General
• 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 3approx for constant functions and hom. lin.
functions.
• Most important direction for future work:
extending to other models of learning, such
as classification.