Incentive Compatible Regression Learning

Transcription

1 Incentive Compatible Regression Learning Ofer Dekel 1 Felix Fischer 2 Ariel D. Procaccia 1 1 School of Computer Science and Engineering The Hebrew University of Jerusalem 2 Institut für Informatik Ludwig-Maximilians-Universität München 19th ACM-SIAM Symposium on Discrete Algorithms

2 Motivation Goal: learn from information held by strategic agents Agents have different interests and different ideas of a good output Manipulation to improve result, leads to bias in the training data Machine learning and (algorithmic) mechanism design Two interrelated issues: sampling and manipulation This talk: regression learning Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 2

3 Outline The Model Regression Learning The Learning Game Mechanism Design (in One Slide) Three Levels of Generality Degenerate Distributions Uniform Distributions General Distributions Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 3

4 Regression Learning Input space X Target function o : X R, distribution ρ over X (both unknown) Hypothesis space F of functions f : X R Loss function l : R 2 R Absolute loss l(a, b) = a b Squared loss l(a, b) = (a b) 2 Risk (disutility) R(f) = E x ρ [l(f(x), o(x))] associated with f F Regression learning Given training set S = {(x j, o(x j ))} j=1,...,m, x j sampled i.i.d. from ρ, find h argmin f F R(f) Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 4

5 Regression Learning with Strategic Agents Input space X Set N of strategic agents Target functions o i : X R, i N Distributions ρ i over X, i N Hypothesis space F Risk R i (f) = E x ρi [l(f(x), o i (x))] Goal: minimize E J [R J (h)], where J is a random variable distributed uniformly over N (i.e., maximize social welfare) Training set? Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 5

6 The Learning Game Agent i controls x ij, j = 1,..., m, sampled i.i.d. from ρ i y ij = o i (x ij ) is private information Agent i reveals ŷ ij, j = 1,..., m True sample S = {S i : i N}, S i = {(x ij, y ij ) : 1 j m} Training set Ŝ = {Ŝ i : i N}, Ŝ i = {(x ij, ŷ ij ) : 1 j m} Empirical Risk ˆR(f, S) = 1 S (x,y) S l(f(x), y) Empirical Risk Minimization (ERM): minimize ˆR(f, Ŝ) Two issues: sampling and manipulation Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 6

7 Mechanism Design Terminology ERM is a social choice function economically efficient, i.e., maximizes social welfare Mechanism: social choice function plus a payment function strategyproof if agent i cannot increase payoff (i.e., decrease sum of risk and payment) by revealing a ŷ ij y ij group strategyproof if no group can (weakly) increase the payoff of all members Why strategyproofness? Otherwise: no well-defined input to the learning problem, no theoretical guarantees Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 7

8 Degenerate Distributions: ERM with Absolute Loss Distribution ρ i degenerate at x i Agent i holds y i = o i (x i ) and reveals ŷ i Ŝ = {(x i, ŷ i ) : i N} h = argmin f F ˆR(f, Ŝ) = argminf F i N l(f(x i ), ŷ i ) Agent i incurs cost R i (h) = ˆR(h, Si ) = l(h(x i ), y i ) Theorem: If l is the absolute loss and F is convex, then ERM is group strategyproof. Actually: if we don t get the (real) best fit, then somebody must have lied and lost Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 8

9 ERM with Superlinear Loss Theorem: If l is superlinear, F is convex, not full on x 1,....x n, and contains at least two functions, then ERM is not strategyproof. Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 9

10 ERM with Superlinear Loss Theorem: If l is superlinear, F is convex, not full on x 1,....x n, and contains at least two functions, then ERM is not strategyproof. Example: X = R, F are the constant functions, l(a, b) = (a b) 2, N = {1, 2} Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 9

11 ERM with Superlinear Loss Theorem: If l is superlinear, F is convex, not full on x 1,....x n, and contains at least two functions, then ERM is not strategyproof. Example: X = R, F are the constant functions, l(a, b) = (a b) 2, N = {1, 2} Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 9

12 ERM with Superlinear Loss Theorem: If l is superlinear, F is convex, not full on x 1,....x n, and contains at least two functions, then ERM is not strategyproof. Example: X = R, F are the constant functions, l(a, b) = (a b) 2, N = {1, 2} Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 9

13 Uniform Distributions Distribution ρ i discrete, uniform over {x i1,..., x im } Agent i holds y ij = o i (x ij ) and reveals ŷ ij, j = 1,..., m ERM computes h = argmin f F ˆR(h, Ŝ) Agent i incurs cost R i (h) = ˆR(h, Si ) = 1 m m j=1 l(h(x ij), y ij ) Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 10

14 Uniform Distributions Distribution ρ i discrete, uniform over {x i1,..., x im } Agent i holds y ij = o i (x ij ) and reveals ŷ ij, j = 1,..., m ERM computes h = argmin f F ˆR(h, Ŝ) Agent i incurs cost R i (h) = ˆR(h, Si ) = 1 m m j=1 l(h(x ij), y ij ) ERM with absolute loss no longer strategyproof 1 0 Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 10

15 Uniform Distributions Distribution ρ i discrete, uniform over {x i1,..., x im } Agent i holds y ij = o i (x ij ) and reveals ŷ ij, j = 1,..., m ERM computes h = argmin f F ˆR(h, Ŝ) Agent i incurs cost R i (h) = ˆR(h, Si ) = 1 m m j=1 l(h(x ij), y ij ) ERM with absolute loss no longer strategyproof 1 0 Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 10

16 Uniform Distributions Distribution ρ i discrete, uniform over {x i1,..., x im } Agent i holds y ij = o i (x ij ) and reveals ŷ ij, j = 1,..., m ERM computes h = argmin f F ˆR(h, Ŝ) Agent i incurs cost R i (h) = ˆR(h, Si ) = 1 m m j=1 l(h(x ij), y ij ) ERM with absolute loss no longer strategyproof 1 0 Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 10

17 VCG to the Rescue General Mechanism due to Vickrey (1961), Clarke (1971), and Groves (1973) Perform ERM (recall: it is economically efficient) Agent i pays j i ˆR j (h, Ŝ), incurs total cost ˆR(h, Si ) + j i ˆR(h, Ŝ j ) Good news: strategyproof for any loss function Bad news in general: not group strategyproof, payments problematic in our case: payments not bounded, no obvious way to ensure individual rationality Is there a (group) strategyproof mechanism without payments? Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 11

18 Mechanisms without Payments Absolute loss function Relax efficiency requirement, consider α-efficient mechanisms Theorem (upper bound): There exists a 3-efficient, group strategyproof mechanism for constant functions over R k and homogeneous linear functions over R Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 12

19 Mechanisms without Payments Absolute loss function Relax efficiency requirement, consider α-efficient mechanisms Theorem (upper bound): There exists a 3-efficient, group strategyproof mechanism for constant functions over R k and homogeneous linear functions over R Theorem (lower bound): There is no (3 ε)-efficient, strategyproof mechanism for constant or homogeneous linear functions over R k for any k and any ε > 0 Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 12

20 Mechanisms without Payments Absolute loss function Relax efficiency requirement, consider α-efficient mechanisms Theorem (upper bound): There exists a 3-efficient, group strategyproof mechanism for constant functions over R k and homogeneous linear functions over R Theorem (lower bound): There is no (3 ε)-efficient, strategyproof mechanism for constant or homogeneous linear functions over R k for any k and any ε > 0 Conjecture: There is no α-efficient, strategyproof mechanism without payments for homogeneous linear functions over R k for k 2 and any α Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 12

21 Generalization Assume that for all f F, (a). for all i N, ˆRi (f, S) R i (f) ε 2 and (b). ˆR(f, S) 1 n i N R i (f) ε 2 Then the following holds for any mechanism: If (group) strategyproof under uniform distributions, then ε-(group) strategyproof under general distributions If α-efficient under uniform distributions, then α-efficient under general distributions up to an additive factor of ε If F has bounded complexity and sample size is Θ( log(1/δ) ε 2 ), then (a) holds with probability 1 δ (b) holds if (a) holds for all i, increasing the sample size by a log N factor Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 13

22 Discussion For m large enough, any loss function, any function class: VCG is ε-strategyproof with probability 1 δ For m large enough, absolute loss, constant functions: there exists a mechanism without payments that is ε-group strategyproof and 3-efficient (up to additive ε) Future work: Settle impossibility conjecture for homogeneous linear functions over R k Extend to other settings, e.g. classification Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 14

23 Thank you for your attention! Dekel, Fischer, Procaccia Incentive Compatible Regression Learning 15