Incentive%20Compatible%20Regression%20Learning

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Incentive Compatible Regression Learning

• Ofer Dekel, Felix A. Fischer and Ariel D.
  Procaccia

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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

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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.

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Jaguar vs. Panthera Onca
Model
Degenerate
Uniform
General
(Jaguar, jaguar.com)
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Regression Learning
Model
Degenerate
Uniform
General

• Input space XRk ((query,answer) pairs).
• Function class FX?R (ranking functions).
• Target function oX?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)), i1,...,m, xi
  sampled from ?.
• R(h)Ex??l (h(x),o(x)).
• Find h?F to minimize R(h).

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Our Setting
Model
Degenerate
Uniform
General

• Input space XRk ((query,answer) pairs).
• Function class F (ranking functions).
• Set of players N1,...,n (experts).
• Target functions oiX?R.
• Distributions ?i over X.
• Training set?

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The Learning Game
Model
Degenerate
Uniform
General

• ?i controls xij, j1,...,m, sampled w.r.t. ?i
  (common knowledge).
• Private info of i oi(xij)yij, j1,...,m.
• Strategies of i yij, j1,...,m.
• h is obtained by learning S(xij,yij)
• Cost of i Ri(h)Ex??i l (h(x),oi(x)).
• Goal Social Welfare (please avg. player).

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Example The learning game with ERM
Model
Degenerate
Uniform
General

• Parameters XR, FConstant Functions, l
  (a,b)a-b, N1,2, o1(x)1, o2(x)2,
  ?1?2uniform dist on 0,1000.
• Learning algorithm Empirical Risk Minimization
  (ERM)
• Minimize R(h,S)1/S ? ?(x,y)?Sl (h(x),y).

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1
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Degenerate Distributions ERM with abs. loss
Model
Degenerate
Uniform
General

• The Game
• Players N1,...n
• ?i degenerate at xi.
• ?i controls xi.
• Private info of i oi(xi)yi.
• Strategies of i yi.
• 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
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 XR, FConstant Functions, l
  (a,b)(a-b)2, N1,2.

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Uniform dist. over samples
Model
Degenerate
Uniform
General

• The Game
• Players N1,...n
• ?i Discrete uniform on xi1,...,xim
• ?i controls xij, j1,...,m
• Private info of i oi(xij)yij.
• Strategies of i yij, j1,...,m.
• Cost of i
  Ri(h) Ri(h,S) 1/m??jl (h(xij),yij).

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ERM with abs. loss is not IC
Model
Degenerate
Uniform
General
1
0
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VCG to the Rescue
Model
Degenerate
Uniform
General

• Use ERM.
• Each player pays ?j?iRj(h,S).
• Each players total cost is
  Ri(h,S)?j?iRj(h,S) ?jRj(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
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.

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Proof of Lower Bound
Model
Degenerate
Uniform
General
3
2
1
0
1-
2-
3-
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Proof of Lower Bound
Model
Degenerate
Uniform
General
3
2
1
0
1-
2-
k
k-1
k
k-1
3-
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Generalization
Model
Degenerate
Uniform
General

• Theorem If ?f,
• (1) ?i, Ri(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.

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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 3-approx for
  constant functions and hom. lin. functions.
• Most important direction for future work
  extending to other models of learning, such as
  classification.