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: This paper studies the boosting framework for classification. Weak classifiers are produced sequentially by an abstract learner. The popular α-CVaR (Conditional Value at Risk) is adopted. The 0/1 loss is considered. First, it is proved that the optimizer (among all the deterministic models) of ERM is also the optimizer of α-CVaR, and vice versa if α-CVaR could be smaller than α by some deterministic model. The authors also prove that with the same 0/1 loss, the randomized model achieves smaller α-CVaR. The optimization structures of α-LPBoost and α-AdaLPBoost are studied to construct randomized classifiers.