Notice: The reproducibility variables underlying each score are classified using an automated LLM-based pipeline, validated against a manually labeled dataset. LLM-based classification introduces uncertainty and potential bias; scores should be interpreted as estimates. Full accuracy metrics and methodology are described in [1].
Fair Data Representation for Machine Learning at the Pareto Frontier
Authors: Shizhou Xu, Thomas Strohmer
JMLR 2023 | Venue PDF | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Numerical simulations underscore the advantages: (1) the pre-processing step is compositive with arbitrary conditional expectation estimation supervised learning methods and unseen data; (2) the fair representation protects the sensitive information by limiting the inference capability of the remaining data with respect to the sensitive data; (3) the optimal aļ¬ne maps are computationally eļ¬cient even for high-dimensional data. |
| Researcher Affiliation | Academia | Shizhou Xu EMAIL Department of Mathematics University of California Davis Davis, CA 95616-5270, USA Thomas Strohmer EMAIL Department of Mathematics Center of Data Science and Artiļ¬cial Intelligence Research University of California Davis Davis, CA 95616-5270, USA |
| Pseudocode | Yes | Algorithm 1: Pseudo-Barycenter Geodesics for Independent Variable Algorithm 2: Dependent (or Post-processing) Pseudo-Barycenter Geodesics |
| Open Source Code | Yes | The code for the results of our experiments is available online at: github.com/xushizhou/fair_data_ representation |
| Open Datasets | Yes | We adopt the following metrics of accuracy and discrimination that are frequently used in fair machine learning experiments on various data sets: (1) For fair classiļ¬cation, the prediction accuracy, and statistical disparity are quantiļ¬ed respectively by AUC (area under the Receiver Operator Characteristic curve) and Discrimination = max z,z Z P( ĖYz = 1) P( ĖYz = 1) 1 as deļ¬ned in [13]. (2) For univariate supervised learning, the prediction error and statistical disparity are quantiļ¬ed respectively by MSE (mean squared error, equivalent to the squared L2 norm on sample probability space) and KS (Kolmogorov-Smirnov) distance as in [18] for indirect comparison purpose. So that readers can compare the proposed methods indirectly with other methods that are tested in [13, 18, 44] and their references. (3) For univariate and multivariate supervised learning, the prediction error and statistical disparity are quantiļ¬ed respectively by L2 and W2 (Wasserstein) distances, which are the quantiļ¬cation the current work adopts to prove the Pareto frontier in the above sections. In addition, we perform tests on four benchmark data sets: CRIME, LSAC, Adult, COMPAS, which are also frequently used in fair learning experiments. |
| Dataset Splits | Yes | For all the test results, we apply 5-fold cross-validation with 50% training and 50% testing split, except for 90% training and 10% testing split in the linear regression test on LSAC due to the high computational cost of the post-processing Wasserstein barycenter method [18]. |
| Hardware Specification | Yes | As shown in the table above, the computational cost of the pseudo-barycenter method is signiļ¬cantly lower than the cost of the known post-processing methods: on average 7836 times faster on LSAC and 21 times faster on CRIME in a single train-test cycle for a single supervised learning model. Furthermore, in model selection or composition, the pre-processing time is a ļ¬xed one-time cost while the post-processing time is additive. (See point 4 below for a more detailed explanation) Now, we show the major advantages of the proposed method compared to the postprocessing ones, such as [18, 28, 24]: Acknowledgement The authors want to thank the referees, whose profound and detailed feedback greatly enhanced the quality and clarity of this paper. The authors acknowledge support from NSF DMS-2027248, NSF DMS-2208356, NSF CCF-1934568, NIH R01HL16351, and DESC0023490. Appendix A. Appendix: Proof of Results in Section 2 A.1 Proof of Lemma 2.1 W2 2(Āµ, Ī½) = Z ||x y||2dĪ³ (x, y) = Z ||((x mĀµ) (y mĪ½)) + (mĀµ mĪ½)||2dĪ³ (x, y) = Z ||(x mĀµ) (y mĪ½)||2dĪ³ (x, y) + ||mĀµ mĪ½||2 W2 2(Āµ , Ī½ ) + ||mĀµ mĪ½||2 = Z ||x y||2d(Ī³ ) (x, y) + ||mĀµ mĪ½||2 = Z ||(x + mĀµ) (y + mĪ½)||2d(Ī³ ) (x, y) where Ī³ and (Ī³ ) denote the optimal transport plan for (Āµ, Ī½) and (Āµ , Ī½ ), respectively. The ļ¬rst inequality results from the fact that Ī³ (x, y) := Ī³ (x mĀµ, y mĪ½) Q(Āµ , Ī½ ), the second inequality from Ī³(x, y) := (Ī³ ) (x + mĀµ, y + mĪ½) Q(Āµ, Ī½), and the equalities from direct expansion. A.2 Proof of Lemma 2.3 Proof Existence and uniqueness follow directly from Theorem 2.1. For the equivalent multi-marginal coupling problem, there exists an optimal solution Ī³ = L({Xz}z). It follows from Remark 2.3 that X = T({Xz}z) where L( X) is the Wasserstein barycenter. Therefore, the Gaussianity of barycenter results from linearity of T in the ļ¬nite |Z| case, and the fact that the set of Gaussian distribution is closed in (P2,ac, W2) when |Z| is inļ¬nite. The Fair Data Representation for Machine Learning at the Pareto Frontier characterization equation is proved in the case of ļ¬nite |Z| in [2]. For inļ¬nite |Z|, the equation still holds due to the continuity of the covariance function on (P2,ac, W2). The suļ¬ciency and necessity of the equation follows from the following characterization of the barycenter via Brenier s maps {T XXz}z derived in [2]: Z Z T XXzdĪ»(z) = Id. (96) It follows from the explicit form of {T XXz}z in Lemma 2.2 that Z Z T XXzdĪ»(z) = Z 1 2 X) 1 2 Ī£ 1 2 X dĪ»(z) = Id 1 2 X) 1 2 dĪ»(z)Ī£ 1 1 2 X) 1 2 dĪ»(z) = Ī£ X. Appendix B. Appendix: Proof of Results in Section 4 B.1 Proof of Lemma 4.1 Proof First, it follows from the triangle inequality that W2(Āµ0, Āµ1) W2(Āµ0, Āµs) + W2(Āµs, Āµt) + W2(Āµt, Āµ1) for any s, t [0, 1]. On the other hand, it follows from the deļ¬nition of Āµt that for s, t [0, 1] W2 2(Āµs, Āµt) Z (Rd)2 ||x y||2d(Ļs) Ī³(x) d(Ļt) Ī³(y) (Rd)2 ||Ļs(x, y) Ļt(x, y)||2dĪ³(x, y) (Rd)2 ||(1 s)x + sy (1 t)x ty||2dĪ³(x, y) (Rd)2 ||(t s)x (t s)y||2dĪ³(x, y) (Rd)2 ||x y||2dĪ³(x, y) = |t s|2W2 2(Āµ0, Āµ1), where the ļ¬rst equation results from deļ¬nition of W2. Given the above two facts, we complete the proof by contradiction. Assume s, t [0, 1] such that W2(Āµs, Āµt) < |t s|W2(Āµ0, Āµ1), then W2(Āµ0, Āµ1) W2(Āµ0, Āµs) + W2(Āµs, Āµt) + W2(Āµt, Āµ1) < |s|W2(Āµ0, Āµ1) + |t s|W2(Āµ0, Āµ1) + |1 t|W2(Āµt, Āµ1) = W2(Āµ0, Āµ1). Fair Data Representation for Machine Learning at the Pareto Frontier B.2 Proof of Theorem 4.1 Proof First, we derive the inequality from the triangle inequality and the optimality of {T( , z)}z: Let f : X Z Y be an arbitrary measurable function. It follows that Z ||E(Y |X, Z)z f(X, Z)z||2 2dĪ»(z)) 1 2 L(f(X, Z)) + ( Z Z ||f(X, Z)z f(X, Z)z||2 2dĪ»(z)) 1 2 L(f(X, Z)) + ( Z Z W2 2(L(f(X, Z)z), L(f(X, Z)z))dĪ»(z)) 1 2 = L(f(X, Z)) + (1 Z2 W2 2(L(f(X, Z)z1), L(f(X, Z)z2))dĪ»(z1)dĪ»(z2)) 1 2 = L(f(X, Z)) + 1 2D(f(X, Z)). Here, the penultimate equation results from the fact that, for any {Ī½z}z P2,ac(Rd), Z Z2 W2 2(Ī½z1, Ī½z2)dĪ»(z1)dĪ»(z2) = 2 Z Z W2 2(Ī½z, Ī½)dĪ»(z), (97) where Ī½ is the Wasserstein barycenter of {Ī½z}z. Now, we show that the lower bound is achieved if and only if f(X, Z) = T(t)(E(Y |X, Z), Z), t [0, 1]. Let t [0, 1], Tz := T( , z), and Āµz := L(E(Y |X, Z)z). It follows from Lemma 4.1 and Remark 4.1 that: Z W2 2(Āµz, Āµ)dĪ»(z)) 1 2 Z W2 2(Āµz, Tz(t) Āµz)dĪ»(z)) 1 2 + ( Z Z W2 2(Tz(t) Āµz, Āµ)dĪ»(z)) 1 2 Z W2 2(Āµz, Āµ)dĪ»(z)) 1 2 + ((1 t)2 Z Z W2 2(Āµz, Āµ)dĪ»(z)) 1 2 = t V + (1 t)V = V. Therefore, the second inequality is an equality where the ļ¬rst term is L(T(t)): L(T(t)) = ( Z Z ||E(Y |X, Z)z Tz(t)(E(Y |X, Z)z)||2 2dĪ»(z)) 1 2 Z W2 2(Āµz, Tz(t) Āµz)dĪ»(z)) 1 2 Z W2 2(Āµz, Āµ)dĪ»(z)) 1 2 = t V. Xu and Strohmer For the second term, we claim that it equals 1 2D(T(t)). To see this, we need to ļ¬rst show Tz(t) Āµz = Āµ. Indeed, if not, then R Z W2 2(Tz(t) Āµz, Tz(t) Āµz)dĪ»(z) is strictly less than R Z W2 2(Tz(t) Āµz, Āµ)dĪ»(z) by the deļ¬nition and uniqueness of Tz(t) Āµz. It follows that Z W2 2(Āµz, Tz(t) Āµz)dĪ»(z)) 1 2 Z W2 2(Āµz, Tz(t) Āµz)dĪ»(z)) 1 2 + ( Z Z W2 2(Tz(t) Āµz, Tz(t) Āµz)dĪ»(z)) 1 2 <L(T(t)) + ( Z Z W2 2(Tz(t) Āµz, Āµ)dĪ»(z)) 1 2 Z W2 2(Āµz, Āµ)dĪ»(z)) 1 2 , which contradicts the deļ¬nition and uniqueness of Āµ. Therefore, D(T(t)) = ( Z Z2 W2 2(Tz1(t) Āµz1, Tz2(t) Āµz2)dĪ»(z1)dĪ»(z2)) 1 2 Z W2 2(Tz(t) Āµz, Tz(t) Āµz)dĪ»(z)) 1 2 Z W2 2(Tz(t) Āµz, Āµ)dĪ»(z)) 1 2 Z W2 2(Āµz, Āµ)dĪ»(z)) 1 2 That completes the proof. Appendix C. Appendix: Proof of Results in Section 5 C.1 Proof of X Z implies E(Yz| X) = E(Y | X, Z)z Proof Let X Z and assume for contradiction that E(Yz| X) = E(Y | X, Z)z. Then, we have ||Y E(Y | X, Z)||2 2 = Z Z ||Yz f ( X, Z)z||2 2dĪ» Z Z ||Yz f ( X, z)||2 2dĪ» Z Z ||Yz E(Yz| X)||2 2dĪ» Z Z ||Yz fz( X)||2 2dĪ» Fair Data Representation for Machine Learning at the Pareto Frontier where the ļ¬rst line follows from disintegration and the fact that there exists a measurable function f : X Z Y such that f ( X, Z) = E(Y | X, Z), the second from X Z, the third line follows from orthogonal projection property of conditional expectation and the assumption, and the forth from the fact that there exists a measurable function fz : X Y such that fz( X) = E(Yz| X). Now, deļ¬ne f : X Z Y by f( , z) := fz for Ī»-a.e. z Z. It follows that ||Y E(Y | X, Z)||2 2 > Z Z ||Yz fz( X)||2 2dĪ» Z Z ||Yz f( X, z)||2 2dĪ» = ||Y f( X, Z)||2 2 = ||Y E(Y | X, Z)||2 2 + ||E(Y | X, Z) f( X, Z)||2 2. That implies ||E(Y | X, Z) f( X, Z)||2 2 < 0, a contradiction. This completes the proof. C.2 Proof of Lemma 5.1 Proof Let X, X { X DX : X Z} satisfy Ļ( X ) Ļ( X). We have ||E(Y |X, Z) E( Y | X, Z)||2 2 ||E(Y |X, Z) E( Y | X , Z)||2 2 =||E(Y |X, Z) E(Y | X, Z)||2 2 ||E(Y |X, Z) E(Y | X , Z)||2 2 Notice that ||E(Y |X, Z) E(Y | X, Z)||2 2 = ||E(Y |X, Z) E(Y | X, Z)||2 2 + Z Z W2 2(Āµz, Āµ)dĪ» where Āµz := L(E(Y | X, Z)z) and Āµ := L(E(Y | X, Z)). Also, we deļ¬ne Āµ z and Āµ analogously to have ||E(Y |X, Z) E(Y | X , Z)||2 2 =||E(Y |X, Z) E(Y | X , Z)||2 2 + Z Z W2 2(Āµ z, Āµ )dĪ» =||E(Y |X, Z) E(Y | X, Z)||2 2 + ||E(Y | X, Z) E(Y | X , Z)||2 2 + Z Z W2 2(Āµ z, Āµ )dĪ». Combining the above, we have ||E(Y |X, Z) E( Y | X, Z)||2 2 ||E(Y |X, Z) E( Y | X , Z)||2 2 Z W2 2(Āµz, Āµ)dĪ» Z Z W2 2(Āµ z, Āµ )dĪ» ||E(Y | X, Z) E(Y | X , Z)||2 2. It remains to show that R Z W2 2(Āµz, Āµ)dĪ» < R Z W2 2(Āµ z, Āµ )dĪ» + ||E(Y | X, Z) E(Y | X , Z)||2 2. Indeed, assume for contradiction that R Z W2 2(Āµ z, Āµ )dĪ» + ||E(Y | X, Z) E(Y | X , Z)||2 2 Xu and Strohmer Z W2 2(Āµz, Āµ)dĪ», then we have Z Z W2 2(Āµz, Āµ )dĪ» ||E(Y | X, Z) E(Y | X , Z)||2 2 + Z Z W2 2(Āµ z, Āµ )dĪ» Z W2 2(Āµz, Āµ)dĪ». This contradicts the optimality and uniqueness of Āµ by Lemma 3.1. Therefore, we prove by contradiction that R Z W2 2(Āµz, Āµ)dĪ» < R Z W2 2(Āµ z, Āµ )dĪ» + ||E(Y | X, Z) E(Y | X , Z)||2 2 and, hence, ||E(Y |X, Z) E( Y | X, Z)||2 2 ||E(Y |X, Z) E( Y | X , Z)||2 2 < 0. That completes the proof. C.3 Proof of Lemma 5.2 Proof We ļ¬rst prove Ļ(( X, Z)) = Ļ((X, Z)). Since L(Xz) P2,ac, it follows from Lemma 3.1 that there exists a measurable map T : X Z X such that T(Xz, z) = Xz Ī»-a.e., where X denotes the Wasserstein barycenter of {Xz}z. Deļ¬ne T Id|Z : X Z X Z, we have T Id|Z is X Z/X Z-measurable and satisļ¬es T Id|Z((X, Z)) = ( X, Z). That implies Ļ(( X, Z)) Ļ((X, Z)). Furthermore, since L( X) P2,ac, it follows from Brenier s theorem [11] that there exists T 1( , z) such that T 1( Xz, z) = Xz. Therefore, we have (T Id|Z) 1 = T 1 Id|Z is X Z/X Z-measurable and satisļ¬es (T Id|Z) 1(( X, Z)) = (X, Z). That implies Ļ((X, Z)) Ļ(( X, Z)). That completes the proof of Ļ(( X, Z)) = Ļ((X, Z)). Now, we show Ļ( X) Ļ( X). From the construction of X, we have Ļ(( X, Z)) Ļ(( X, Z)) = Ļ((X, Z)). But X Z implies that, for any BX BX , we can construct BX Z BX BZ. In addition, due to Ļ(( X, Z)) Ļ(( X, Z)), there exists B XZ BX BZ such that ( X, Z) 1(B XZ) = (X, Z) 1(BX Z). Lastly, X Z also implies that there exists B X BX satisfying B XZ = B X Z. It follows that X 1(BX) = ( X, Z) 1(BX Z) = (X, Z) 1(B X Z) = X 1(B X) (98) Since our choice of BX BX is arbitrary, it follows that Ļ( X) Ļ( X). Finally, since our choice of X { X D|X : X Z} is arbitrary, we are done. [1] B. L. Adamson. Ricci v. De Stefano: Procedural Activism (?). 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Bias, Fairness, and Accountability with AI and ML Algorithms. ar Xiv preprint ar Xiv:2105.06558, 2021. |
| Software Dependencies | No | The paper mentions software components like "logistic regression", "random forest", "linear regression", and "artiļ¬cial neural networks (ANN)" but does not provide specific version numbers for these or other libraries/frameworks used. |
| Experiment Setup | Yes | The two supervised learning methods we use are linear regression and artiļ¬cial neural networks (ANN with 4 linearly stacked layers where each of the ļ¬rst three layers has 32 units all with Re Lu activation while the last has 1 unit with linear activation). |