On the Fundamental Trade-offs in Learning Invariant Representations

Many applications of representation learning , such as privacy-preservation , al-1 gorithmic fairness and domain adaptation , desire explicit control over semantic2 information being discarded . This goal is often formulated as satisfying two po-3 tentially competing objectives : maximizing utility for predicting a target attribute4 while simultaneously being independent or invariant with respect to a known seman-5 tic attribute . In this paper , we identify and determine two fundamental trade-offs6 between utility and semantic dependence induced by the statistical dependencies7 between the data and its corresponding target and semantic attributes . We derive8 closed-form solutions for the global optima of the underlying optimization prob-9 lems under mild assumptions , which in turn yields closed formulae for the exact10 trade-offs . We also derive empirical estimates of the trade-offs and show their11 convergence to the corresponding population counterparts . Finally , we numeri-12 cally quantify the trade-offs on representative problems and compare the solutions13 achieved by baseline representation learning algorithms.14 1 Introduction15 Real-world applications of representation learning algorithms often have to contend with objectives16 beyond predictive performance . These include cost functions pertaining to , invariance ( e.g. , to17 photometric or geometric variations ) , semantic independence ( e.g. , w.r.t to age or race for face18 recognition systems ) , privacy ( e.g. , mitigating leakage of sensitive information [ 1 ] ) , algorithmic19 fairness ( e.g. , demographic parity [ 2 ] ) , and generalization across multiple domains [ 3 ] , to name a few.20 At its core , the underlying goal of the aforementioned formulations of representation learning is to21 satisfy two competing objectives , extracting as much information necessary to predict a target label22 y ( e.g. , face identity ) while intentionally and permanently suppressing information pertaining to a23 desired semantic attribute s ( e.g. , age , gender or race ) . When y is independent of s , one can learn a24 representation that is independent of s with no loss of performance , i.e. , no trade-off exists between25 the two objectives . However , when the two attributes y and s are correlated , attaining semantic26 independence will necessarily reduce the performance of the target predictor , i.e. , there is a trade-off27 between the two objectives . The trade-off is unknown yet is important for understanding the limits of28 existing and future representation learning algorithms that involve semantic independence constraints.29 Let z = f ( x ) be a representation of input data x , and f ( · ) be the encoder ( see Fig 1 ( a ) ) . Invariant30 learning requires that prediction of the target label , ŷ = gY ( z ) be independent of a semantic attribute31 s i.e. , ŷ ⊥⊥ s for all possible downstream target predictors gY ( · ) . This independence condition is32 satisfied if and only if ( iff ) , the representation z is independent of s i.e. , z ⊥⊥ s. Therefore , Invariant33 representation learning ( IRL ) seeks to optimize two objectives : i ) the degree of dependence between34 data representation z and semantic attribute s , and ii ) target task utility . These two objectives can be35 combined into one , with a parameter τ controlling the trade-off.36 Submitted to 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Do not distribute . In this paper , we identify and analytically determine two fundamental trade-offs in the invariant37 representation learning setting introduced above , namely Data Space Trade-Off and Label Space38 Trade-Off . These trade-offs are illustrated in Figure 1 ( b ) and formally defined next.39 Definition 1 . Data Space Trade-Off arises from the statistical dependence between the target attribute40 y and the semantic attribute s conditioned on the given input data x . When the learner ’ s hypothesis41 class contains all Borel-measurable functions1 we have:42 inf f ( · ) measurable { ( 1− τ ) inf gY ( · ) measurable Ex , y [ LY ( gY ( f ( x ) ) , y ) ] + τ dep ( f ( x ) , s ) } . ( 1 ) where f ( · ) is the encoder that extracts representation z from x , gY ( · ) predicts ŷ from the repre-43 sentation z , LY ( · , · ) is the loss for the desired task of predicting the task label y . The function44 dep ( · , · ) ≥ 0 is a parametric or non-parametric measure of statistical dependence i.e. , dep ( q , r ) = 045 means q and r are independent , and dep ( q , r ) > 0 means q and r are dependent with larger values46 indicating greater degrees of dependence . The scalar τ ∈ [ 0 , 1 ) is a hyper-parameter that controls47 the trade-off between the two objectives , with τ = 0 being the standard approach that enforces no48 independence to the attribute s , while τ → 1 enforces representation z to be independent of s.49 Including all measurable functions in the hypothesis class of the encoder f ( · ) and target predic-50 tor gY ( · ) ensures that the best possible trade-off is included within the feasible solution space.51 For example , when τ = 0 and LY ( · , · ) is the mean-squared error , the optimal Bayes estimator,52 gY ( f ( x ) ) = Ey [ y |x ] is reachable . This definition corresponds to the trade-off D in Figure 1 ( b ) .53 Definition 2 . Label Space Trade-Off arises by ignoring the data x and is purely determined by the54 statistical dependence between the target feature y and the semantic attribute s. Such a trade-off can55 be defined as:56 inf z∈L2 { ( 1− τ ) inf gY ( · ) measurable Ex , y [ LY ( gY ( z ) , y ) ] + τ dep ( z , s ) } , ( 2 ) where L2 is the space of all random vectors with finite second-order moment ( i.e. , Ez [ ‖z‖2 ] < ∞ ) 57 on the same probability space in which the joint variable ( s , y ) comes from.58 1More specifically , we consider square-integrable Borel-measurable functions for boundedness . This definition corresponds to the optimal trade-off obtained by an ideal representation z that is not59 constrained by the learnability of the encoder f ( · ) . For example , if τ = 0 , the ideal representation60 z is perfectly aligned with the target label y i.e. , z = y and gY ( · ) is the identity function , perfect61 prediction of target attribute is feasible . Therefore , this trade-off corresponds to the best trade-off that62 any combination of data x and learnable encoder f ( · ) can aspire to . This definition corresponds to63 the trade-off L in Figure 1 ( b ) , and it necessarily dominates the Data Space Trade-Off D.64 Contributions : i ) Identify two fundamental trade-offs in invariant representation learning . ii ) Obtain65 closed-form solution for the corresponding optimization problems , and consequently determine the66 trade-offs exactly . iii ) Provide consistent empirical closed-form solution for the representations that67 achieve optimal trade-offs . iv ) Numerically quantify the trade-offs defined here and compare them to68 those obtained by existing solutions.69 Implications : i ) Our closed-form empirical estimators for the optimal representations lend themselves70 to practical invariant representation learning algorithms . ii ) Theoretically elucidating and empirically71 quantifying the intrinsic limits of invariant representations will enable researchers and practitioners72 alike to identify the feasible and infeasible solution space for the trade-offs and lead to informed73 development and deployment of optimal IRL methods . iii ) Our theoretical analysis sheds light on the74 utility-semantic independence trade-off , the role of statistical dependency between target label y , the75 semantic attribute s , and the input data x , and the hypothesis class adopted for the learners.76 2 Related Work77 Trade-Offs in Representation Learning : While there are abundant empirical approaches for the78 representation learning applications considered in this paper , to the best of our knowledge , there79 is no prior work that exactly characterizes and empirically quantifies the trade-offs inherent to80 representation learning with semantic independence constraints.81 Prior work primarily sought to either obtain lower or upper bounds or characterize the extreme82 points of the trade-off in specific contexts such as fair representation learning . For instance , [ 5 ] 83 uses information theoretic tools and characterizes the utility-fairness trade-off in terms of a lower84 bounds when both y and s are binary labels . Later [ 6 ] provided both upper and lower bound for the85 binary labels . By leveraging Chernoff bound [ 7 ] proposed a construction method to generate an ideal86 representation beyond input data to achieve perfect fairness while maintaining the best performance87 on target task for equalized odds . In the case of categorical features , a lower bound on utility-fairness88 trade-off has been provided by [ 8 ] . The notion of Pareto optimality was used by [ 9 ] to minimize89 the maximum possible error among sensitive attributes where both target and sensitive features are90 categorical . In contrast to this body of work , our trade-off analysis is applicable to multi-dimensional91 discrete and/or continuous attributes where we find the exact optimal trade-offs.92 The only prior work that investigates fundamental trade-offs in a general setting where both y and s93 can be continuous or discrete features , are [ 4 ] and [ 10 ] . [ 4 ] considers only linear dependence between94 the representation and semantic attribute and proposed a closed-form solution for the utility-fairness95 trade-off . Even though [ 10 ] considers non-linear dependencies , optimal losses have been derived only96 for the extremes of the trade-off ( i.e. , τ → 0 and τ → 1 ) . In a more general setting where 0 < τ < 1,97 [ 10 ] only provides a lower bound on utility-invariance trade-off through information plane analysis.98 In contrast to the foregoing , we take a functional analysis approach and utilize covariance operator99 based measures of dependence that account for all non-linear dependence relations . We exactly100 characterize and quantify the utility-invariance trade-offs , while also providing a means to empirically101 estimate the encoder that achieves said optimal trade-off . Lastly , in addition to the Data Space102 Trade-Off , we also introduce and determine the Label Space Trade-Off which is the ideal trade-off103 that any unrestricted learning algorithm can aspire to.104 Invariant , Fair , Privacy-Preserving Representation Learning : The basic idea of representation105 learning that discards unwanted semantic information has been explored under different contexts like106 invariant , fair , or privacy-preserving learning . In domain adaptation [ 11 , 12 , 13 ] , the goal is to learn107 features that are independent of the data domain . In fair learning [ 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23,108 2 , 24 , 25 , 26 , 27 , 4 ] , the goal is to discard the demographic information that leads to unfair outcomes.109 Similarly , there is a growing interest in mitigating unintended leakage of private information from110 data representations [ 28 , 29 , 1 , 30 , 31 ] . A vast majority of this body of work is empirical in nature.111 These methods implicitly look for a single or more points in the trade-off between utility and fairness112 and do not explicitly seek to characterize the whole trade-off front . Overall , these approaches are113 not concerned ( or aware ) about the feasibility and limitations on the utility-invariance trade-off . In114 contrast , this paper determines the fundamental theoretical limits of controlling independence to115 semantic attributes , and proposes practical learning algorithms that achieve this limit.116 Adversarial Representation Learning : Most practical approaches for learning fair , invariant , do-117 main adaptive or privacy-preserving representations discussed above are based on adversarial repre-118 sentation learning ( ARL ) . This learning problem is typically formulated as,119 inf f∈Hx { ( 1− τ ) inf gY ∈Hy Ex , y [ LY ( gY ( f ( x ) ) , y ) ] − τ inf gS∈Hs Ex , s [ LS ( gS ( f ( x ) ) , s ) ] } , ( 3 ) whereLS ( · , · ) is the loss function of a hypothetical adversary gS ( · ) who intends to extract the semantic120 attribute s through the best predictor within the hypothesis classHs . ARL is a special case of the Data121 Space Trade-Off in ( 1 ) where the negative loss of the adversary , − inf gS∈Hs Ex , s [ LS ( gS ( f ( x ) ) , s ) ] 122 plays the role of dep ( f ( x ) , s ) . However , this form of adversarial learning suffers from a fundamental123 drawback as also noted in [ 32 , 33 ] . The measure of dependence induced by ARL does not account124 for all modes of non-linear dependence between s and the representation z . The next theorem states125 this observation precisely,126 Theorem 1 . 2 Let Hs contain all Borel-measurable functions and LS ( · , · ) be mean squared error127 ( MSE ) loss . Then,128 z ∈ arg sup { inf gS∈Hs Ex , s [ LS ( gS ( z ) , s ) ] } ⇔ E [ s | z ] = E [ s ] . This theorem implies that an optimal adversary does not necessarily lead to a representation z that129 is statistically independent of s ( i.e. , p ( s|z ) = p ( s ) ) , but rather leads to s being mean independent130 of representation z i.e. , independence with respect to first order moment only . In other words,131 adversarially learned measure of dependence is not a complete measure of dependence and hence132 does not account for all modes of non-linear dependence between two random variables . As such , ARL133 is inherently incapable of attaining the trade-offs achievable by complete measures of dependence.134 3 Theoretical Results135 3.1 Problem Setting136 Consider the probability space ( Ω , F , P ) , where Ω is the sample space , F is a σ−algebra on Ω , and137 P is a probability measure on F . We assume that the joint random vector ( x , y , s ) , containing the138 input data x ∈ Rdx , the target label y ∈ Rdy and the sensitive attribute s ∈ Rds , is a random vector139 on ( Ω , F ) with joint distribution pxys.140 Assumption 1 . We assume that the encoder consists of r functions in an L2-universal RKHS141 ( Hx , kx ( · , · ) ) ( e.g. , Gaussian kernel ) , where L2−universality guarantees thatHx can approximate142 any Borel-measurable function with arbitrary precision [ 34 ] .143 Now , the representation vector z can be expressed as144 z = f ( x ) : = [ f1 ( x ) , · · · , fr ( x ) ] T ∈ Rr , fj ( · ) ∈ Hx ∀j = 1 , . . . , r. ( 4 ) where r is the dimensionality of the representation z . As discussed in Corollary 5.1 , unlike common145 practice where it is chosen arbitrarily , r itself is an object of interest for optimization . We consider a146 general scenario where both y and s can be continuous or discrete , or one of y or s is continuous147 while the other is discrete . To do this , we substitute3 the target loss , inf gY Ex , y [ LY ( gY ( z ) , y ) ] in ( 1 ) 148 with the negative of a non-parametric measure of dependence i.e. , −dep ( z , y ) . Furthermore , in149 2We defer the proofs of all lemmas , theorems and corollaries to the supplementary material . 3Many standard loss functions can be written in term of dependence measures [ 35 ] that capture all non- linear dependencies i.e , Ex , y [ LY ( fT ( f ( x ) ) , y ) ] ∝ −dep ( f ( x ) , y ) . For example , the mean squared error is proportional to 1− ρ ( f ( x ) , y ) , where ρ is the Pearson correlation coefficient , a plausible dependence measure . unsupervised settings , when there is no target attribute y , the target dependence dep ( z , y ) can be150 replaced with dep ( z , x ) , which implicitly forces the representation z to retain as much information151 as is necessary for reconstructing the input data x . This scenario is of practical interest when a data152 producer aims to provide a representation of data that is independent of a desired semantic attribute153 for any arbitrary downstream task.154 We start by designing dep ( z , s ) , and dep ( z , y ) follows similarly . A key desiderata of dependence155 measures is that they should be able to account for all possible non-linear dependence relations156 between the random variables ( or vectors ) . Examples of such measures include information theoretic157 measures such as mutual information ( e.g. , MINE [ 36 ] ) or covariance operator based measures such158 as Hilbert-Schmidt Independence Criterion [ 37 ] , Constrained Covariance [ 38 ] and Kernel Canonical159 Correlation [ 39 ] . The underlying principle behind the latter class of dependence measures is that160 finite dimensional spaces with non-linear dependencies behave as linearly dependent spaces when161 mapped appropriately to higher dimensional spaces . In this paper we adopt the covariance operator162 based measures as our choice of dependence measure for analytical tractability.163 Principally , z and s are independent iff Cov ( α ( z ) , βs ( s ) ) is zero for all α ( · ) and βs ( · ) belong-164 ing to some universal RKHSs [ 38 ] . Since z = f ( x ) and f ( · ) ∈ Hx , Cov ( α ( z ) , βs ( s ) ) =165 Cov ( α ( f ( x ) ) , βs ( s ) ) , which necessitates application of a kernel on top of another kernel . This166 limits the analytical tractability of our solution . However , as we argue below , it is almost sufficient to167 consider transformation on s , only , in which case it reduces to Cov ( f ( x ) , βs ( s ) ) . Let ( Hs , ks ( · , · ) ) 168 and ( Hy , ky ( · , · ) ) be separable4 RKHSs of functions defined on Rds and Rdy , respectively . Consider169 the bi-linear functional,170 h ( · , · ) : Hx ×Hs → R , hj ( fj , βs ) : = Covx , s ( fj ( x ) , βs ( s ) ) . ( 5 ) Assumption 2 . We assume in the rest of this paper that the positive definite kernel functions are171 bounded , i.e.,172 Ex [ kx ( x , x ) ] < ∞ , Es [ ks ( s , s ) ] < ∞ , and Ey [ ky ( y , y ) ] < ∞ . ( 6 ) The assumptions in ( 6 ) guarantee that h ( · , · ) in ( 5 ) is bounded [ 40 ] and therefore , invoking Riesz173 representation theorem [ 41 ] , there exists a unique and bounded linear operator Σsx , such that174 h ( f , βs ) = Covx , s ( f ( x ) , βs ( s ) ) = 〈βs , Σsxf〉Hs ∀f ∈ Hx , ∀βs ∈ Hs . ( 7 ) Based on h ( · , · ) , we define the linear operator hf , s : Hs → Rr as175 hf , s ( βs ) : = Covx , s ( f1 ( x ) , βs ( s ) ) ... Covx , s ( fr ( x ) , βs ( s ) ) = 〈βs , Σsxf1〉Hs ... 〈βs , Σsxfr〉Hs . The operator hf , s captures all modes of non-linear dependence , since the distribution of a low-176 dimensional projection of high-dimensional data is approximately normal [ 42 ] , [ 43 ] . In other words,177 we assume that ( f ( x ) , βs ( s ) ) is an approximately Gaussian random vector.178 4By separable we mean having a countable orthonormal basis set . Among the different dependence measures that have been defined through the covariance operator179 we adopt the Hilbert-Schmidt Independence Criterion ( HSIC ) [ 37 ] which is defined as the Hilbert-180 Schmidt norm ( HS-norm ) of the covariance operator,181 dep ( z , s ) : = ‖hf , s‖2HS = ∑ βs∈Us ‖hf , s ( βs ) ‖22= ∑ βs∈Us r∑ j=1 h2 ( fj , βs ) ( 8 ) where Us is a countable orthonormal basis set for Hs . Note that , based on this definition , if the182 distribution ( f ( x ) , βs ( s ) ) fails to be a normal distribution , we end up measuring mean dependency183 of z = f ( x ) from s which is still much stronger than the linear dependency between z and s [ 44 ] .184 Even under this assumption , empirically ( Section 4 ) we observe that trade-offs we obtain significantly185 dominate those from existing invariant representation learning algorithms.186 The following Lemma introduces a well-defined population expression for dep ( z , s ) in ( 8 ) .187 Lemma 2. dep ( z , s ) = r∑ j=1 { Ex , s , x′ , s′ [ fj ( x ) fj ( x ′ ) ks ( s , s ′ ) ] + Ex [ fj ( x ) ] Ex′ [ fj ( x′ ) ] Es , s′ [ ks ( s , s′ ) ] −2Ex , s [ fj ( x ) Ex′ [ fj ( x′ ) ] Ey′ [ ks ( s , s′ ) ] ] } where ( x , s ) and ( x′ , s′ ) are independently drawn from the joint distribution pxs.188 In practice , it is necessary to empirically estimate dep ( z , s ) , since the population distributions are189 typically unknown in most real-world scenarios.190 Definition 3 . Let D = { ( x1 , s1 , y1 ) , · · · , ( xn , sn , yn ) } be the training data , containing n i.i.d.191 realizations from the joint distribution pxsy . Using , the representer theorem [ 45 ] , it follows that192 f ( x ) = ΘE [ kx ( x1 , x ) , · · · , kx ( xn , x ) ] T , where Θ ∈ Rr×n is a free parameter matrix.193 Lemma 3 . Let an empirical estimation of covariance be194 Covx , s ( fj ( x ) , βs ( s ) ) ≈ 1 n n∑ i=1 fj ( xi ) βs ( si ) − 1 n2 n∑ i=1 n∑ k=1 fj ( xi ) βs ( sk ) . Then , the empirical estimator of dep ( z , s ) is given by195 depemp ( z , s ) : = 1 n2 ‖ΘKxHLs‖2F , ( 9 ) where Kx , Ks ∈ Rn×n are Gram matrices corresponding to Hx and Hs , respectively , H =196 I − 1n11T , and Ls is a full column-rank matrix in which LsLTs = Ks ( Cholesky factorization ) .197 This empirical estimator in ( 9 ) has a bias of O ( n−1 ) and a convergence rate of O ( n−1/2 ) .198 The population and empirical dependence measures between z and y i.e. , dep ( z , y ) and depemp ( z , y ) ,199 respectively , can be defined and obtained similarly.200 3.2 Trade-Off D201 We now turn to the the optimization problem corresponding to the trade-off D in ( 1 ) . Recall that202 z = f ( x ) is r-dimensional , where the dimensionality r is a free variable . A common desiderata of203 learned representations is that of compactness [ 46 ] in order to avoid learning representations with204 redundant information where different dimensions are highly correlated with each other . Therefore,205 going beyond the assumption that each component of f ( · ) ( i.e. , fj ( · ) ) belongs to a L2−universal206 RKHS Hx , we impose additional constraints on the representation . Specifically , we constrain the207 search space of the encoder f ( · ) to learn a disentangled representation [ 46 ] as follows,208 Ar : = { ( f1 ( · ) , · · · , fr ( · ) ) ∣∣∣ fi , fj ∈ Hx , Covx ( fi ( x ) , fj ( x ) ) + γ〈fi , fj〉Hx = δi , j } , ( 10 ) where the regularization term γ〈fi , fj〉Hx , encourages orthogonality and boundedness , which in turn209 forces the representation to be compact or non-redundant . Such disentangled representations have210 been studied in the context of independent component analysis ( ICA ) [ 39 ] . Now , the optimization211 problem in ( 1 ) reduces to,212 sup f∈Ar { J ( f ( x ) ) : = ( 1− τ ) dep ( f ( x ) , y ) − τ dep ( f ( x ) , s ) } , 0 ≤ τ < 1 , ( 11 ) where as justified earlier the target loss function inf fY Ex , y [ LY ( fT ( f ( x ) ) , y ) ] is substituted by213 −dep ( f ( x ) , y ) . Fortunately , the above optimization problem lends itself to a closed-form solu-214 tion as given by the next theorem.215 Theorem 4 . A solution5 to the optimization problem in ( 11 ) is the eigenfunctions corresponding to r216 largest eigenvalues of the following generalized problem217 ( ( 1− τ ) Σ∗yxΣyx − τ Σ∗sxΣsx ) f = λΣxxf , ( 12 ) where Σsx and Σyx are the covariance operators defined in ( 7 ) , and Σ∗sx and Σ ∗ yx are the adjoint218 operators of Σsx and Σyx , respectively.219 Remark . If the trade-off parameter τ = 0 ( i.e. , no semantic independence constraint is imposed ) , the220 solution in Theorem 4 resembles a supervised version of ICA in [ 39 ] which is essentially a kernelized221 dimensionality reduction supervised by the target attribute y . On the other hand , if τ → 1 ( i.e.,222 utility is ignored and only semantic independence is considered ) , the solution in Theorem 4 is the223 eigenfunctions corresponding to the negative eigenvalues of Σ∗sxΣsx , which are the directions that224 are least explanatory of the semantic attribute s.225 An empirical version of ( 11 ) is the following optimization problem226 sup f∈Ar { J emp ( f ( x ) ) : = ( 1− τ ) depemp ( f ( x ) , y ) − τ depemp ( f ( x ) , s ) } , 0 ≤ τ < 1 ( 13 ) where depemp ( f ( x ) , s ) and depemp ( f ( x ) , y ) are given in ( 9 ) .227 Theorem 5 . Consider the Cholesky factorization Kx = LxLTx , where Lx is a full column-rank228 matrix . A solution to ( 13 ) is229 f opt = Θopt [ kx ( x1 , · ) , · · · , kx ( xn , · ) ] T where Θopt = UT ( Lx ) † and the columns of U are eigenvectors corresponding to r largest eigenval-230 ues , λ1 , · · · , λr of the following generalized problem,231 ( LTx ( ( 1− τ ) K̃y − τK̃s ) Lx ) u = λ ( LTxHLx + nγI ) u ( 14 ) where γ is the regularization parameter from ( 10 ) and the supremum value of ( 13 ) is ∑r j=1 λj .232 Corollary 5.1 . Embedding Dimensionality : A useful corollary of Theorem 5 is optimal embedding233 dimensionality:234 arg sup r { sup f∈Ar { J emp ( f ( x ) ) : = ( 1− τ ) depemp ( f ( x ) , y ) − τ depemp ( f ( x ) , s ) } } , which is the number of positive eigenvalues of the generalized eigenvalue problem in ( 14 ) . To235 intuitively examine this result , consider two extreme cases : i ) If there is no semantic independence236 constraint ( i.e. , τ = 0 ) , adding more dimensions to the optimum r will not harm the representation237 power of z. ii ) If we only care about semantic independence and ignore the target task ( i.e. , τ → 1 ) ,238 the optimal r would be equal to zero , indicating that a null representation is the best for discarding all239 semantic information . In this case , adding more dimension to z will necessarily violate the semantic240 independence constraint . More discussion can be found in the supplementary material.241 In the following Theorem , we prove that the empirical solution converges to its population counterpart.242 Theorem 6 . Assume that ks ( · , · ) and ky ( · , · ) are bounded by one and f2k ( xi ) is bounded by M for243 any k = 1 , . . . , r and i = 1 , . . . , n for which f = ( f1 , . . . , fr ) ∈ Ar . For any n > 1 and 0 < δ < 1,244 with probability at least 1− δ , we have245 ∣∣∣ sup f∈Ar J ( f ( x ) ) − sup f∈Ar J emp ( f ( x ) ) ∣∣∣ ≤ rM √ log ( 6/δ ) a2n +O ( 1 n ) , where 0.22 ≤ a ≤ 1 is a constant.246 5The term ’ solution ’ in any optimization problem in this paper refers to a global optima . 3.3 Trade-Off L247 We recall that label space trade-off arises when the representation z is ideal and is free to be designed248 optimally i.e. , it does not necessarily depend on the input data x or the encoder ’ s hypothesis class.249 However , we assume that the representation z is a direct effect of the target and sensitive variables ( y250 and s ) . Following [ 47 ] , we use an additive noise model as251 z = fL ( y , s ) + e , e ⊥⊥ y , e ⊥⊥ s ( 15 ) where fL ( · , · ) : Rdy × Rds → Rr is a Borel-measurable function . Following Section 3.1,252 we deploy −dep ( z , y ) , defined similar to dep ( z , s ) in ( 8 ) , as a proxy for the loss function253 inf gY ∈Hy Ex , y [ LT ( gT ( z ) , y ) ] . Recall that , the desired optimization problem is given in ( 2 ) . Instead254 of directly optimizing over z ∈ L2 , we optimize over all Borel-measurable functions fL ( · , · ) by255 ignoring e since it is independent of both y and s:256 sup fL∈Ar ( y , s ) { ( 1− τ ) dep ( fL ( y , s ) , y ) − τ dep ( fL ( y , s ) , s ) } , ( 16 ) where Ar ( y , s ) is defined similar to Ar in ( 10 ) by using ( y , s ) instead of x in the definition . Recall257 that Ar ( y , s ) ensures that z will not contain highly correlated ( entangled ) dimensions , and thus be258 minimally redundant or maximally compact.259 Remark . The optimization problem in ( 16 ) and its empirical counterpart can be solved similar to260 that of trade-off D in Theorems 5 and 6 where x is replaced with ( y , s ) .261 3.4 Trade-Off F262 Here we define and discuss the trade-off achievable by practical realizations of representation learning263 algorithms with either fairness , invariance or semantic independence constraints.264 Definition 4 . Feasible Space Trade-Off arises from the statistical dependence between the target265 feature y and the sensitive attribute s conditioned on the given input data x , the choice of hypothesis266 class for the learners involved , and the choice of dependence measure adopted . This setting can be267 formalized as,268 inf f∈Hx { ( 1− τ ) inf gY ∈Hy Ex , y [ LY ( gY ( f ( x ) ) , y ) ] + τ d̃ep ( f ( x ) , s ) } , 0 ≤ τ < 1 , ( 17 ) whereHx andHy are the hypothesis class for the encoder network and target predictor , respectively,269 LY ( · , · ) denotes the loss function of target task , and d̃ep ( f ( x ) , s ) is a parametric or non-parametric270 surrogate measure of dependency quantifying the dependency between representation vector z =271 f ( x ) and the sensitive attribute s.272 This setting corresponds to the trade-off F in Figure 1 ( b ) , and is necessarily dominated by the273 Data Space Trade-Off D. Multiple factors may lead to such sub-optimal trade-offs . These include,274 hypothesis classes that are not universal RKHSs ( e.g. , [ 4 ] considered the case whereHx is universal,275 butHs andHy are linear RKHSs ) , the surrogate dependence measure d̃ep ( f ( x ) , s ) does not account276 for all non-linear dependencies ( e.g. , [ 3 , 2 , 21 , 4 ] which consider adversarially learned dependence277 measures ) , sub-optimal optimization of ( 17 ) in terms of achieving only local optima but not the278 global optima ( e.g. , when the hypothesis class is deep neural networks that are optimized through279 stochastic gradient descent , or through stochastic gradient descent-ascent in the case of adversarial280 representation learning [ 3 , 21 , 2 ] ) , and combinations thereof.281 4 Numerical Estimation of Trade-Offs282 In this section , we demonstrate the practical utility of the analytical results developed in the paper283 and validate our theoretical insights . For this purpose , we design an illustrative toy example that284 conforms to the setting studied in the paper and numerically quantify the trade-offs that we introduced.285 Experimental validation on more tasks can be found in the supplementary material.286 Consider the following Gaussian mixture model from which we generate 4000 , 2000 , and 2000287 v = [ v1 , v2 ] ∼ 1 2 ( N ( m , Σ ) +N ( m′ , Σ ) ) , m = [ 0 , 1 ] , m′ = [ 1 , 1 ] , Σ = diag ( 0.12 , 0.12 ) independent samples for training , validation and testing , respectively . Figure 3 ( a ) shows the test288 samples where the samples generated with m and m′ are in blue and red , respectively . The input data289 x is set to v1 ( the first entry of v ) , the sensitive attribute s is v31 , and the target attribute y is [ v1 , v 3 2 ] .290 In this problem both input data and target attribute are dependent on the sensitive attribute . We choose291 all three RKHSHx , Hy , andHs to be Gaussian , which is a universal RKHS . The optimal z is learned292 for the trade-off D through the closed-form solution in Theorem 5 for different invariance parameter293 values τ in [ 0 , 1 ) . Then , this optimal embedding is fed to a target task predictor which is a multi-layer294 perceptron ( MLP ) with two hidden layers , and 4 , 8 neurons and optimize the mean-squared error295 ( MSE ) . The x-axis is a normalized version of the dependence measure used in our optimization , while296 the y-axis quantifies utility normalized to [ 0 , 1 ] as exp ( −MSE ) . The same procedure is implemented297 for trade-off L , except that the input data is v , instead of x . These trade-offs are shown in Figure 3 ( b ) .298 We choose the input data to be v instead of ( y , s ) for trade-off L since ( y , s ) is fully generated from299 v and therefore , v perfectly explains ( y , s ) . For τ = 0 and τ = 0.5 , the optimal embeddings are300 illustrated in Figure 3 , ( c ) and ( d ) , respectively . Since the sensitive attribute is only related to v1,301 an invariant embedding should collapse the corresponding dimension and cause the two colors to302 overlap with each other.303 We make the following observations , ( a ) Trade-off L dominates trade-off D as expected . ( b ) The304 trade-offs F obtained by the baselines are dominated by trade-off D. Adversarial representation305 learning [ 3 , 21 , 2 ] uses sub-optimal optimization ( SGDA ) , while Spectral-ARL [ 4 ] uses a global306 optimum solution but restricts the hypothesis class in ( 3 ) to linear RKHS . As such , the baselines are307 unable to match the global optimal solution of ( 13 ) , and ( c ) At τ = 0.5 the embedding does indeed308 collapse v1 to an extent leading to partial overlap between the two mixtures.309 5 Conclusions and Societal Impact310 This paper developed the theoretical underpinnings for identifying and determining the fundamental311 trade-offs and limits of representation learning under competing objectives . These trade-offs included312 i ) label space trade-off which is solely induced by the statistical relation between target task and313 semantic attribute ; ii ) data space trade-off which is due to the statistical dependence between the314 input data and both target and semantic attributes . Further , we found closed-from solutions for the315 global optima , both the population and empirical versions , for the underlying optimization problems,316 and thus quantify the trade-offs exactly . 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Summary: This paper studies the tradeoffs between accurately predicting the target and being invariant with respect to a known sensitive attribute in representation learning. When the features are functions in a RKHS, the paper derives closed-form solutions to the tradeoff optimization program. Finite sample guarantees are also given.