Original: C OST -S ENSITIVE H IERARCHICAL C LASSIFICATION THROUGH L AYER - WISE ABSTENTIONS
Revised: C OST -S ENSITIVE H IERARCHICAL C LASSIFICATION THROUGH L AYER - WISE ABSTENTIONS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: We study the problem of cost-sensitive hierarchical classiﬁcation where a label taxonomy has a cost-sensitive loss associated with it, which represents the cost of (wrong) predictions at different levels of the hierarchy.
Revised: We study the problem of cost-sensitive hierarchical classiﬁcation where a label taxonomy has a cost-sensitive loss associated with it, which represents the cost of (wrong) predictions at different levels of the hierarchy.

Original: Directly optimizing the cost-sensitive hierarchical loss is hard, due to its non-convexity, especially when the size of the taxonomy is large.
Revised: Directly optimizing the cost-sensitive hierarchical loss is hard, due to its non-convexity, especially when the size of the taxonomy is large.

Original: In this paper, we propose a L ayer-wise A bstaining
Revised: In this paper, we propose a L ayer-wise A bstaining

Original: Loss M inimization method (LAM), a tractable method that breaks the hierarchical learning problem into layer-by-layer learning-to-abstain sub-problems.
Revised: Loss M inimization method (LAM), a tractable method that breaks the hierarchical learning problem into layer-by-layer learning-to-abstain sub-problems.

Original: We prove that there is a bijective mapping between the original hierarchical cost-sensitive loss and the set of layer-wise abstaining losses under symmetry assumptions.
Revised: We prove that there is a bijective mapping between the original hierarchical cost-sensitive loss and the set of layer-wise abstaining losses under symmetry assumptions.

Original: We employ the distributionally robust learning framework to solve the learningto-abstain problems in each layer.
Revised: We employ the distributionally robust learning framework to solve the learningto-abstain problems in each layer.

Original: We conduct experiments on large-scale bird dataset as well as on cell classiﬁcation problems.
Revised: We conduct experiments on large-scale bird dataset as well as on cell classiﬁcation problems.

Original: Our results demonstrate that
Revised: Our results demonstrate that

Original: LAM achieves a lower hierarchical cost-sensitive loss in high accuracy regions, compared to previous methods and their modiﬁed versions for a fair comparison, even though they are not directly optimizing this loss.
Revised: LAM achieves a lower hierarchical cost-sensitive loss in high accuracy regions, compared to previous methods and their modiﬁed versions for a fair comparison, even though they are not directly optimizing this loss.

Original: For each layer, we also achieve higher accuracy when the overall accuracy is kept ﬁxed across different methods.
Revised: For each layer, we also achieve higher accuracy when the overall accuracy is kept ﬁxed across different methods.

Original: Furthermore, we also show the ﬂexibility of LAM by proposing a perclass loss-adjustment heuristic to achieve a performance proﬁle.
Revised: Furthermore, we also show the ﬂexibility of LAM by proposing a perclass loss-adjustment heuristic to achieve a performance proﬁle.

Original: This can be used for cost design to translate user requirements into optimizable cost functions.
Revised: This can be used for cost design to translate user requirements into optimizable cost functions.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: In many real-world machine learning tasks, the class space has a tree-structured taxonomy, e g. the organization of species.
Revised: In many real-world machine learning tasks, the class space has a tree-structured taxonomy, e.g. the organization of species.

Original: Biologists have built datasets that include labels ranging from coarse-level domains to the most ﬁne-grained species (Van Horn et al., 2018) .
Revised: Biologists have built datasets that include labels ranging from coarse-level domains to the most ﬁne-grained species (Van Horn et al., 2018) .

Original: Similar hierarchical structures also exist in the classiﬁcation of cells (Keren et al., 2018), objects (Yang et al., 2015) and remote sensing images (Li et al., 2017).
Revised: Similar hierarchical structures also exist in the classiﬁcation of cells (Keren et al., 2018), objects (Yang et al., 2015) and remote sensing images (Li et al., 2017).

Original: When there is a label taxonomy, the ground-truth labels correspond to the leaves of the tree and predicting these labels is known as ﬁne-grained classiﬁcation.
Revised: When there is a label taxonomy, the ground-truth labels correspond to the leaves of the tree and predicting these labels is known as ﬁne-grained classiﬁcation.

Original: In general, we can predict at any level of the tree, and it is natural to incorporate a cost of such predictions.
Revised: In general, we can predict at any level of the tree, and it is natural to incorporate a cost of such predictions.

Original: For example, given an image of “owl”, classifying it as either “bird” or “animal” are both correct, but the cost associated with them should be different, since the predicted labels have different levels of speciﬁcity.
Revised: For example, given an image of “owl”, classifying it as either “bird” or “animal” are both correct, but the cost associated with them should be different, since the predicted labels have different levels of speciﬁcity.

Original: In addition, hard examples are generally easier to distinguish at a coarser level, e.g. in animal mimicry in Fig.
Revised: In addition, hard examples are generally easier to distinguish at a coarser level, e.g. in animal mimicry in Fig.

Original: 1 (a).
Revised: 1 (a).

Original: In many practical settings, the user is able to specify the cost of different types of errors in a classiﬁcation problem.
Revised: In many practical settings, the user is able to specify the cost of different types of errors in a classiﬁcation problem.

Original: The cost can be formulated as an asymmetric matrix with the columns as the labels at the leaves and the rows as the predictions at any level.
Revised: The cost can be formulated as an asymmetric matrix with the columns as the labels at the leaves and the rows as the predictions at any level.

Original: If we have a label taxonomy, as in Fig.
Revised: If we have a label taxonomy, as in Fig.

Original: 1 (a), an example of the cost matrix is in Fig.
Revised: 1 (a), an example of the cost matrix is in Fig.

Original: 1 (b).
Revised: 1 (b).

Original: Cost-sensitive loss is hard to optimize since it is non-smooth and non-convex.
Revised: Cost-sensitive loss is hard to optimize since it is non-smooth and non-convex.

Original: Previous works usually rely on its approximation using convex surrogates losses (Zhou & Liu, 2005;
Revised: Previous works usually rely on its approximation using convex surrogates losses (Zhou & Liu, 2005;

Original: Tu & Lin, 2010; Chung et al., 2016; Liu et al., 2019; Khan et al., 2017).
Revised: Tu & Lin, 2010; Chung et al., 2016; Liu et al., 2019; Khan et al., 2017).

Original: However, in hierarchical classiﬁcation, a convex surrogate for the entire taxonomy is not easy to ﬁnd.
Revised: However, in hierarchical classiﬁcation, a convex surrogate for the entire taxonomy is not easy to ﬁnd.

Original: Previous work on hierarchical classiﬁcation usually takes a top-down strategy to decouple the problem into O ( n ) binary classiﬁcation problems for a n node tree (Fan et al., 2015; Charuvaka & Rangwala, 2015; Chen & Warren, 2013).
Revised: Previous work on hierarchical classiﬁcation usually takes a top-down strategy to decouple the problem into O ( n ) binary classiﬁcation problems for a n node tree (Fan et al., 2015; Charuvaka & Rangwala, 2015; Chen & Warren, 2013).

Original: However, there is no guaranteed way to optimize the resulting cost-sensitive loss.
Revised: However, there is no guaranteed way to optimize the resulting cost-sensitive loss.

Original: The most relevant work that deals with both cost-sensitive and hierarchical learning takes a reward maximization perspective and
Revised: The most relevant work that deals with both cost-sensitive and hierarchical learning takes a reward maximization perspective and

Original: [Figure]
Revised: [Figure]

Original: 45a
Revised: 45a

Original: [Figure]
Revised: [Figure]

Original: Details uses a post-training heuristic to ﬁnd the desired trade-off between accuracy and speciﬁcity, given an overall accuracy (Deng et al., 2012).
Revised: Details uses a post-training heuristic to ﬁnd the desired trade-off between accuracy and speciﬁcity an overall al

Original: However, it requires anaccurate estimation of probabilities (conﬁdence levels) for the leaf nodes, which is not possible in standard deep neural networks.
Revised: However requires an estimation of probabilities (conﬁdence levels) for the leaf nodes, which is not possible in standard deep neural networks.

Original: This is because DNNs tend to be overconﬁdent even when they are wrong (Guo et al., 2017).
Revised: This is because DNNs tend to be overconﬁdent even when they are wrong (Guo et al., 2017).

Original: Distributionally robust learning framework can be incorporated to optimize a given cost-sensitive loss directly (Asif et al., 2015), which involves solving a linear programming problem at each inference step.
Revised: Distributionally robust learning framework can be incorporated to optimize a given cost-sensitive loss directly (Asif et al., 2015), which involves solving a linear programming problem at each inference step.

Original: However, this scales poorly as the number of classes grows since the time of solving linear programs numerically scales polynomially with the number of classes (Spielman & Teng, 2004).
Revised: However, this scales poorly as the number of classes grows since the time of solving linear programs numerically scales polynomially with the number of classes (Spielman & Teng, 2004).

Original: Instead of using a numerical solution, analytical solution to the linear program is available for certain special classes including the abstaining losses when the loss value is no greater than 0.5 (assuming loss is in [0 , 1] )
Revised: Instead of using a numerical solution, analytical solution to the linear program is available for certain special classes including the abstaining losses when the loss value is no greater than 0.5 (assuming loss is in [0 , 1] )

Original: (Fathony et al., 2016; 2018).
Revised: (Fathony et al., 2016; 2018).

Original: Abstaining means that the predictor rejects to predict any of the classes and instead pays a given abstaining cost, which is typically not higher than the cost of wrong prediction.
Revised: Abstaining means that the predictor rejects to predict any of the classes and instead pays a given abstaining cost, which is typically not higher than the cost of wrong prediction.

Original: However, these special cases of cost sensitive losses do not incorporate a hierarchy.
Revised: However, these special cases of cost sensitive losses do not incorporate a hierarchy.

Original: Moreover, most of the previous methods only discuss the case of low-dimensional linear features and do not incorporate deep neural networks (Asif et al., 2015; Fathony et al., 2016; 2018).
Revised: Moreover, most of the previous methods only discuss the case of low-dimensional linear features and do not incorporate deep neural networks (Asif et al., 2015; Fathony et al., 2016; 2018).

Original: Our approach: In this paper, we take a bottom-up approach to break down the hierarchical costsensitive loss into the abstaining losses at each layer.
Revised: Our approach: In this paper, we take a bottom-up approach to break down the hierarchical costsensitive loss into the abstaining losses at each layer.

Original: Here, abstaining means that the predictor decides not to predict the leaf nodes and instead explores predicting labels at higher levels of the tree.
Revised: Here, abstaining means that the predictor decides not to predict the leaf nodes and instead explores predicting labels at higher levels of the tree.

Original: The key insight is that each mistake made by the predictor is due to a series of abstaining and prediction decisions along the hierarchy, which results in a cumulative cost.
Revised: The key insight is that each mistake made by the predictor is due to a series of abstaining and prediction decisions along the hierarchy, which results in a cumulative cost.

Original: Fig.
Revised: Fig.

Original: 1 (a) demonstrates learning-to-abstain losses in each layer that are equivalent to the hierarchical cost-sensitive loss in
Revised: 1 (a) demonstrates learning-to-abstain losses in each layer that are equivalent to the hierarchical cost-sensitive loss in

Original: Fig.
Revised: Fig.

Original: 1 (b).
Revised: 1 (b).

Original: The corresponding prediction mechanism is shown in Fig.
Revised: The corresponding prediction mechanism is shown in Fig.

Original: 2 (b), where a prediction is ﬁrst made on the leaves and the abstained data is evaluated on its parents in the tree.
Revised: 2 (b), where a prediction is ﬁrst made on the leaves and the abstained data is evaluated on its parents in the tree.

Original: We then solve the learning-to-abstain problems in each layer using a distributionally robust learning (DRL) method.
Revised: We then solve the learning-to-abstain problems in each layer using a distributionally robust learning (DRL) method.

Original: Applying DRL to each layer of the tree involves retraining of the neural network for different number of classes.
Revised: Applying DRL to each layer of the tree involves retraining of the neural network for different number of classes.

Original: In practice, retraining is efﬁcient since only the heads are replaced and the rest of the representation is shared.
Revised: In practice, retraining is efﬁcient since only the heads are replaced and the rest of the representation is shared.

Original: Therefore, we train the higher-level models with a different head and parameters from the lower-level ones as the initialization (Fig. 2 (a)).
Revised: Therefore, we train the higher-level models with a different head and parameters from the lower-level ones as the initialization (Fig. 2 (a)).

Original: Our contribution can be summarized as follows :
Revised: Our contribution can be summarized as follows :

Original: We propose to decompose the cost-sensitive hierarchical classiﬁcation problem into learning-toabstain sub-problems at each layer.
Revised: We propose to decompose the cost-sensitive hierarchical classiﬁcation problem into learning-toabstain sub-problems at each layer.

Original: We show the bijective correspondence when the cost-sensitive losses satisfy symmetry conditions (Sec. 3.1).
Revised: We show the bijective correspondence when the cost-sensitive losses satisfy symmetry conditions (Sec. 3.1).

Original: We propose to solve the learning-to-abstain problems using a distributionally robust learning (DRL) approach that directly minimizes the abstaining loss at each of the layers.
Revised: We propose to solve the learning-to-abstain problems using a distributionally robust learning (DRL) approach that directly minimizes the abstaining loss at each of the layers.

Original: Our method also allows for efﬁcient training of the model in different layers.
Revised: Our method also allows for efﬁcient training of the model in different layers.

Original: We call our method Layer-wise Abstaining Loss Minimization method (LAM).
Revised: We call our method Layer-wise Abstaining Loss Minimization method (LAM).

Original: We conduct experiments on large-scale bird recognition with 244 nodes (iNat2017 (Van Horn et al., 2018)) and cell classiﬁcation tasks (Keren et al., 2018).
Revised: We conduct experiments on large-scale bird recognition with 244 nodes (iNat2017 (Van Horn et al., 2018)) and cell classiﬁcation tasks (Keren et al., 2018).

Original: Our results
Revised: Our results

Original: demonstrate that LAM achieves lower hierarchical cost-sensitive loss in high accuracy regions, compared to previous
Revised: demonstrate that LAM achieves lower hierarchical cost-sensitive loss in high accuracy regions, compared to previous

Original: [Figure]
Revised: [Figure]

Revised: Predicted

Revised: [Figure]

Original: methods, even though it is not directly optimizing this loss.
Revised: methods, even though it is not directly optimizing this loss.

Original: This is because the regime of high accuracy corresponds to low abstaining costs and our method is proven to correspond to the original hierarchical loss in this regime.
Revised: This is because the regime of high accuracy corresponds to low abstaining costs and our method is proven to correspond to the original hierarchical loss in this regime.

Original: For each layer, we can also achieve higher accuracy when the overall accuracy is kept ﬁxed across different methods.
Revised: For each layer, we can also achieve higher accuracy when the overall accuracy is kept ﬁxed across different methods.

Original: Our method is also ﬂexible enough to achieve a user-deﬁned performance proﬁle efﬁciently.
Revised: Our method is also ﬂexible enough to achieve a user-deﬁned performance proﬁle efﬁciently.

Original: We provide a heuristic based on per-class loss adjustment to show the potential of our method in achieving arbitrary performance proﬁles.
Revised: We provide a heuristic based on per-class loss adjustment to show the potential of our method in achieving arbitrary performance proﬁles.

Original:  B ACKGROUND
Revised:  B ACKGROUND

Original: Hierarchical Classiﬁcation Given input x ∈ X and labels y ∈ Y , we are interested in learning a classiﬁer f ( x ) : x → y .
Revised: Hierarchical Classiﬁcation Given input x 2 X and labels y 2 Y , we are interested in learning a classiﬁer f ( x ) : x ! y .

Original: We know that y has a taxonomy as a tree structure, which means each x can be classiﬁed as a set of correct labels, from ﬁne-grained to coarse labels.
Revised: We know that y has a taxonomy as a tree structure, which means each x can be classiﬁed as a set of correct labels, from ﬁne-grained to coarse labels.

Original: Assume the most ﬁne-grained class among all the correct labels for x is y ∗ ( x ) , then as long as f ( x ) ∈ P ( y ∗ ( x )) , the prediction is considered as correct, where P ( y ∗ ( x )) represents the “path” from the leaf node y ∗ ( x ) to the root of the tree.
Revised: Assume the most ﬁne-grained class among all the correct labels for x is y ⇤ ( x ) , then as long as f ( x ) 2 P ( y ⇤ ( x )) , the prediction is considered as correct, where P ( y ⇤ ( x )) represents the “path” from the leaf node y ⇤ ( x ) to the root of the tree.

Original: The classiﬁer predicts a coarse label by abstaining from the more ﬁne-grained leaf nodes.
Revised: The classiﬁer predicts a coarse label by abstaining from the more ﬁne-grained leaf nodes.

Original: However, not all correct labels provide the same correctness.
Revised: However, not all correct labels provide the same correctness.

Original: The user deﬁnes a cost matrix C that represents the importance of the different types of errors and abstentions when evaluating the performance of f ( x ) .
Revised: The user deﬁnes a cost matrix C that represents the importance of the different types of errors and abstentions when evaluating the performance of f ( x ) .

Original: Assume we have m most ﬁne-grained level classes and n total labels in the tree.
Revised: Assume we have m most ﬁne-grained level classes and n total labels in the tree.

Original: The matrix has m columns and n rows.
Revised: The matrix has m columns and n rows.

Original: The ( i, j ) -th entry C ( i, j ) in the cost matrix represents the cost the classiﬁer suffers when the ground-truth label of x is j but the prediction is i .
Revised: The ( i, j ) -th entry C ( i, j ) in the cost matrix represents the cost the classiﬁer suffers when the ground-truth label of x is j but the prediction is i .

Original: Cost-sensitive Learning Assume our training data is drawn from a data distribution P ( x , y ) , cost-sensitive learning aims to ﬁnd a classiﬁer that minimizes the following expected loss function: 
Revised: Cost-sensitive Learning Assume our training data is drawn from a data distribution P ( x , y ) , cost-sensitive learning aims to ﬁnd a classiﬁer that minimizes the following expected loss function: f

Revised: [Equation]

Original: f = arg min E ( x ,y ) ∼ P ( x ,y )
Revised: arg min E ( x ,y ) ⇠ P ( x ,y )

Original: [ C ( y, f ( x ))] .
Revised: [ C ( y, f ( x ))] .

Original: Under the framework of empirical risk minimization, the expectation is approximated using ﬁnite samples.
Revised: Under the framework of empirical risk minimization, the expectation is approximated using ﬁnite samples.

Original: Assume, we have M data points ( x i , y i ) drawn from training set D train , then the problem is converted to: f = arg min 1 M (cid:80) ( x i ,y i ) ∼ D train C ( y i , f ( x i )) .
Revised: Assume, we have M data points ( x i , y i ) drawn from training set D train , then the problem is converted to: f = arg min 1 M [Equation]

Original: Conventional machine learning usually approximates the expected cost-sensitive loss using a convex surrogate.
Revised: Conventional machine learning usually approximates the expected cost-sensitive loss using a convex surrogate.

Original: Different losses induce different classiﬁers.
Revised: Different losses induce different classiﬁers.

Original: For example, the 0-1 loss is a special case of the cost-sensitive loss when there is no label taxonomy.
Revised: For example, the 0-1 loss is a special case of the cost-sensitive loss when there is no label taxonomy.

Original: Both log loss and hinge loss are convex surrogates of 0-1 loss.
Revised: Both log loss and hinge loss are convex surrogates of 0-1 loss.

Original: Logistic regression and support vector machines are derived by optimizing log loss and hinge loss, respectively.
Revised: Logistic regression and support vector machines are derived by optimizing log loss and hinge loss, respectively.

Original: Learning to abstain Learning-to-abstain problems aim to minimize the abstaining losses using training data.
Revised: Learning to abstain Learning-to-abstain problems aim to minimize the abstaining losses using training data.

Original: Abstaining loss is a special case of general cost-sensitive losses.
Revised: Abstaining loss is a special case of general cost-sensitive losses.

Original: It is an asymmetric matrix with k +rows and k columns if k is the total number of classes.
Revised: It is an asymmetric matrix with k +rows and k columns if k is the total number of classes.

Original: The additional row represents the abstaining cost for each of the ground truth labels.
Revised: The additional row represents the abstaining cost for each of the ground truth labels.

Original: We use C a to denote an abstaining loss.
Revised: We use C a to denote an abstaining loss.

Original: The abstaining cost should be smaller than the cost of making a wrong prediction.
Revised: The abstaining cost should be smaller than the cost of making a wrong prediction.

Original: Otherwise, the classiﬁer will never abstain.
Revised: Otherwise, the classiﬁer will never abstain.

Original: Then if the ﬁrst k × k block of C a is exactly the 0-1 loss, than the entries α k needs to satisfy: 0 < α k < 
Revised: Then if the ﬁrst k ⇥ k block of C a is exactly the 0-1 loss, than the entries ↵ k needs to satisfy: 0 < ↵ k < 1 .

Original: 2. D ISTRIBUTIONALLY R OBUST C OST -S ENSITIVE C LASSIFICATION
Revised: 2. D ISTRIBUTIONALLY R OBUST C OST -S ENSITIVE C LASSIFICATION

Original: We now introduce the background in distributionally robust learning (DRL) for optimizing the cost-sensitive loss.
Revised: We now introduce the background in distributionally robust learning (DRL) for optimizing the cost-sensitive loss.

Original: The formulation: Fathony et al.
Revised: The formulation: Fathony et al.

Original: (2018) proposed a distributionally robust learning method that incorporate a minimax game to robustly optimize the expected cost-sensitive loss.
Revised: (2018) proposed a distributionally robust learning method that incorporate a minimax game to robustly optimize the expected cost-sensitive loss.

Original: Assume g ( x ) and h ( x ) are valid conditional label distributions.
Revised: Assume g ( x ) and h ( x ) are valid conditional label distributions.

Original: Formally,
Revised: Formally,

Original: [Equation]
Revised: [Equation]

Original: Here, d , C and ∆ denote the input dimension, class number, and probabilistic simplex, respectively.
Revised: Here, d , C and  denote the input dimension, class number, and probabilistic simplex, respectively.

Original: Instead of empirical risk minimization, the DRL approach optimizes the following:
Revised: Instead of empirical risk minimization, the DRL approach optimizes the following:

Original: [Equation]
Revised: [Equation]

Original: where g ( x ) , h ( x ) ∈ R C are the conditional label distributions given an input x and G , H are the entire distribution over all the input.
Revised: where g ( x ) , h ( x ) 2 R C are the conditional label distributions given an input x and G , H are the entire distribution over all the input.

Original: Σ is a constraint set h needs to also reside in.
Revised: ⌃ is a constraint set h needs to also reside in.

Original: The idea of this formulation is that g , as a predictor player, plays a two-player adversarial risk minimization game
Revised: The idea of this formulation is that g , as a predictor player, plays a two-player adversarial risk minimization game

Original: Gr¨unwald et al.
Revised: Gr¨unwald et al.

Original: with an adversarial player h .
Revised: with an adversarial player h .

Original: The predictor player minimizes the expected cost, while the adversary player maximizes the expected cost.
Revised: The predictor player minimizes the expected cost, while the adversary player maximizes the expected cost.

Original: The adversary is allowed to perturb the ground-truth labels, subject to certain feature-matching constraints to ensure data compatibility.
Revised: The adversary is allowed to perturb the ground-truth labels, subject to certain feature-matching constraints to ensure data compatibility.

Original: In particular, the adversary H is constrained by the empirical training data features:
Revised: In particular, the adversary H is constrained by the empirical training data features:

Original: [Equation]
Revised: [Equation]

Original: where x i ∼ D train , h y is the y -th dimension of h and φ is a feature function.
Revised: where x i ⇠ D train , h y is the y -th dimension of h and  is a feature function.

Original: Eq.
Revised: Eq.

Original: 3 is a necessary but not sufﬁcient condition for H equals to the ground truth P
Revised: 3 is a necessary but not sufﬁcient condition for H equals to the ground truth P

Original: ( Y | X ) , when P ( Y | X ) is represented using ﬁnite training data.
Revised: ( Y | X ) , when P ( Y | X ) is represented using ﬁnite training data.

Original: The insights here is that given a predeﬁned feature function φ , when the adversary perturbs the labels, certain aggregate function of φ on g should equal to the counterpart on the empirical source data.
Revised: The insights here is that given a predeﬁned feature function  , when the adversary perturbs the labels, certain aggregate function of  on g should equal to the counterpart on the empirical source data.

Original: The derived prediction form: Solving Eq. 2 involves solving a two-player zero-sum game with constraints.
Revised: The derived prediction form: Solving Eq. 2 involves solving a two-player zero-sum game with constraints.

Original: Fathony et al.
Revised: Fathony et al.

Original: (2018) provides analytical solutions for G when the cost-sensitive loss C takes some special cases.
Revised: (2018) provides analytical solutions for G when the cost-sensitive loss C takes some special cases.

Original: In these cases, G would be parameterized with dual parameters θ .
Revised: In these cases, G would be parameterized with dual parameters ✓ .

Original: Here, we omit the derivation but show an example of the analytical solution when we incorporate C a in Eq.
Revised: Here, we omit the derivation but show an example of the analytical solution when we incorporate C a in Eq.

Original: 2 with α k ≤ 0 . 
Revised: 2 with ↵ k  0 . 5 .

Original: Theorem 1.
Revised: Theorem 1.

Original: (Theorem 13 in (Fathony et al., 2018)) Let θ ∗ be the learned parameter, and f ( x ) be the potential vector for all classes where f i ( x ) 
Revised: (Theorem 13 in (Fathony et al., 2018)) Let ✓ ⇤ be the learned parameter, and f ( x ) be the potential vector for all classes where f i ( x ) =

Original: = θ ∗ (cid:124) φ i ( x ) , i ∈ [0 , C ] .
Revised: ✓ ⇤ |  i ( x ) , i 2 [0 , C ] .

Original: Given a new data point x , let i ∗ = arg max
Revised: Given a new data point x , let i ⇤ = arg max

Original: i f i ( x ) (break tie arbitrarily), j ∗ = arg max j (cid:54){+
+}
Revised: i f i ( x ) (break tie arbitrarily), j ⇤ = arg max j {+
+}

Original: = i ∗ f j ( x ) , and e i ∗ ∈ R C be the i ∗ -th cannonical vector.
Revised: = i ⇤ f j ( x ) , and e i ⇤ 2 R C be the i ⇤ -th cannonical vector.

Original: Then the predictor G can be directly computed as, for each x :
Revised: Then the predictor G can be directly computed as, for each x :

Original: [Equation]
Revised: [Equation]

Original: [Equation] The parameter learning: Learning the parameters in DRL involves computing the (sub)-gradients of θ with respect to the dual objective of Eq. 2.
Revised: The parameter learning: Learning the parameters in DRL involves computing the (sub)-gradients of ✓ with respect to the dual objective of Eq. 2.

Original: In this paper, we investigate using deep neural networks as the feature function φ ( x ) in DRL.
Revised: In this paper, we investigate using deep neural networks as the feature function  ( x ) in DRL.

Original:  C OST -S ENSITIVE H IERARCHICAL C LASSIFICATION
Revised:  C OST -S ENSITIVE H IERARCHICAL C LASSIFICATION

Original: In this section, we describe our proposed distributionally robust learning method for cost-sensitive learning in hierarchical classiﬁcation.
Revised: In this section, we describe our proposed distributionally robust learning method for cost-sensitive learning in hierarchical classiﬁcation.

Original: We ﬁrst break the cost-sensitive loss on the full taxonomy into abstaining losses in the learning-to-abstain sub-problems in each layer (Sec. 3.1).
Revised: We ﬁrst break the cost-sensitive loss on the full taxonomy into abstaining losses in the learning-to-abstain sub-problems in each layer (Sec. 3.1).

Original: We then propose to use deep DRL to solve the learning-to-abstain problems.
Revised: We then propose to use deep DRL to solve the learning-to-abstain problems.

Original: We provide details on the gradient (Sec.
Revised: We provide details on the gradient (Sec.

Original: Finally, we propose the algorithms for training and evaluating our cost-sensitive hierarchical classiﬁcation method (Sec. 3.3).
Revised: Finally, we propose the algorithms for training and evaluating our cost-sensitive hierarchical classiﬁcation method (Sec. 3.3).

Original: 3. C OST - SENSITIVE LOSS AS AN A CCUMULATIVE LOSS
Revised: 3. C OST - SENSITIVE LOSS AS AN A CCUMULATIVE LOSS

Original: [Table]
Revised: [Table]

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Original: We take a bottom-up approach and observe that the prediction on the taxonomy takes a series of decisions.
Revised: We take a bottom-up approach and observe that the prediction on the taxonomy takes a series of decisions.

Original: In Fig.
Revised: In Fig.

Original: 1, if an “owl” is classiﬁed as a “insect”, it means the classiﬁer abstains on the leaf layer and then makes a wrong prediction in the “bird” vs. “insect” layer.
Revised: 1, if an “owl” is classiﬁed as a “insect”, it means the classiﬁer abstains on the leaf layer and then makes a wrong prediction in the “bird” vs. “insect” layer.

Original: Therefore, we can break the overall classiﬁcation problem into smaller layer-by-layer learning-to-abstain problems.
Revised: Therefore, we can break the overall classiﬁcation problem into smaller layer-by-layer learning-to-abstain problems.

Original: Here we analyze whether there is a one-on-one correspondence between a
Revised: Here we analyze whether there is a one-on-one correspondence between a

Original: C h and a set of C a in hierarchical classiﬁcation.
Revised: C h and a set of C a in hierarchical classiﬁcation.

Original: Given a tree taxonomy with m leaves and n total number of nodes in the label space, we assume P ( y ) represents the “path” from the leaf node y to the root of the tree.
Revised: Given a tree taxonomy with m leaves and n total number of nodes in the label space, we assume P ( y ) represents the “path” from the leaf node y to the root of the tree.

Original: We also assume D ( y ) represents the index of the layers of each node.
Revised: We also assume D ( y ) represents the index of the layers of each node.

Original: Assume the leaf nodes has
Revised: Assume the leaf nodes has

Original: D = 0 .
Revised: D = 0 .

Original: Then we have the following:
Revised: Then we have the following:

Original: Lemma 1.
Revised: Lemma 1.

Original: Assume an abstaining loss C a is associated with each of the layers with α i as the abstaining loss for each of the nodes, i ∈ [1 , n − 1] (except the root node), a hierarchical costsensitive loss C h is uniquely constructed by C h ( i, j ) 
Revised: Assume an abstaining loss C a is associated with each of the layers with ↵ i as the abstaining loss for each of the nodes, i 2 [1 , n  1] (except the root node), a hierarchical costsensitive loss C h is uniquely constructed by C h ( i, j ) =

Original: = (cid:80) k ∈ P ( j ) ∩ k>i
Revised: P [Equation] +

Original: α k
Revised: [Equation]

Original: + I ( i ∈ P ( j )) .
Revised: [Equation] P ( j )) .

Original: Lemma 2.
Revised: Lemma 2.

Original: Given a tree taxonomy, assume the overall cost-sensitive loss is C h .
Revised: Given a tree taxonomy, assume the overall cost-sensitive loss is C h .

Original: Let L denote the set of leaf nodes in the tree.
Revised: Let L denote the set of leaf nodes in the tree.

Original: D ( y ) is the depth of a node.
Revised: D ( y ) is the depth of a node.

Original: There is a unique C a for each layer such that
Revised: There is a unique C a for each layer such that

Revised: C h ( i, j ) =

Original: [Equation]
Revised: [Equation]

Original: I ( i ∈ P( j )) , if C h satisﬁes the following
Revised: P j )) , if C h satisﬁes the following

Original: ∀ y ∈ L , if i ∈ P ( y ) , C h( i, y ) < min k (cid:54)∈ P ( y ) C h ( k, y ) ; 2. ∀ i (cid:54)∈ L , C h
Revised: 1. 8 y 2 L , if i 2 P ( y ) , C h

Original: ( i, j ) [-≤ 
-]
Revised: ( i, y ) [-≤ 
-]<

Original:  + D ( i ) ; 3. ∀ i (cid:54)∈ L , C h ( i, j ) =
Revised: min k 62 P ( y ) C h ( k, y ) ; 2. 8 i 62 L , C h ( i, j )  1 + D ( i ) ; 3. 8 i 62 L , C h ( i, j ) =

Original: C h
Revised: C h

Original: ( i, j (cid:48) ) , for ∀ j, j (cid:48) satisfying i ∈ P ( j ) ∩ P ( j (cid:48) ) ∀ i (cid:54)∈ L , C h ( i, j ) =
Revised: ( i, j 0 ) , for 8 j, j 0 satisfying i 2 P ( j ) \ P ( j 0 ) ; 4. 8 i 62 L , C h ( i, j ) =

Original: C h
Revised: C h

Original: ( i, j (cid:48) ) , for ∀ j, j (cid:48) satisfying i (cid:54)∈ P ( j ) ∪ P ( j (cid:48) ) .
Revised: ( i, j 0 ) , for 8 j, j 0 satisfying i 62 P ( j ) [ P ( j 0 ) .

Original: Remarks: Lemma 1 is trivial by construction.
Revised: Remarks: Lemma 1 is trivial by construction.

Original: Lemma 2 can be proved easily by induction.
Revised: Lemma 2 can be proved easily by induction.

Original: Constraints in lemma 2 deﬁnes the structure the cost-sensitive loss needs to possess.
Revised: Constraints in lemma 2 deﬁnes the structure the cost-sensitive loss needs to possess.

Original: Without loss of generality, this lemma shows that our method is applicable to a large set of cost-sensitive losses.
Revised: Without loss of generality, this lemma shows that our method is applicable to a large set of cost-sensitive losses.

Original: Here we explain each of the conditions as follows:
Revised: Here we explain each of the conditions as follows:

Original: The cost of a mistake in any layer is higher than a correct prediction.
Revised: The cost of a mistake in any layer is higher than a correct prediction.

Original: This is naturally required in any meaningful cost-sensitive losses.
Revised: This is naturally required in any meaningful cost-sensitive losses.

Original: The abstaining cost is not greater than the cost of making a mistake in any layer.
Revised: The abstaining cost is not greater than the cost of making a mistake in any layer.

Original: This makes sure that there is no harm to abstain, compared to predicting a wrong label.
Revised: This makes sure that there is no harm to abstain, compared to predicting a wrong label.

Original: Otherwise, no abstention will be made and all predictions and mistakes will be made in the leaf nodes.
Revised: Otherwise, no abstention will be made and all predictions and mistakes will be made in the leaf nodes.

Original: If two leaf nodes share the same relative position in the tree, i.e., they are the children of the same father, there is symmetry in their cost if they are abstained on.
Revised: If two leaf nodes share the same relative position in the tree, i.e., they are the children of the same father, there is symmetry in their cost if they are abstained on.

Original: For example, in Fig.
Revised: For example, in Fig.

Original: 3, if a predictor makes abstention decisions on the leaf layer but predicts node 5, it suffers the same cost when the ground truth is node 2 or 3: C h (5 , 3) =
Revised: 3, if a predictor makes abstention decisions on the leaf layer but predicts node 5, it suffers the same cost when the ground truth is node 2 or 3: C h (5 , 3) =

Original: C h (5 , 2) .
Revised: C h (5 , 2) .

Original: The symmetry also holds if a mistake is made in the non-leaf layers.
Revised: The symmetry also holds if a mistake is made in the non-leaf layers.

Original: Following the same logic, in Fig.
Revised: Following the same logic, in Fig.

Original: 3, if a predictor makes abstention decisions on the leaf layer but predicts node 4, it suffers the same cost when the ground truth is node2 or 3: C h (4 , 3) =
Revised: 3, if a predictor makes abstention decisions on the leaf layer but predicts node 4, it suffers the same cost when the ground truth is node or 3: C h (4 , 3) =

Original: C h (4 , 2) .
Revised: C h (4 , 2) .

Original: 3. DEEP D ISTRIBUTIONALLY R OBUST L EARNING FOR LEARNING - TO - ABSTAIN PROBLEMS
Revised: 3. D ISTRIBUTIONALLY R OBUST L EARNING FOR LEARNING - TO - ABSTAIN PROBLEMS

Original: We present thedeep distributionally robust learning method for the layer-by-layer learning-to-abstain problems.
Revised: We present the distributionally robust learning method for the layer-by-layer learning-to-abstain problems.

Original: We ﬁrst review the (sub-)gradient of the θ in the original DRL formulation.
Revised: We ﬁrst review the (sub-)gradient of the ✓ in the original DRL formulation.

Original: We then discuss how to incorporate deep neural networks and how parameters in the neural networks are optimized.
Revised: We then discuss how to incorporate deep neural networks and how parameters in the neural networks are optimized.

Original: Finally, we present an algorithm for optimizing the abstaining loss in each layer in Alg. 1.
Revised: Finally, we present an algorithm for optimizing the abstaining loss in each layer in Alg. 1.

Original: Parameter learning in DRL:
Revised: Parameter learning in DRL:

Original: According to Fathony et al.
Revised: According to Fathony et al.

Original: (2018), the subdifferential of the objective with respect to the parameter θ can be fully characterized by the following.
Revised: (2018), the subdifferential of the objective with respect to the parameter ✓ can be fully characterized by the following.

Original: For each data sample:
Revised: For each data sample:

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: where H ∗ is the set of optimal solution of the following:
Revised: where H ⇤ is the set of optimal solution of the following:

Original: [Equation]
Revised: [Equation]

Original: In general, we need solve a linear program to compute h ∗ .
Revised: In general, we need solve a linear program to compute h ⇤ .

Original: When α ≤ 0 . 5 in C a , there exists an analytical form of the solution such that the gradient can be computed efﬁciently.
Revised: When ↵  0 . 5 in C a , there exists an analytical form of the solution such that the gradient can be computed efﬁciently.

Original: Incorporation of deep neural networks: Instead of low-dimensional features, we incorporate deep neural networks to learn representation Φ ( x , θ , w ) for constraining the adversarial h in the DRL framework.
Revised: Incorporation of deep neural networks: Instead of low-dimensional features, we incorporate deep neural networks to learn representation  ( x , ✓ , w ) for constraining the adversarial h in the DRL framework.

Original: The last layer of the network is a linear layer parameterized by θ , the other layers are parameterized with w .
Revised: The last layer of the network is a linear layer parameterized by ✓ , the other layers are parameterized with w .

Original: Formally, Φ ( x , θ , w ) =
Revised: Formally,  ( x , ✓ , w ) =

Original: θ T φ ( x , w ) .
Revised: ✓ T  ( x , w ) .

Original: Then after we update θ using Eq. 7 and
Revised: Then after we update ✓ using Eq. 7 and

Original: Eq. 6, the sub-gradient of φ ( x , w ) can be computed as follows:
Revised: Eq. 6, the sub-gradient of  ( x , w ) can be computed as follows:

Original: [Equation]
Revised: [Equation]

Original: where 1 ( y ) is the one-hot encoding of y .
Revised: where 1 ( y ) is the one-hot encoding of y .

Original: Then we use Eq.
Revised: Then we use Eq.

Original: 8 to back-prop to compute gradients of w in the networks.
Revised: 8 to back-prop to compute gradients of w in the networks.

Original: We present the algorthim in Alg.1. Algorithm 1: Input : D
Revised: We present the algorthim in Alg.

Original: , cost-sensitive loss C h , label taxonomy with T layers, T DNN: M 0 , ..., M T − 1 .

Original: 2: Decompose C h into T layer-by-layer abstaining loss C 0 a , ..., C T − 1 a , pretrain Musing C 0 a ; 3: For t ∈ [0 , T − 1] : 4:

Original: Use pretrained M 0 to initialize

Original: M 1 , ..., M T − 1 ; 5: M t = Algorithm 1 ( C ta , D train ) ; 6: 7: //Making predictions: 8: For t ∈ 0 , T − 1] : 9: ( D abt , ˆ Y t ) =
Revised: 1. derived gradients

Original: Predict ( M t , D test , C ta ) ; 10:

Original: D test =

Original: D abt ; 11: Output : Trained models M 0 , ..., M T − 1 , predictions for all the test data.

Original: 3. T HE OVERALL ALGORITHM
Revised: 3. T HE OVERALL ALGORITHM

Original: We now present the training and prediction techniques for the whole taxonomy.
Revised: We now present the training and prediction techniques for the whole taxonomy.

Original: Training: We run Alg. 1 for each layer of the taxonomy using the corresponding C a in that layer.
Revised: Training: We run Alg. 1 for each layer of the taxonomy using the corresponding C a in that layer.

Original: However, since the only difference in the data is the relabeling of the labels using nodes in different layers, we initialize the networks in the father layers using pre-trained networks in the leaf layer to achieve efﬁcient training.
Revised: However, since the only difference in the data is the relabeling of the labels using nodes in different layers, we initialize the networks in the father layers using pre-trained networks in the leaf layer to achieve efﬁcient training.

Original: Making predictions: After the models for each of the layers are trained, the prediction follows a bottom-up procedure.
Revised: Making predictions: After the models for each of the layers are trained, the prediction follows a bottom-up procedure.

Original: We ﬁrst use the model in the leaf layer to make predictions on all the samples, the abstained samples are collected for predictions using the father layer’s model.
Revised: We ﬁrst use the model in the leaf layer to make predictions on all the samples, the abstained samples are collected for predictions using the father layer’s model.

Original: Note that the prediction also requires the loss C a since it involves solving the game, like in Theorem 1.
Revised: Note that the prediction also requires the loss C a since it involves solving the game, like in Theorem 1.

Original: This process goes on until we reach the root of the tree.
Revised: This process goes on until we reach the root of the tree.

Original: Eventually, every data sample is assigned to a node in the taxonomy.
Revised: Eventually, every data sample is assigned to a node in the taxonomy.

Original: We summarize the whole algorithm in Alg. 2.
Revised: We summarize the whole algorithm in Alg. 2.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: In this section, we evaluate the effectiveness of our proposed algorithm in two real-world tasks: bird classiﬁcation and cell classiﬁcation.
Revised: In this section, we evaluate the effectiveness of our LAM algorithm in two real-world tasks: bird classiﬁcation and cell classiﬁcation.

Revised: We also compare one-layer LAM with previous baselines on real-world image classiﬁcation datasets: CIFAR10 and SVHN.

Original: For the bird data, we use a subset of the Aves supercategory data from iNaturalist (iNat)
Revised: For the bird data, we use a subset of the Aves supercategory data from iNaturalist (iNat)

Original: Competition.
Revised: Competition.

Original: The Aves dataset contains images of bird species in the Aves kingdom with labels on the hierarchical taxonomy of the animal in the image: from kingdom to order, family, genus, and species (ground-truth).
Revised: The Aves dataset contains images of bird species in the Aves kingdom with labels on the hierarchical taxonomy of the animal in the image: from kingdom to order, family, genus, and species (ground-truth).

Original: Thus, there are ﬁve layers in the Aves data tree with the Aves kingdom as the  root and species as leaf nodes.
Revised: Thus, there are ﬁve layers in the Aves data tree with the Aves kingdom as the [Figure] root and species as leaf nodes.

Original: The iNat data was pre-split into training and validation images, where the training data exhibits a long-tailed distribution across classes, and the validation set exhibits a uniform distribution across classes.
Revised: The iNat data was pre-split into training and validation images, where the training data exhibits a long-tailed distribution across classes, and the validation set exhibits a uniform distribution across classes.

Original: We chose the Aves supercategory since it has the most number of training images and the third most number of validation images in the iNat 2018 competition dataset. We selected the top 100 species/classes from the Aves supercategory data by the number of training and validation images per class to construct a subset to train and evaluate LAM on. The hierarchy of the Aves subset has 244 nodes intotal: 100 classes at the species level (leaf nodes),classes at the genus level, 41 classes at the family level, 17 classes at the order level, and 1 class at the Aves kingdom level (root). To construct our training and validation sets, 200 random images were selected from each class, where a random 100 of those images were used for the validation set and the other 100 were used for the test set.
Revised: We include more dataset details in the appendix.

Original: The resulting training dataset contains 53,561 images, and the validation and test sets each contain 10,000 images.

Original: [Table]
Revised: [Table]

Original: Our cell data comes from the triple-negative breast cancer (TNBC) dataset introduced inKeren et al.
Revised: Our cell data comes from the triple-negative breast cancer (TNBC) dataset introduced in Keren et al.

Original: This data comes from multiplexed imaging, a technique that quantiﬁes the abundance of biomarkers within a sample while still capturing the spatial organization of the tissue.
Revised: This data comes from multiplexed imaging, a technique that quantiﬁes the abundance of biomarkers within a sample while still capturing the spatial organization of the tissue.

Original: After some processing, the dimensions of the image were 1988 x 1988 x 23. The channel dimension of 23 refers to the quantiﬁcation of 23 different biomarkers. We segment the cells from the tissue sample Greenwald et al.
Revised: Table 1 gives the cell types in the dataset and their respective counts in the training and validation sets. Note that in this data, not all the leaf nodes are in the same level.

Original: and use the biomarkers as features for classiﬁcation. The subset of the data we used contains eight patients for training and four in our validation set.
Revised: We then add dummy intermediate nodes to the taxonomy, which does not affect the cost structure. More dataset details can be found in the appendix.

Original: There are 24,443 cells in the training data and 11,095 cells in the validation set.
Revised: CIFAR10 and SVHN are both non-hierarchical or one layer datasets.

Original: Table 1 gives the cell types in the dataset and their respective counts in the training and validation sets.
Revised: We use these datasets to validate our advantage on one-layer learning-to-abstain problems over previous baseline Deep Gambler (Liu et al., 2019).

Original:  4. C OMPETING M ETHODS
Revised: To be consistent, we use the same training, validation, and test split as in Liu et al. 4. C OMPETING M ETHODS

Original: We use the same abstaining loss for each layer and implement LAM models with α =[0 . 1 , 0 . 2 , ..., 0 . 9] in Aves classiﬁcation.
Revised: We use the same abstaining loss for each layer and implement LAM models with ↵ = 2 , ..., 0 . 9] in Aves classiﬁcation.

Original: In cell classiﬁcation, we set α =
Revised: In cell classiﬁcation, we set ↵ =

Original: We compare our method with DARTS (Deng et al., 2012).
Revised: We compare our method with DARTS (Deng et al., 2012).

Original: DARTS uses the information gain as the reward for each node in the tree.
Revised: DARTS uses the information gain as the reward for each node in the tree.

Original: After a model is trained on the leaf nodes, a hyperparameter that adjusts the balance between accuracy and speciﬁcity is tuned using binary search.
Revised: After a model is trained on the leaf nodes, a hyperparameter that adjusts the balance between accuracy and speciﬁcity is tuned using binary search.

Original: Again, it is not clear what cost-sensitive loss is optimized.
Revised: Again, it is not clear what cost-sensitive loss is optimized.

Original: In order to make the comparison fair, we implement two versions of DARTS.
Revised: In order to make the comparison fair, we implement two versions of DARTS.

Original: 1) DARTS (reward):
Revised: 1) DARTS (reward):

Original: We use the original reward but set the overall accuracy to be the same as every DRL model.
Revised: We use the original reward but set the overall accuracy to be the same as every DRL model.

Original: 2) DARTS (abstain loss):
Revised: 2) DARTS (abstain loss):

Original: We modify the reward in DARTS using a “negative cost” so that each node in the tree is associated with a reward that is either − α for nodes in the leaf layer or summation of previous abstaining losses for non-leaf layers.
Revised: We modify the reward in DARTS using a “negative cost” so that each node in the tree is associated with a reward that is either  ↵ for nodes in the leaf layer or summation of previous abstaining losses for non-leaf layers.

Original: Then we tune the modiﬁed version of DARTS using the same overall accuracy achieved by our method.
Revised: Then we tune the modiﬁed version of DARTS using the same overall accuracy achieved by our method.

Original: Also, sometimes DARTS cannot achieve the exact required accuracy.
Revised: Also, sometimes DARTS cannot achieve the exact required accuracy.

Original: We then compare to the closest similar accuracy we can ﬁnd using DARTS.
Revised: We then compare to the closest similar accuracy we can ﬁnd using DARTS.

Original: Note that this correspondence between abstaining losses and rewards is only possible when we use the same α for all the nodes in the same layer.4. I MPLEMENTATION D ETAILS
Revised: Note that this correspondence between abstaining losses and rewards is only possible when we use the same ↵ for all the nodes in the same layer.

Original: For Aves data, we use a ResNeXt-50-32x4d model pre-trained on ImageNet with an SGD optimizer.

Original: We ﬁrst train the ResNeXt model on cross-entropy loss on the Aves classes (leaf nodes) for 21 epochs to use as a base model.

Original: The initial learning rate is 0 . 0005 and is decayed by a factor of 10 every 10 epochs.

Original: We train and evaluate LAM with α = 0 . 9 in increments of 0 . 1 on the Aves dataset.
Revised: We also compare our method against Deep Gamblers (DG) (Liu et al., 2019).

Original: One LAM model is trained per layer of the Aves hierarchy, and we employ transfer learning to expedite the training process by using the weights of the model trained on cross-entropy loss and replacing the last linear layer with a linear layer of the correct number of classes.

Original: To solve the game for training and validating LAM models with α > 0 . 5 , we use the Gurobi Optimizer library. For TNBC tabular data, we use an artiﬁcial neural network with 3 hidden layers and a ﬁnal softmax layer trained using an Adam optimizer on cross entropy loss  ona reduced set of the full set of TNBC classes. The network is trained for 30 epochs with a learning rate of 0 .
Revised: DG minimizes an abstaining loss function based on the doubling rate of gambling–inspired by portfolio theory–in an empirical risk minimization framework.

Original: 001 and early stopping afterepochs of no improvement. This model is trained and evaluated for LAM using the same method as described for Aves data, but with α = 0 . 3 to achieve an even distribution of accuracies. Values of α above 0.3 were found to result in the same accuracy as the base model, making their results uninformative.
Revised: We include more details in the appendix.

Original: [Figure]

Original: 4. R ESULTS AND ANALYSIS
Revised: 4. R ESULTS AND ANALYSIS

Original: How abstaining loss affects the learning-to-abstain performance?
Revised: How abstaining loss affects the learning-to-abstain performance?

Original: We plot the overall error rate and abstaining rate of LAM models using different abstaining losses in the Aves classiﬁcation task’s leaf layer in Fig.
Revised: We plot the overall error rate and abstaining rate of LAM models using different abstaining losses in the Aves classiﬁcation task’s leaf layer in Fig.

Original: We see that the larger the abstaining cost is, the more aggressive the classiﬁer is, thus the larger the error rate and the lower the abstaining rate.
Revised: We see that the larger the abstaining cost is, the more aggressive the classiﬁer is, thus the larger the error rate and the lower the abstaining rate.

Original: This validates the working mechanism of the abstaining cost.
Revised: This validates the working mechanism of the abstaining cost.

Original: Imagine the abstaining cost is closer to the cost of making a wrong prediction.
Revised: Imagine the abstaining cost is closer to the cost of making a wrong prediction.

Original: The classiﬁer does not gain much by rejecting to make a prediction.
Revised: The classiﬁer does not gain much by rejecting to make a prediction.

Original: On the other hand, if the abstaining cost is small, the classiﬁer would favor abstentions more.
Revised: On the other hand, if the abstaining cost is small, the classiﬁer would favor abstentions more.

Revised: [Figure]

Original: How does the overall algorithm perform?
Revised: How does the overall algorithm perform?

Original: We show the comparison  of non-abstained accuracy on the Aves classiﬁcation in Fig.
Revised: We show the comparison with DARTS of non-abstained accuracy on the Aves classiﬁcation in Fig.

Original: 6 (a) and (b), which are the correct samples over all the predicted samples.
Revised: 5 (a) and (b), which are the correct samples over all the predicted samples.

Original: We can see that, by achieving the same overall accuracy in the whole taxonomy, our method achieves higher non-abstained accuracy consistently in each of the layers.
Revised: We can see that, by achieving the same overall accuracy in the whole taxonomy, our method achieves higher non-abstained accuracy consistently in each of the layers.

Original: We also compare the cost-sensitive loss achieved by our method and the modiﬁed version of DARTS in Fig.
Revised: We also compare the cost-sensitive loss achieved by our method and the modiﬁed version of DARTS in Fig.

Original: 6 (c).
Revised: 5 (c).

Original: We can see that our cost is much lower in the high accuracy domain.
Revised: We can see that our cost is much lower in the high accuracy domain.

Revised: We also show the comparison with DG of cost-sensitive loss on CIFAR10 and SVHN in Fig.

Original: Can our method be used to achieve a speciﬁc performance proﬁle?
Revised: 7 (a) and (b). We see that LAM (with only one-layer) achieves better cost-sensitive loss when the same accuracy is obtained. In

Revised: Fig 7 (c) and (d), we compare cumulative overall accuracy of equivalent LAM and DG models on

Revised: Aves classiﬁcation.

Revised: The overall accuracy LAM achieves is signiﬁcantly larger than DG at every level at which a model is trained–when we set DG’s loss to be the same as LAM.

Revised: Note that level 4 denotes the singular root node of the Aves hierarchy, and all inputs abstained on in level 3 are automatically deemed correct in level 4.

Original:  Since our method can assign different abstaining losses to each class (different α k ), we show how we can achieve a similar performance proﬁle by using a loss-adjustment heuristic.
Revised: Can our method be used to achieve a speciﬁc performance proﬁle? Since our method can assign different abstaining losses to each class (different ↵ k ), we show how we can achieve a similar performance proﬁle by using a loss-adjustment heuristic.

Original: Here, the user-deﬁned performance proﬁle means the tuple of overall accuracy, error rate and abstaining rate.
Revised: Here, the user-deﬁned performance proﬁle means the tuple of overall accuracy, error rate and abstaining rate.

Original: In Fig.
Revised: In Fig.

Original: 6 (d), we use the leaf layer in the Aves taxonomy and start with α = 0 .
Revised: 5 (d), we use the leaf layer in the Aves taxonomy and start with ↵ = 0 .

Original: 2 LAM.
Revised: 2 LAM.

Original: Our goal is to achieve a similar performance proﬁle achieved by DARTS with overall accuracy 0 .
Revised: Our goal is to achieve a similar performance proﬁle achieved by DARTS with overall accuracy 0 .

Original: 9 in the leaf layer.
Revised: 9 in the leaf layer.

Original: After four iterations of loss-adjusting, we achieve a very similar performance proﬁle with DARTS.
Revised: After four iterations of loss-adjusting, we achieve a very similar performance proﬁle with DARTS.

Original: The loss-adjustment method is a binary search that takes as inputs the error per class proﬁles of the LAM model (in the ﬁrst iteration, α = 0 .
Revised: The loss-adjustment method is a binary search that takes as inputs the error per class proﬁles of the LAM model (in the ﬁrst iteration, ↵ = 0 .

Original: 2 LAM) and target model (DARTS accuracy 0 . 9 ).
Revised: 2 LAM) and target model (DARTS accuracy 0 . 9 ).

Original: In every iteration, the LAM model is retrained using an alpha vector that is updated based on a comparison of the error per class between the LAM and target model: if, for a class k , the error rate is less than the error rate of the same class achieved by the target model, then α k for that class is increased to be ( α k, current +1) / 2 .
Revised: In every iteration, the LAM model is retrained using an alpha vector that is updated based on a comparison of the error per class between the LAM and target model: if, for a class k , the error rate is less than the error rate of the same class achieved by the target model, then ↵ k for that class is increased to be ( ↵ k, current +1) / 2 .

Original: An increase in α corresponds to a lower abstaining and higher error rate as seen in Fig.
Revised: An increase in ↵ corresponds to a lower abstaining and higher error rate as seen in Fig.

Original: If the error rate is greater than the error rate achieved by the target model, then α is updated as ( α k, current ) / 2 .
Revised: If the error rate is greater than the error rate achieved by the target model, then ↵ is updated as ( ↵ k, current ) / 2 .

Original: The updated α values are compiled into an updated vector to retrain the LAM model on.
Revised: The updated ↵ values are compiled into an updated vector to retrain the LAM model on.

Original: This iterative process stops via a ”lookback” condition: at iteration i , if the overall accuracy, error, and abstain rates of the previous iteration i − 1 are all closer in magnitude to the target rates than the current iteration, we stop and use the LAM model trained at iteration i − 1 as the closest model that achieves the target proﬁle.
Revised: This iterative process stops via a ”lookback” condition: at iteration i , if the overall accuracy, error, and abstain rates of the previous iteration i  1 are all closer in magnitude to the target rates than the current iteration, we stop and use the LAM model trained at iteration i  1 as the closest model that achieves the target proﬁle.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: Deep Cost-Sensitive Learning Traditional cost-sensitive learning works usually focus on transforming data samples (Domingos, 1999), changing the prediction procedure (Langford & Beygelzimer,
Revised: Deep Cost-Sensitive Learning Traditional cost-sensitive learning works usually focus on transforming data samples (Domingos, 1999), changing the prediction procedure (Langford & Beygelzimer,

Original: [Figure]
Revised: [Figure]

Original: 2005; Zadrozny et al., 2003), and modifying the existing learning models to include cost-sensitive information (Kukar et al., 1998;
Revised: 2005; Zadrozny et al., 2003), and modifying the existing learning models to include cost-sensitive information (Kukar et al., 1998;

Original: Tu & Lin, 2010; Zhou & Liu, 2005; Cao et al., 2013).
Revised: Tu & Lin, 2010; Zhou & Liu, 2005; Cao et al., 2013; Gangrade et al., 2021).

Original: The investigation of the cost-sensitive perspective in deep learning is still limited.
Revised: The investigation of the cost-sensitive perspective in deep learning is still limited.

Original: Existing works usually optimize a convex surrogate (Chen & Warren, 2013; Chung et al., 2016; Khan et al., 2017).
Revised: Existing works usually optimize a convex surrogate (Chen & Warren, 2013; Chung et al., 2016; Khan et al., 2017; Charoenphakdee et al., 2021).

Original: Learning to abstain can be regarded as a special case of cost-sensitive learning.
Revised: Learning to abstain can be regarded as a special case of cost-sensitive learning.

Original: There are several different strategies.
Revised: There are several different strategies.

Original: Besides the naive dropout-based method (Gal & Ghahramani, 2016), Geifman & El-Yaniv (2017; 2019) proposed to change the architecture to have a network for deciding abstain or not, while Liu et al.
Revised: Besides the naive dropout-based method (Gal & Ghahramani, 2016), Geifman & El-Yaniv (2017; 2019) proposed to change the architecture to have a network for deciding abstain or not, while Liu et al.

Original: (2019) utilized a tunable weight to incorporate the rejection option in the loss function.
Revised: (2019) utilized a tunable weight to incorporate the rejection option in the loss function.

Original: In our paper, we investigate cost-sensitive loss on a tree taxonomy and decompose the loss into tractable layer-by-layer abstaining losses, which can be optimized using DRL.
Revised: In our paper, we investigate cost-sensitive loss on a tree taxonomy and decompose the loss into tractable layer-by-layer abstaining losses, which can be optimized using DRL.

Original: Deep Hierarchical Classiﬁcation For learning with hierarchical classes, most of the work takes the top-down strategy (Zheng & Zhao, 2020; Guo & Zhao, 2021), which suffers from error propagation.
Revised: Deep Hierarchical Classiﬁcation For learning with hierarchical classes, most of the work takes the top-down strategy (Babbar et al., 2013; 2016; Zheng & Zhao, 2020; Guo & Zhao, 2021), which suffers from error propagation.

Original: Others seek to add an adjustment component (Fan et al., 2015) or hierarchy-related regularization (Charuvaka & Rangwala, 2015) to the model to make reasonable hierarchical decisions.
Revised: Others seek to add an adjustment component (Fan et al., 2015) or hierarchyrelated regularization (Gopal & Yang, 2013; Charuvaka & Rangwala, 2015) to the model to make reasonable hierarchical decisions.

Original: Deng et al.
Revised: Deng et al.

Original: (2012) is a post-processing method that uses posterior probabilities obtained on a validation set to tune parameters according to information-gain reward and target accuracy.
Revised: (2012) is a post-processing method that uses posterior probabilities obtained on a validation set to tune parameters according to information-gain reward and target accuracy.

Original: These methods heavily depend on an accurate probabilistic classiﬁer, which is usually not trained under using hierarchical information.
Revised: These methods heavily depend on an accurate probabilistic classiﬁer, which is usually not trained under using hierarchical information.

Original: Direct improvements over DARTS have included Bayesian methods learning (Jia et al., 2013) and integrating the uncertainty of predicting the parent nodes (Wang et al., 2017), as well as an approach that uses modiﬁed top-down and ﬂatten methods to recognize novel classes (Lee et al., 2018).
Revised: Direct improvements over DARTS have included Bayesian methods learning (Jia et al., 2013) and integrating the uncertainty of predicting the parent nodes (Wang et al., 2017), as well as an approach that uses modiﬁed top-down and ﬂatten methods to recognize novel classes (Lee et al., 2018).

Original: Recently, Zheng & Zhao (2021) proposes to decompose the hierarchical problem in a similar way.
Revised: Recently, Zheng & Zhao (2021) proposes to decompose the hierarchical problem in a similar way.

Original: However, it stills uses the post-hoc thresholding method for each of the layers.
Revised: However, it stills uses the post-hoc thresholding method for each of the layers.

Original: Distributionally robust learning Distributionally robust learning methods have been developed to robustly minimize the general loss functions (Fathony et al., 2016; 2017; 2018).
Revised: Distributionally robust learning Distributionally robust learning methods have been developed to robustly minimize the general loss functions (Fathony et al., 2016; 2017; 2018).

Original: In the most general case, the minimax game is solved using linear programming, which is hard to scale to large number of classes.
Revised: In the most general case, the minimax game is solved using linear programming, which is hard to scale to large number of classes.

Original: More advanced techniques have been developed to provide analytical solutions (Fathony et al., 2018) and deal with combinatorially large size of the game (Wang et al., 2015) to optimize the general performance metric.
Revised: More advanced techniques have been developed to provide analytical solutions (Fathony et al., 2018) and deal with combinatorially large size of the game (Wang et al., 2015) to optimize the general performance metric.

Original: Recently, this framework has been further developed to incorporate deep neural works for optimizing the logarithmic loss under covariate shift (Wang et al., 2021) and make general losses differetiable (Fathony & Kolter, 2020).
Revised: Recently, this framework has been further developed to incorporate deep neural works for optimizing the logarithmic loss under covariate shift (Wang et al., 2021) and make general losses differetiable (Fathony & Kolter, 2020).

Original: Our work studies the relation between the general cost-sensitive loss and the learning-to-abstain losses and proposes feasible solutions using the DRL framework to deal with real-world high-dimensional data.
Revised: Our work studies the relation between the general cost-sensitive loss and the learning-to-abstain losses and proposes feasible solutions using the DRL framework to deal with real-world high-dimensional data.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper, we propose to break the overall cost-sensitive learning problem into learning-to-abstain sub-problems in each layer in the tree taxonomy.
Revised: In this paper, we propose to break the overall cost-sensitive learning problem into learning-to-abstain sub-problems in each layer in the tree taxonomy.

Original: We provide a distributionally robust learning method for optimizing the abstaining losses in each layer.
Revised: We provide a distributionally robust learning method for optimizing the abstaining losses in each layer.

Original: We show the bijective correspondence between the hierarchical cost-sensitive loss and the set of abstaining losses.
Revised: We show the bijective correspondence between the hierarchical cost-sensitive loss and the set of abstaining losses.

Original: We evaluate our method on real-world datasets from applications in bird classiﬁcation and cell classiﬁcation.
Revised: We evaluate our method on real-world datasets from applications in bird classiﬁcation and cell classiﬁcation.

Original: Our method outperforms the competing methods by achieving lower cumulative cost and higher accuracy in each layer.
Revised: Our method outperforms the competing methods by achieving lower cumulative cost and higher accuracy in each layer.

Original: We also provide a heuristic for adjusting abstaining losses to achieve a desired performance proﬁle.
Revised: We also provide a heuristic for adjusting abstaining losses to achieve a desired performance proﬁle.