Boosted CVaR Classification

1 Introduction . As machine learning continues to find broader usage , there is an increasing understanding of the importance of tail performance of models , in addition to their average performance . For instance , in datasets with highly imbalanced classes , the tail performance is the accuracy over the minority classes which have much fewer samples than the others . In the field of algorithmic fairness , where a dataset contains several demographic groups , the tail performance is the accuracy over certain underrepresented groups that normal machine learning models often neglect . In all these examples , it is crucial to design models with good tail performance that perform well across all parts/groups of the data domain , instead of just performing well on average . Owing to its importance , a number of recent works have designed techniques to learn models with high tail performance [ HSNL18 , SKHL20 , SRKL20 ] . Maximizing the tail performance is sometimes referred to as learning under subpopulation shift , in the sense that the testing distribution could consist of just a subpopulation of the training distribution . Most of the works on subpopulation shift fall into two categories . In the first , also referred to as the domain-aware setting , the dataset is divided into several predefined groups , and the goal is to maximize the worst-group performance , i.e . the minimum performance over all the groups . A number of methods , such as importance weighting [ Shi00 ] and Group DRO [ SKHL20 , SRKL20 ] , have been proposed for domain-aware subpopulation shift . However , the domain-aware setting is not always applicable , either because the groups can be hard to define , or because the group labels are not available . Thus , in the second category of 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . work on subpopulation shift , also referred to as the domain-oblivious setting , there are no pre-defined groups , and the goal is to maximize the model ’ s performance over the worst-off samples in the dataset . Most previous work on domain-oblivious subpopulation shift [ HSNL18 , DN18 , HNSS18 , LBC+20 , MHN21 ] measure the tail performance using the Distributionally Robust Optimization ( DRO ) loss , which is defined as the model ’ s maximum loss over all distributions within a divergence ball around the training distribution . A popular instance is the α-CVaR loss , defined as the model ’ s average loss over the worst α ∈ ( 0 , 1 ) fraction of the samples incurring the highest losses in the dataset . Naturally one might think of maximizing a model ’ s tail performance by directly minimizing the DRO loss or the α-CVaR loss , which has been used by many previous work [ HSNL18 , DN18 , XDKR20 ] . However , [ HNSS18 ] proved the negative result that for classification tasks where models are evaluated by the zero-one loss , empirical risk minimization ( ERM ) achieves the lowest possible DRO loss given that the model is deterministic and the DRO loss is defined by some f -divergence function . We extend their result to the α-CVaR loss which can be written as the limit of Rényi-divergence DRO losses ( with a more direct and simpler proof due to the specialized case of CVaR ) . This is a very pessimistic result since it entails that there is no hope to get a better classifier than ERM so long as models are evaluated by a DRO ( or CVaR ) loss . So some previous work [ HNSS18 , LBC+20 , MHN21 ] proposed to avoid this issue by making extra assumptions on the testing distribution ( specifically , that the testing subpopulation can be represented by a parametric model ) , and changing the evaluation metric correspondingly to something other than a f -divergence DRO loss . In this work , we take a different approach and show that no extra assumption is needed provided that we use randomized models . While the case of general DRO is more complicated , the reason why ERM achieves the lowest possible CVaR zero-one loss in the deterministic case is very simple : there exists a linear relationship between the CVaR zero-one loss and the average zero-one loss , so the former is non-decreasing with the latter . For randomized models , however , such a monotonic relationship no longer exists . Note that for any single test sample , the zero-one loss of the deterministic model is either 0 or 1 , while the expected zero-one loss of the randomized model is a real number in [ 0 , 1 ] , so that the randomized model can typically achieve lower α-CVaR loss than the deterministic model . In fact , we can prove that if the two models have the same average accuracy , then the α-CVaR zero-one loss of the randomized model is consistently lower than the deterministic one . Motivated by the above analysis , we propose the framework of Boosted CVaR Classification to train ensemble models via Boosting . The key observation we make is that minimizing the α-CVaR loss is equivalent to maximizing the objective of α-LPBoost , a subpopulation-performance counterpart of a classical boosting variant known as LPBoost [ DBST02 ] . Thus training with respect to the α-CVaR loss can be related to a goal of boosting an unfair learner , which always produces a model with low average loss on any reweighting of the training set , to obtain a fair ensemble classifier with low α-CVaR loss for a fixed α . Note that this is in contrast to the classical boosting , which boosts a weak learner to produce an ensemble model with better average performance . We can thus show that α-CVaR training is equivalent to a sequential min-max game between a Boosting algorithm and an unfair learner , in which the Boosting algorithm provides the sample weights and the unfair learner provides base models with low average loss with respect to these weights . After all base models are trained , we compute the optimal model weights . At inference time , we first randomly sample a base model according to the model weights , and then predict with the model . Thus , the final ensemble model is a linear combination of all the base models . This paper is organized as follows : In Section 2 , we provide the necessary background of subpopulation shift and CVaR , and show that ERM achieves the lowest CVaR zero-one loss in classification tasks with deterministic classifiers . In Section 3 we show how to boost an unfair learner : we first show that minimizing the CVaR loss is equivalent to maximizing the LPBoost objective in Section 3.1 ; based on this observation , we propose the Boosted CVaR Classification framework and implement an algorithm that uses LPBoost for CVaR classification in Section 3.2 ; Then , in order to improve computation efficiency , we implement another algorithm called α-AdaLPBoost in Section 3.3 . Finally , in Section 4 we empirically evaluate the proposed method on popular benchmark datasets . 1.1 Related Work . Caveats of DRO . DRO was first applied to domain-oblivious subpopulation shift tasks in [ HSNL18 ] , in which the authors proved that the DRO loss is an upper bound of the worst-case loss over K groups ( group CVaR ) for fairness problems . [ DN18 ] analyzed the convergence rate of DRO for the Cressie-Read family of f -divergence . However , [ HNSS18 ] showed that minimizing f -divergence DRO over deterministic function classes , with respect to the zero-one classification loss , yields the same minimizer as that of ERM , provided that the former has loss less than one . [ SKHL20 ] further showed that for highly flexible classifiers ( such as many modern neural models ) that achieve zero error on each training sample , both empirical average risks and DRO risks are zero , so that the model is not specifically focusing on population DRO objective . [ SRKL20 ] made the related point that the Group DRO objective with respect to the zero one loss is prone to overfit . Boosting . Boosting is a classic algorithm in machine learning dating back to AdaBoost proposed in [ Sch90 ] . See [ Sch03 ] for a survey of the early works . Motivated by the success of AdaBoost , many later works proposed variants of Boosting , such as AdaBoost∗v [ RWST05 ] , AdaBoost ` 1 [ SL09 ] , LPBoost [ DBST02 ] , SoftBoost [ RWG07 ] and entropy regularized LPBoost [ WGV08 ] . There are some previous works that apply ensemble methods to tasks related to subpopulation shift : see [ GFB+11 ] for a survey of Bagging , Boosting and hybrid methods for the class imbalance problem , while [ IN19 , BHL19 ] used AdaBoost to improve algorithmic fairness . 2 Preliminaries . In this section we provide the necessary background on subpopulation shift , DRO and CVaR , in the context of classification , which is the focus of the paper . Particularly , we will demonstrate that in classification tasks , ERM achieves the lowest CVaR loss , so there is no gain in using CVaR compared to ERM . Then , we show that using randomized models can circumvent this problem . Denote the input space by X , the label space by Y , and the data domain by Z = X × Y . We are given a dataset { zi = ( xi , yi ) } ni=1 that i.i.d . sampled from some underlying data distribution . We assume that any input x has only one true label y , i.e . for any x ∈ X there exists y ∈ Y such that P ( y | x ) = 1 . Given a family of classifiers F , the goal of a subpopulation shift task is to train a classifier F : X → Y ∈ F with high tail performance . Given a loss function ` : Y × Y → R , the standard training algorithm is empirical risk minimization ( ERM ) which minimizes the empirical risk defined as R̂ ` ( F ) = 1 n n∑ i=1 ` ( F ( xi ) , yi ) . ( 1 ) Denote the set of minimizers of the empirical risk by F ∗ERM ` = arg minF∈F R̂ ` ( F ) . For classification tasks , the model performance is evaluated by the zero-one loss ` 0/1 ( ŷ , y ) = 1 { ŷ 6=y } . Although we can use different surrogate loss functions during training , at test time we always use the zero-one loss because we care about the accuracy , and the zero-one loss is equal to one minus the accuracy . To quantitatively measure the tail performance of model F , we can use the α-CVaR ( Conditional Value at Risk ) loss . For a fixed α ∈ ( 0 , 1 ) , the α-CVaR loss is defined as CVaR ` α ( F ) = max w∈∆n , w4 1αn n∑ i=1 wi ` ( F ( xi ) , yi ) ( 2 ) where ∆n = { ( x1 , · · · , xn ) : xi ≥ 0 , x1 + · · ·+xn = 1 } is the unit simplex in Rn . The α-CVaR loss measures how well a model performs over the worst α fraction of the dataset . For instance , ifm = αn is an integer , then the α-CVaR loss is the average loss over the m samples that incur the highest losses . We use CVaR classification to refer to classification tasks where models are evaluated by the CVaR loss . Denote the set of minimizers of the α-CVaR loss by F ∗CVaR ` α = arg minF∈F CVaR ` α ( F ) . The CVaR loss can be written as the limit of DRO ( Distributional Robust Optimization ) losses . For some divergence function D between distributions , the DRO loss measures the model ’ s performance over the worst-case distribution Q P 1 within a ball w.r.t . divergence D around the training distribution P . Formally , the DRO loss of model F is defined as DRO ` D , ρ ( F ) = sup Q P { EQ [ ` ( F ( x ) , y ) ] : D ( Q ‖ P ) ≤ ρ } ( 3 ) If we denote the Rényi-divergence by Dβ ( P ‖ Q ) = 1β−1 log ∫ ( dPdQ ) βdQ , then the α-CVAR loss is equal to the limit of DRO ` Dβ , − logα as β → ∞ ( see Example 3 in [ DN18 ] ) . Many previous works 1Q is absolute continuous to P ( i.e . Q P ) if for any event A , P ( A ) = 0⇒ Q ( A ) = 0. proposed to train a model with high tail performance by minimizing the CVaR loss or the DRO loss . However , the following result shows that for classification tasks , any model in F ∗ ERM ` 0/1 is also the minimizer of the CVaR loss if F only contains deterministic models , i.e . every F ∈ F is a deterministic mapping F : X 7→ Y ( all proofs can be found in Appendix A ) : Proposition 1 . If F only contains deterministic models , then for any model F ∈ F and any F ∗ ∈ F ∗ ERM ` 0/1 , we have CVaR ` 0/1 α ( F ) ≥ CVaR ` 0/1 α ( F ∗ ) . Moreover , if minF∈F R̂ ` 0/1 ( F ) < α , then we have F ∗ ERM ` 0/1 = F ∗ CVaR ` 0/1 α . [ HNSS18 ] showed that a counterpart of the above result holds for any f -divergence DRO loss . As noted earlier , we can write the α-CVaR loss as the limit of the Rényi family of f -divergence DRO losses . However our proof is much more direct and simple , and proceeds by showing the following simple monotonic relationship between the average zero-one loss and the α-CVaR zero-one loss : CVaR ` 0/1 α ( F ) = min { 1 , 1αR̂ ` 0/1 ( F ) } , so the α-CVaR loss is non-decreasing with the ERM loss . This is a very pessimistic result , since it entails that there is no hope to obtain a better model than ERM no matter what learning algorithm we use for CVaR classification . For the DRO context , some previous papers [ HNSS18 , LBC+20 , MHN21 ] propose to avoid this issue by making extra assumptions on the testing distribution , so as to change the evaluation metric to some function other than the f -divergence DRO loss . In this work , however , we take a completely different approach : we show that the above difficulty can be circumvented without any extra assumptions by using randomized models2 . For a randomized model F , its empirical risk and α-CVaR loss is defined as the expectation of ( 1 ) and ( 2 ) , where the expectation is taken over the randomness of F . For randomized models , the monotonic relationship in Proposition 1 does not exist , and they can achieve lower α-CVaR zero-one loss than deterministic models . For example , if we have a deterministic model with average accuracy 90 % , then the 10 % -CVaR zero-one loss of this model is 1 . However , if we have 5 deterministic models with average accuracy 90 % , such that each sample is classified correctly by at least 4 of the 5 models , then the 10 % -CVaR zero-one loss of the average of the 5 models is only 0.2 , though its average accuracy is still 90 % . Furthermore , we can prove that : Proposition 2 . Let F be a deterministic model , and F ′ be any randomized model whose average zero-one loss is the same as that of F . Then , for any α ∈ ( 0 , 1 ) , CVaR ` 0/1α ( F ′ ) ≤ CVaR ` 0/1 α ( F ) . This result implies that the tail performance of a randomized model is consistently higher than a deterministic model with the same average performance . In a nutshell , we have proved that for CVaR classification , ERM is the best deterministic model learning algorithm , and randomized models can achieve higher performance than deterministic models .

Summary: The paper attacks the problem of "fair classification", where the goal is to train a classifier that is "good" on any small fraction of the samples. Obviously, any deterministic classifier having less than 100% accuracy is arbitrarily bad on a small fraction of the distribution. The authors propose the usage of randomised classifiers to bypass this. i.e we prefer a classifier that is correct on all samples 80% of the time over a classifier that is correct on 80% of the samples wrong on the rest 20% all the time.   Firstly the authors observe the superiority of randomised classifiers for the CVaR loss over deterministic classifiers.  Then they propse the usage of a boosting framework to aggregate individual "good but unfair" classifiers into "equally good but fair" classifiers. In particular, the classic LP-boost algorithm is observed to be a good fit for this problem.  The authors then propose a more efficient version of the LP-boost algorithm where classifiers need not be trained for different levels of fairness (the worst fraction of samples being optimised for).   The algorithms are validated on classic classification datasets where they do better than standard baselines.