An Axiomatic Theory of Provably-Fair Welfare-Centric Machine Learning

1 Introduction . It is now well-understood that contemporary ML systems exhibit differential accuracy across gender , race , and other protected groups , for tasks like facial recognition [ 5–7 ] , in medical settings [ 2 , 17 ] , and many others . This exacerbates existing inequality , as facial recognition in policing yields disproportionate false-arrest rates , and medical ML yields disproportionate health outcomes . Welfarecentric ML methods encode both accuracy and fairness in a welfare function defined on a set of groups , and optimize or constrain welfare to learn fairly . This addresses differential accuracy and bias issues across groups by ensuring that ( 1 ) each group is seen and considered during training , and ( 2 ) an outcome is incentivized that is desirable overall , ideally according to some mutually-agreed-upon welfare function . Unfortunately , welfare metrics require a notion of ( positive ) utility , and we argue that this is not natural to many ML tasks , wherein we instead minimize some negatively connoted loss value . We thus define a complementary measure to welfare , which we term malfare , measuring societal harm ( rather than wellbeing ) . In particular , malfare arises naturally when one applies the standard axioms of cardinal welfare to loss values rather than utility values . We then cast fair ML as a direct malfare minimization problem , and study its computational and statistical aspects . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Perhaps surprisingly , minimizing a malfare function is not equivalent to maximizing some welfare function while taking utility to be negative loss , except in the egalitarian and utilitarian edge cases . This is because nearly every function satisfying the standard axioms of cardinal welfare requires nonnegative inputs , and it is in general impossible to contort a loss function into a utility function while satisfying this requirement . For example , while minimizing the 0-1 loss , which simply counts the number of mistakes a classifier makes , is isomorphic to maximizing the 1-0 gain , which counts number of correct classifications , minimizing some malfare function defined on 0-1 loss over groups is not in general equivalent to maximizing any welfare function defined on 1-0 gain . Building upon these concepts , we develop a generic notion of fair ML , termed fair-PAC ( FPAC ) learning , where the goal is to learn models for which finite training samples may guarantee ( additively ) ε-optimal malfare , with probability at least 1− δ , for any ( axiomatically justified ) malfare concept . This definition extends Valiant ’ s [ 31 ] classic PAC-learning formalization , and we show that , with appropriate modifications , many ( standard ) PAC-learners may be converted to FPAC learners . In particular , we show via a constructive polynomial reduction that realizable FPAC-learning reduces to realizable PAC-learning . Furthermore , we show , non-constructively , that for learning problems where PAC-learnability implies uniform convergence , it is equivalent to FPAC-learnability . We also show that when training is possible via convex optimization under standard assumptions , then training ε-δ malfare-optimal models , like training risk-optimal models , requires polynomial time . We briefly summarize our contributions below . 1 . We derive in section 2 the malfare concept , extending welfare to measure negatively-connoted attributes , and show that malfare-minimization naturally generalizes risk-minimization to produce fairness-sensitive ML objectives that consider multiple protected groups . 2 . Section 3 extends PAC-learning to FPAC-learning , where we consider minimization of malfare rather than risk ( expected loss ) objectives . Both PAC and FPAC learning are parameterized by a learning task ( model space and loss function ) , and we explore the resulting learnability hierarchy . 3 . We show that for many loss functions , PAC and FPAC learning are statistically equivalent , and convexity conditions for computationally efficient PAC-learning are also sufficient for FPAC-learning . 1.1 Related Work . Constraint-based notions of algorithmic fairness have risen to prominence in fair ML , with the potential to ensure fairness ( i.e. , via parity constraints between the various groups , such as equalized odds , demographic parity , equality of opportunity , equality of outcome , etc . ) , thus correcting for some forms of data or algorithmic bias . While noble in intent and intuitive by design , fairness via such statistical constraints has several prominent flaws : most notably , several popular parity constraints are mutually unsatisfiable [ 16 ] , and their constraint-based formulation inherently puts accuracy and fairness at odds , where additional tolerance parameters are required to strike a balance between the two . Furthermore , recent works [ 14 , 15 ] have shown that welfare and even disadvantaged group utility can be harmed by such fairness constraints , calling into question whether these constraints actually benefit those that they purport to aid . Perhaps in response to these issues , some recent work has trended toward welfare-based fairnessconcepts , wherein both accuracy and fairness are encoded in a welfare function defined on a group of subpopulations . Welfare is then directly optimized [ 14 , 23 , 28 ] or constrained [ 13 , 29 ] to promote fair learning across all groups . Perhaps the most similar to our work is a method of [ 14 ] , wherein they directly maximize empirical welfare over linear classifiers ; however , as with other previous works , an appropriate utility function must be selected , which we avoid by instead using malfare . We argue that empirical welfare maximization is an effective strategy when an appropriate and natural measure of utility is available , but in ML contexts , there is no “ correct ” or clearly neutral way to generically convert loss to utility . We avoid this issue by working directly with malfare and loss . Our most poignant contrast to existing theoretical work is the Seldonian learner [ 30 ] framework , which essentially extends PAC-learning to learning problems with both constraints and arbitrary nonlinear objectives . We argue that this generality is harmful to the utility of the concept as a mathematical or practical object , as nearly any ML problem can be posed as a constrained nonlinear optimization task . The utility in FPAC learning is that it is sophisticated enough to handle fairness issues with an axiomatically-justified objective , but remains simple enough to study as a mathematical object , leading to informative reductions between various PAC and FPAC learnable classes . A similar framework , termed “ multi-group agnostic PAC learning ” [ 3 , 24 ] , also considers per-group risk values , but there regret ( maximum relative dissatisfaction ) , rather than malfare , is minimized . 2 Aggregating Sentiment within Populations . A generic aggregator function M ( S ; w ) summarizes some sentiment vector S ∈ Rg0+ , across a population of g groups , weighted by a probability vector w ∈ Rg+ s.t.‖w‖1 = 1 . When S measures a desirable quantity , generally termed utility , the aggregator function is a measure of cardinal welfare [ 19 ] , and thus quantifies overall wellbeing . We also consider the inverse-notion , that of overall illbeing , termed malfare , in terms of an undesirable S , generally loss or risk , which naturally extends the welfare concept . We show an equivalent axiomatic justification for malfare , and argue that its use is more natural in many situations , particularly when considering or optimizing loss functions . Definition 2.1 ( Aggregator Functions : Welfare and Malfare ) . An aggregator function M ( S ; w ) measures the overall sentiment of a population , given sentiment vector S and probability vector w. If S denotes a desirable quantity ( e.g. , utility ) , we call M ( S ; w ) a welfare function , written W ( S ; w ) , and inversely , if it is undesirable ( e.g. , disutility , loss , or risk ) , we call M ( S ; w ) a malfare function , written W ( S ; w ) . For now , think of the term aggregator function as signifying that an entire population , with diverse and subjective desiderata , is considered and summarized into an aggregate value M ( S ; w ) , rather than a single group ’ s perspective ( i.e. , some Si ) . As we introduce axioms and show consequent properties , the appropriateness of the term shall become more apparent . We use the term sentiment to refer to S with neutral connotation , but when discussing welfare or malfare , we often refer to S as utility or risk , respectively , as in these cases , S describes a well-understood preëxisting concept . We shall see that often aggregator functions on utility values and risk values are mathematically interchangeable , however , in order to promote fairness , the desirable axioms of malfare and welfare functions differ slightly . The notation reflects the distinction ; M ( S ; w ) is M for mean , whereas W ( S ; w ) is W for welfare , and W ( S ; w ) is W ( inverted W ) , to emphasize its inverted nature . 2.1 Axioms of Cardinal Welfare and Malfare . Definition 2.2 ( Axioms of Cardinal Welfare and Malfare ) . We define the aggregator-function axioms for aggregator function M ( S ; w ) below . For each item , assume ( if necessary ) that the axiom applies ∀S , S ′ ∈ Rg0+ , scalars α , β ∈ R0+ , and probability vector w ∈ R g + . 1 . Strict Monotonicity : S ′ 6= 0 =⇒ M ( S ; w ) < M ( S + S ′ ; w ) . 2 . Weighted Symmetry:1 Suppose S ′ ∈ Rg ′ 0+ and probability vector w ′ ∈ Rg ′ 0+ , such that for all u ∈ R0+ it holds ∑ i : Si=uwi = ∑ i : Si=uw ′ i . Then M ( S ; w ) = M ( S ′ ; w′ ) . 3 . Continuity : M ( S ; w ) is a continuous function in both S and w. 4 . Independence of Unconcerned Agents : M ( 〈S1 : g−1 , α〉 ; w ) ≤ M ( 〈S ′1 : g−1 , α〉 ; w ) =⇒ M ( 〈S1 : g−1 , β〉 ; w ) ≤ M ( 〈S ′1 : g−1 , β〉 ; w ) . 5 . Independence of Common Scale : M ( S ; w ) ≤ M ( S ′ ; w ) =⇒ M ( αS ; w ) ≤ M ( αS ′ ; w ) . 6 . Multiplicative Linearity : M ( αS ; w ) = αM ( S ; w ) . 7 . Unit Scale : M ( 1 ; w ) = M ( 〈1 , . . . , 1〉 ; w ) = 1 . 8 . Pigou-Dalton Transfer Principle : Suppose µ = w · S = w · S ′ , and for all i ∈ { 1 , . . . , g } : ∣∣µ− S ′i∣∣ ≤|µ− Si| . Then W ( S ′ ; w ) ≥W ( S ; w ) . 9 . Negated Pigou-Dalton Transfer Principle : Suppose as in 8 , but conclude W ( S ′ ; w ) ≤ W ( S ; w ) . We take a moment to comment on each axiom , to preview their purpose and assure the reader of their necessity . Axioms 1-5 are essentially the standard axioms of cardinal welfarism [ 22 , 25 ] . Together , they imply ( via the Debreu-Gorman theorem [ 10 , 12 ] ) that any aggregator function can be decomposed as a monotonic function of a sum ( over groups ) of logarithm or power functions . Axiom 6 is a natural and useful property , and one which enables dimensional analysis on mean functions ; in particular , the units of aggregator function match those of sentiment values . Note that axiom 6 implies axiom 5 , and is thus a simple strengthening of a traditional cardinal welfare axiom . We will also see that it is essential to show convenient statistical and learnability properties . Axiom 7 1In the unweighted case , it is standard to define symmetry as simply M ( S ) = M ( π ( S ) ) for all permutations π over { 1 , . . . , g } , but with weightings , the simple extension M ( S ; w ) = M ( π ( S ) ; π ( w ) ) is not quite sufficient . In particular , in this weaker form , nowhere is additive decomposability , wherein a group may be decomposed into multiple groups of the same sentiment value and total weight without impacting the aggregate , codified . furthers this theme , as it ensures that not only do units of means match those of S , but scale does as well ( making comparisons like “ Si is above the population ’ s welfare ” meaningful ) , and also enabling comparison across populations ( i.e. , comparing averages rather than sums ) . Finally , axiom 8 , the Pigou-Dalton transfer principle [ 9 , 21 ] , is also standard in cardinal welfare theory , as it ensures fairness in the sense that welfare is higher when utility values are more uniform ( i.e. , incentivizing equitable redistribution of “ wealth ” in welfare ) . Its antithesis , axiom 9 , encourages the opposite ; in the context of welfare , this perversely incentivizes inequality , but for malfare , which we generally wish to minimize , the opposite occurs , thus this axiom characterizes fairness in the context of malfare . We thus state axioms 8-9 specifically for welfare or malfare , respectively , whereas axioms 1-7 apply equally well to both welfare and malfare . Axioms 1-5 , are modified from their standard presentation to include the weighting w , wherein continuity ( 3 ) must also hold over w , and weighted symmetry ( 2 ) specifies additionally that weight may be transferred between groups with equal sentiment value . Axioms 6-7 are novel to this work , and are key in strengthening the Debreu-Gorman theorem to ensure that all welfare and malfare functions are power-means in the sequel . Axiom 9 is also novel , as it is necessary to flip the inequality of axiom 8 when the sense of the aggregator function is inverted from welfare to malfare ; in particular , the semantic meaning shifts from requiring that “ redistribution of utility is desirable ” to “ redistribution of disutility is not undesirable. ” The Power-Mean We now define the p-power-mean Mp ( · ; · ) , for any p ∈ R∪±∞ , which we shall use to quantify both malfare and welfare . Power-means arise when analyzing aggregator functions obeying the various axioms of definition 2.2 , and are thus particularly important to this work . Definition 2.3 ( Power-Mean Welfare and Malfare ) . Suppose p ∈ R∪±∞ , sentiment vector S ∈ Rg0+ and probability vector w ∈ Rg+ . We then define the p-weighted-power-mean as Mp ( S ; w ) . = p √√√√ g∑ i=1 wiSpi , M0 ( S ; w ) . = exp g∑ i=1 wi lnSi , M±∞ ( S ; w ) .= ± max i∈1 , ... , g ±Si , where p ∈ { −∞ , 0 , ∞ } attain their limits , the minimum , geometric mean , and maximum , respectively . Theorem 2.4 ( Properties of the Power-Mean ) . Suppose sentiment vectors S , S ′ ∈ Rg0+ and probability vector w ∈ Rg+ . The following then hold . 1 . Monotonicity : Mp ( S ; w ) is weakly-monotonically-increasing in p. 2 . Subadditivity : ∀p ≥ 1 : Mp ( S + S ′ ; w ) ≤ Mp ( S ; w ) + Mp ( S ′ ; w ) . 3 . Contraction : ∀p ≥ 1 : ∣∣Mp ( S ; w ) −Mp ( S ′ ; w ) ∣∣ ≤ Mp ( ∣∣S − S ′∣∣ ; w ) ≤∥∥S − S ′∥∥∞ . 4 . Curvature : Mp ( S ; w ) is concave in S for p ≤ 1 and convex in S for p ≥ 1 .

Summary: This paper investigates an important challenge in welfare-centric fair ML, that is, how to connect the standard loss minimization view of ML with the utility maximization view of welfare-based fairness. The authors define malfare to measure societal harm: malfare is a complementary metric of welfare, and it is justified with axioms of cardinal welfare (and extensions to malfare). Utilizing the malfare metric, the authors cast fair ML that seeks equitable risk distribution to groups as malfare minimization over the expected losses of all groups. They further build up on the malfare minimization framework to extend PAC learning to fair-PAC (FPAC) learning, which is proposed as a formal notion of fair learnability. Their analysis of FPAC learning shows that standard PAC-learners may be converted to FPAC-learners under some conditions, and these connections further yield theoretical results on the statistical and computational learnability with FPAC learners.