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 authors aim to provide a framework, extended from Blackwell approachability, to study online learning with fairness constraints, in the presence of stochastic contexts. The authors assume that the distribution of contexts is unknown and needs to be estimated over time. They use the approachability framework to provide conditions under which certain learning objectives are compatible with certain definitions of fairness.