Incentive Compatible Regression Learning

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 Model Degenerate Uniform General (“Jaguar”, jaguar.com)

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 General • 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).

ERM with abs. loss is not IC Model Degenerate Uniform General 1 0

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.

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

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

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

Incentive Compatible Regression Learning