A Unified Approach to Fair Online Learning via Blackwell Approachability

1 Introduction . Classically , the goal of the decision maker in sequential environment is purely performance driven — she wants to obtain as high reward as if she has had a complete information about the environment . In contrast , algorithmic fairness shifts the attention from the performance-driven behavior by taking into account additional ethical considerations . The latter is often formalized via the notion of fairness constraint [ 10 , 26 , 6 ] on the decision maker ’ s strategies . The goal of this work is to bring to light Blackwell ’ s approachability theory as a suitable theoretical formalism for fair online learning under group fairness constraints . The appealing feature of this theory is two-fold : first , it gives explicit criteria when learning is possible ; second , if this criteria is met , it comes with an explicit strategy . It is well known that Blackwell ’ s approachability theory may be used to characterize online learning problems that are tractable and to design strategies to solve them—for instance , for no-regret learning or calibration . Extensive references to such uses may be found in Cesa-Bianchi and Lugosi [ 5 ] , Perchet [ 23 ] and Abernethy et al . [ 1 ] . Actually , as noted by the latter two references , no-regret learning , calibration , and approachability imply each other in some sense . The main first achievement of this article is to extend this use to online learning under fairness constraints . This idea , though natural and intuitive , requires some extensions to Blackwell ’ s approachability theory , like ignoring the target set and having to estimate it . Related works in fair online learning . Several frameworks have been proposed to tackle various problems of fairness arising in online learning . Blum et al . [ 4 ] consider the problem of online prediction with experts and define fairness via ( approximate ) equality of average payoffs . HébertJohnson et al . [ 16 ] , Gupta et al . [ 14 ] consider the problem of group-wise calibration . ( In passing , we may note that Gupta et al . [ 14 ] consider some techniques with a flavor of approachability . ) Bechavod et al . [ 2 ] consider the problem of online binary classification with partial feedback and 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . equal opportunity constraint [ 15 ] . We treat the above works as sources of inspiration ; they all differ in the specific ( sensitive and non-sensitive ) information that the Player may or may not access before or after taking an action . We apply the general formalism of approachability theory to give new insights into online learning under fairness constraints , and approach this goal in a unified ( and geometric ) way . In particular , the generality of this formalism allows to derive ( im ) possibility results nearly effortlessly . But we also go beyond such a mere compatibility/incompatibility check between the learning objectives and fairness constraints , and note that approachability theory also gives a clear strategy for the study of trade-offs between incompatible learning objectives and fairness constraints , which often arise in batch setup [ 6 ] . Outline . We describe our approachability setting in Section 2 and provide some learning objectives ( noregret and calibration ) and fairness constraints ( group-wise controls , demographic parity , equalized average payoffs ) that fit our framework . A slight extension of the classical result of Blackwell [ 3 ] is required and discussed in Section 3 . We then support the generality of our framework by deriving ( im ) possibility results for some objective–constraint pairs in Section 4 . We also illustrate in Section 5 how this formalism can be used to derive optimal trade-offs ( Pareto frontiers ) between performance and fairness for incompatible objective–constraint pairs ; as an example , we deal with group-wise calibration ( studied by [ 16 , 14 ] ) under demographic parity constraint . For the sake of exposition , we deal in Sections 2–5 with stochastic sensitive contexts whose distribution is known ; Section 6 explains how to overcome this and develops a theory of approachability relying on ignoring but estimating the target set . Notation . The Euclidean norm is denoted by ‖ · ‖ , while the ` 1 norm is denoted by ‖ · ‖1 . Given a convex closed set C ⊂ Rd , we denote by ProjC ( · ) the projection operator onto C in Euclidean norm . 2 Fair online learning cast as an approachability problem . In this section , we propose a setting for fair online learning based on approachability—a theory introduced by Blackwell [ 3 ] ( see also the more modern expositions by Perchet [ 25 ] or Mertens et al . [ 22 ] ) . More precisely , we consider the following repeated game between a Player and Nature , with stochastic contexts . The existence of these contexts is a ( minor ) variation on the classical statement of the approachability problem . The Player and Nature have respective finite action sets A and B . The sets of sensitive and nonsensitive contexts are respectively denoted by S and X . The set X is a general Borel set , while S is a finite set with cardinality denoted by |S| . Typical choices are S = { 0 , 1 } and X = Rm for some m ∈ N. A joint distribution Q on X × S is fixed and is unknown to the Player . Finally , a ( bounded ) Borel-measurable vector-valued payoff function m : A × B × X × S → Rd , as well as a closed target set C ⊆ Rd , are given and known by the Player . At each round t > 1 the pair of non-sensitive and sensitive contexts ( xt , st ) ∼ Q is generated independently from the past . The Player observes only the non-sensitive context xt ; while Nature also observes xt , it may or may not observe the sensitive context st. Then , Nature and the Player simultaneously pick ( possibly in a randomized fashion ) bt ∈ B and at ∈ A , respectively . The Player finally accesses the obtained reward m ( at , bt , xt , st ) and the sensitive context st , while Nature has a more complete monitoring and may observe at and st. We introduce an observation operation G to indicate whether Nature observes xt only—i.e. , G ( xt , st ) = xt , the case of Nature ’ s unawareness—or whether Nature observes both contexts—i.e. , G ( xt , st ) = ( xt , st ) , the case of Nature ’ s awareness . We consider the short-hand notation mt : = m ( at , bt , xt , st ) , mT : = 1 T T∑ t=1 m ( at , bt , xt , st ) , and cT = ProjC ( mT ) = arg min v∈C ‖mT − v‖ for the instantaneous and average payoffs of the player , as well as the Euclidean projection of the latter onto the closed set C , respectively . The distance of mT to C thus equals dT : = ‖mT − cT ‖ . The game protocol is summarized on the next page . We recall that the Player does not know the context distribution Q . Definition 1 . A target set C is called m–approachable by the Player under the distribution Q if there exists a strategy of the Player such that , for all strategies of the Nature , mT → C a.s. PROTOCOL 2.1 Parameters : Observation operator G for Nature ; distribution Q on X × S For t = 1 , 2 , . . . 1 . Contexts ( xt , st ) are sampled according to Q , independently from the past ; 2 . Simultaneously , • Nature observes G ( xt , st ) and picks bt ∈ B ; • the Player observes xt and picks an action at ∈ A ; 3 . The Player observes the reward m ( at , bt , xt , st ) and the sensitive context st , while Nature observes ( at , bt , xt , st ) . Aim : The Player wants to ensure that mT → C a.s. , i.e. , dT = ‖mT − cT ‖ → 0 a.s . Remark 1 ( Awareness for the Player ) . We are mostly interested in a Player unaware of the sensitive contexts st ( Gajane and Pechenizkiy [ 13 ] ) . However , the setting above also covers the case of a Player aware of these contexts : simply consider the lifted non-sensitive contexts x′t = ( xt , st ) . We now describe payoff functions and target sets corresponding to online learning objectives or online fairness constraints . They may be combined together . For instance , vanilla calibration corresponds below to the mcal–approachability of a set Ccal , demographic parity , to the mDP–approachability of a set CDP , so that vanilla calibration under a demographic parity constraint translates into the ( mcal , mDP ) –approachability of the product set Ccal × CDP . We therefore consider each objective and each constraint as some elementary brick , to be combined with one or several other bricks . We recall that S is a finite set and will indicate the cases where we only consider S = { 0 , 1 } . We discuss two objectives : no-regret and approximate calibration , as well as three fairness constraints : group-wise ( per-group ) control , demographic parity , and equal average payoffs . 2.1 Statement of the objectives . For the sake of a more compact exposition , we define the objectives in two forms : global objectives ( the vanilla form of objectives ) and group-wise objectives . We denote γs = P ( st = s ) , so that ( γs ) s∈S corresponds to the marginal of Q on S. Objective 1 : ( Vanilla and group-wise ) no-regret . The definition is based on some payoff function r , possibly taking contexts into account : at each round t , the Player obtains the payoff r ( at , bt , xt , st ) . The aim is to get , on average , almost as much payoff as the best constant action , all things equal . The vanilla ( average ) regret equals RT = min a∈A 1 T T∑ t=1 ( r ( at , bt , xt , st ) − r ( a , bt , xt , st ) ) , while the group-wise ( average ) regret equals Rgr , T = min s∈S min a′s∈A 1 T T∑ t=1 ( r ( at , bt , xt , st ) − r ( a′s , bt , xt , st ) ) I { st = s } . The aim is that lim inf RT > 0 a.s. ( no-regret ) and lim inf Rgr , T > 0 a.s. ( group-wise no-regret ) , respectively . We could replace the 1/T factor by a 1/ ( γsT ) factor in the definition of Rgr , T , as we will do for the CT calibration criterion , but given the wish of a non-negative limit , this is irrelevant . Denote by N = |A| the cardinality of A. No-regret corresponds to the mreg–approachability of ( [ 0 , +∞ ) ) N , with the global payoff function mreg ( a , b , x , s ) = ( r ( a , b , x , s ) − r ( a′ , b , x , s ) ) a′∈A . We also duplicate mreg into the group-wise payoff function mgr-reg ( a , b , x , s ) = ( mreg ( a , b , x , s ) I { s′ = s } ) s′∈S . Group-wise no-regret then corresponds to the mgr-reg–approachability of Cgr-reg = ( [ 0 , +∞ ) ) N |S| . Objective 2 : Approximate ( vanilla or group-wise ) calibration . Online calibration was first solved by Foster and Vohra [ 12 ] and Foster [ 11 ] ; see the monograph by Cesa-Bianchi and Lugosi [ 5 , Section 4.8 ] for references to other solutions and extensions . For simplicity , we focus on binary outcomes bt ∈ { 0 , 1 } and ask the Player to provide at each round forecasts at in [ 0 , 1 ] , and even in a discretization of [ 0 , 1 ] based on a fixed number N > 2 of points : A = { a ( k ) : = ( k − 1/2 ) /N , k ∈ { 1 , . . . , N } } . Each x ∈ [ 0 , 1 ] can be approximated by some a ( k ) ∈ A with |x− a ( k ) | 6 1/ ( 2N ) . At each round , the Player picks kt ∈ { 1 , . . . , N } and forecasts at = a ( kt ) . The action set A can thus be identified with { 1 , . . . , N } . This problem is actually called 1/N–calibration or approximate calibration . The global ( vanilla ) form of the criterion reads CT = N∑ k=1 ∣∣∣∣∣ 1T T∑ t=1 ( a ( k ) − bt ) I { kt = k } ∣∣∣∣∣ , while the approximate group-wise calibration criterion is defined as Cgr , T = ∑ s∈S N∑ k=1 ∣∣∣∣∣ 1γsT T∑ t=1 ( a ( k ) − bt ) I { kt = k } I { st = s } ∣∣∣∣∣ . The aim is that lim supCT 6 1/N a.s. or lim supCgr , T 6 1/N , respectively . Note that unlike vanilla calibration , its group-wise version requires to be calibrated on each sensitive attribute s ∈ S. In particular , the classical 1/T factor is replaced by 1/ ( γsT ) , the expected number of appearances of st = s for t = 1 , . . . , T . Mannor and Stoltz [ 18 ] and Abernethy et al . [ 1 ] rewrote the problem of approximate calibration as an approachability problem as follows : introduce the global payoff function mcal ( k , b ) = ( ( a ( 1 ) − b ) I { k = 1 } , . . . , ( a ( N ) − b ) I { k = N } ) , and duplicate it into the group-wise payoff function as follows : mgr-cal ( k , b , s ) = ( mcal ( k , b ) I { s = s′ } /γs′ ) s′∈S . The calibration criteria CT and Cgr , T can now be rewritten as the ` 1–norms of the average payoff vectors mcal , T and mgr-cal , T . Approximate vanilla calibration thus corresponds to the mcal– approachability of Ccal = { v ∈ RN : ‖v‖1 6 1/N } , while approximate group-wise calibration corresponds to the mgr-cal–approachability of Cgr-cal = { v ∈ RN |S| : ‖v‖1 6 1/N } . Note that non-sensitive contexts play no role in the calibration objectives , but the Player can ( and must ) leverage these non-sensitive contexts to possibly infer sensitive contexts when handling group-wise calibration .

Summary: The paper looks at the online learning problem with different fairness objectives/constraints from the lens of Blackwell Approachability. Doing so allows one to see what fairness objectives (e.g. group-wise calibration/regret) with possibly additional fairness constraints (e.g. statistical parity) are compatible and immediately give an algorithm, if compatible.   Also, they have given a pretty tight characterization of the pareto frontier for group-wise calibration under demographic parity constraint.  Finally, they give how to try to approach a set where some of the parameters characterizing the set is unknown.