Large-scale Optimization of Partial AUC in a Range of False Positive Rates

The area under the receiver operating characteristic (ROC) curve (AUC) is one of the most widely used performance measures for classifiers in machine learning, especially when the data is imbalanced between the classes (Hanley and McNeil, 1982; Bradley, 1997). Typically, the classifier produces a score for each data point. Then a data point is classified as positive if its score is above a chosen threshold; otherwise, it is classified as negative. Varying the threshold will change the true positive rate (TPR) and the false positive rate (FPR) of the classifier. The ROC curve shows the TPR as a function of the FPR that corresponds to the same threshold. Hence, maximizing the AUC of a classifier is essentially maximizing the classifier’s average TPR over all FPRs from zero to one. However, for some applications, some FPR regions have no practical relevance. So does the TPR over those regions. For example, in clinical practice, a high FPR in diagnostic tests often results in a high monetary cost, so people may only need to maximize the TPR when the FPR is low (Dodd and Pepe, 2003; Ma et al., 2013; Yang et al., 2019). Moreover, since two models with the same AUC can still have different ROCs, the AUC does not always reflect the true performance of a model that is needed in a particular production environment (Bradley, 2014).

As a generalization of the AUC, the partial AUC (pAUC) only measures the area under the ROC curve that is restricted between two FPRs. A probabilistic interpretation of the pAUC can be found in Dodd and Pepe (2003). In contrast to the AUC, the pAUC represents the average TPR only over a relevant range of FPRs and provides a performance measure that is more aligned with the practical needs in some applications.

In literature, the existing algorithms for training a classifier by maximizing the pAUC include the boosting method (Komori and Eguchi, 2010) and the cutting plane algorithm (Narasimhan and Agarwal, 2013b; Narasimhan and Agarwal, 2013a, 2017). However, the former has no theoretical guarantee, and the latter applies only to linear models. More importantly, both methods require processing all the data in each iteration and thus, become computationally inefficient for large datasets.

In this paper, we proposed an approximate gradient method for maximizing the pAUC that works for nonlinear models (e.g., deep neural networks) and only needs to process randomly sampled positive and negative data points of any size in each iteration. In particular, we formulated the maximization of the pAUC as a non-smooth difference-of-convex (DC) program (Tao and An, 1997; Le Thi and Dinh, 2018). Due to non-smoothness, most existing DC optimization algorithms cannot be applied to our formulation. Motivated by Sun and Sun (2021), we approximate the two non-smooth convex components in the DC program by their Moreau envelopes and obtain a smooth approximation of the problem, which will be solved using the gradient descent method. Since the gradient of the smooth problem cannot be calculated explicitly, we approximated the gradient by solving the two proximal-point subproblems defined by each convex component using the stochastic block coordinate descent (SBCD) method. Our method, besides its low per-iteration cost, has a rigorous theoretical guarantee, unlike the existing methods. In fact, we show that our method finds a nearly $\epsilon$-critical point of the pAUC optimization problem in $\tilde{O}(\epsilon^{-6})$ iterations with only small samples of positive and negative data points processed per iteration.¹¹1Throughout the paper, $\tilde{O}(\cdot)$ suppresses all logarithmic factors. This is the main contribution of this paper.

Note that, for non-convex non-smooth optimization, the existing stochastic methods (Davis and Grimmer, 2019; Davis and Drusvyatskiy, 2018) find an nearly $\epsilon$-critical point in $O(\epsilon^{-4})$ iterations under a weak convexity assumption. Our method needs $O(\epsilon^{-6})$ iterations because our problem is a DC problem with both convex components non-smooth which is much more challenging than a weakly non-convex minimization problem. In addition, our iteration number matches the known best iteration complexity for non-smooth non-convex min-max optimization (Rafique et al., 2021; Liu et al., 2021) and non-smooth non-convex constrained optimization (Ma et al., 2020).

In addition to pAUC optimization, our method can be also used to minimize the sum of ranked range (SoRR) loss, which can be viewed as a special case of pAUC optimization. Many machine learning models are trained by minimizing an objective function, which is defined as the sum of losses over all training samples (Vapnik, 1992). Since the sum of losses weights all samples equally, it is insensitive to samples from minority groups. Hence, the sum of top-$k$ losses (Shalev-Shwartz and Wexler, 2016; Fan et al., 2017) is often used as an alternative objective function because it provides the model with robustness to non-typical instances. However, the sum of top-$k$ losses can be very sensitive to outliers, especially when $k$ is small. To address this issue, Hu et al. (2020) proposed the SoRR loss as a new learning objective, which is defined as the sum of a consecutive sequence of losses from any range after the losses are sorted. Compared to the sum of all losses and the sum of top-$k$ losses, the SoRR loss maintains a model’s robustness to a minority group but also reduces the model’s sensitivity to outliers. See Fig.1 in Hu et al. (2020) for an illustration of the benefit of using the SoRR loss over other ways of aggregating individual losses.

To minimize the SoRR loss, Hu et al. (2020) applied a difference-of-convex algorithm (DCA) (Tao and An, 1997; An and Tao, 2005), which linearizes the second convex component and solves the resulting subproblem using the stochastic subgradient method. DCA has been well studied in literature; but when the both components are non-smooth, as in our problem, only asymptotic convergence results are available. To establish the total number of iterations needed to find an $\epsilon$-critical point in a non-asymptotic sense, most existing studies had to assume that at least one of the components is differentiable, which is not the case in this paper. Using the approximate gradient method presented in this paper, one can find a nearly $\epsilon$-critical point of the SoRR loss optimization problem in $\tilde{O}(\epsilon^{-6})$ iterations.

The pAUC has been studied for decades (McClish, 1989; Thompson and Zucchini, 1989; Jiang et al., 1996; Yang and Ying, 2022). However, most studies focused on its estimation (Dodd and Pepe, 2003) and application as a performance measure, while only a few studies were devoted to numerical algorithms for optimizing the pAUC. Efficient optimization methods have been developed for maximizing AUC and multiclass AUC by Ying et al. (2016) and Yang et al. (2021a), but they cannot be applied to pAUC. Besides the boosting method (Komori and Eguchi, 2010) and the cutting plane algorithm (Narasimhan and Agarwal, 2013b; Narasimhan and Agarwal, 2013a, 2017) mentioned in the previous section, Ueda and Fujino (2018); Yang et al. (2022, 2021b); Zhu et al. (2022) developed surrogate optimization techniques that directly maximize a smooth approximation of the pAUC or the two-way pAUC (Yang et al., 2019). However, their approaches can only be applied when the FPR starts from exactly zero. On the contrary, our method allows the FPR to start from any value between zero and one. Wang and Chang (2011) and Ricamato and Tortorella (2011) developed algorithms that use the pAUC as a criterion for creating a linear combination of multiple existing classifiers while we consider directly train a classifier using the pAUC.

DC optimization has been studied since the 1950s (Alexandroff, 1950; Hartman, 1959). We refer interested readers to Tuy (1995); Tao and An (1998, 1997); Pang et al. (2017); Le Thi and Dinh (2018), and the references therein. The actively studied numerical methods for solving a DC program include DCA (Tao and An, 1998, 1997; An and Tao, 2005; Souza et al., 2016), which is also known as the concave-convex procedure (Yuille and Rangarajan, 2003; Sriperumbudur and Lanckriet, 2009; Lipp and Boyd, 2016), the proximal DCA (Sun et al., 2003; Moudafi and Maingé, 2006; Moudafi, 2008; An and Nam, 2017), and the direct gradient methods (Khamaru and Wainwright, 2018). However, when the two convex components are both non-smooth, the existing methods have only asymptotic convergence results except the method by Abbaszadehpeivasti et al. (2021), who considered a stopping criterion different from ours. When at least one component is smooth, non-asymptotic convergence rates have been established with and without the Kurdyka-Łojasiewicz (KL) condition (Souza et al., 2016; Artacho et al., 2018; Wen et al., 2018; An and Nam, 2017; Khamaru and Wainwright, 2018).

The algorithms mentioned above are deterministic and require processing the entire dataset per iteration. Stochastic algorithms that process only a small data sample per iteration have been studied (Mairal, 2013; Nitanda and Suzuki, 2017; Le Thi et al., 2017; Deng and Lan, 2020; He et al., 2021). However, they all assumed smoothness on at least one of the two convex components in the DC program. The stochastic methods of Xu et al. (2019); Thi et al. (2021); An et al. (2019) can be applied when both components are non-smooth but their methods require an unbiased stochastic estimation of the gradient and/or value of the two components, which is not available in the DC formulation of the pAUC maximization problem in this paper.

The technique most related to our work is the smoothing method based on the Moreau envelope (Ellaia, 1984; Gabay, 1982; Hiriart-Urruty, 1985, 1991; Sun and Sun, 2021; Moudafi, 2021). Our work is motivated by Sun and Sun (2021); Moudafi (2021), but the important difference is that they studied deterministic methods and assumed either that one function is smooth or that the proximal-point subproblems can be solved exactly, which we do not assume. However, Sun and Sun (2021); Moudafi (2021) consider a more general problem and study the fundamental properties of the smoothed function such as its Lipschitz smoothness and how its stationary points correspond to those of the original problems. We mainly focus on partial AUC optimization which has a special structure we can utilize when solving the proximal-point subproblems. Additionally, Sun and Sun (2021) developed an algorithm when there were linear equality constraints, which we do not consider in this paper.

We consider a classical binary classification problem, where the goal is to build a predictive model that predicts a binary label $y\in\{1,-1\}$ based on a feature vector ${\mathbf{x}}\in\mathbb{R}^{p}$. Let $h_{\mathbf{w}}:\mathbb{R}^{p}\rightarrow\mathbb{R}$ be the predictive model parameterized by a vector ${\mathbf{w}}\in\mathbb{R}^{d}$, which produces a score $h_{\mathbf{w}}({\mathbf{x}})$ for ${\mathbf{x}}$. Then ${\mathbf{x}}$ is classified as positive ($y=1$) if $h_{\mathbf{w}}({\mathbf{x}})$ is above a chosen threshold and classified as negative ($y=-1$), otherwise.

Let ${\mathcal{X}}_{+}=\{{\mathbf{x}}_{i}^{+}\}_{i}^{N_{+}}$ and ${\mathcal{X}}_{-}=\{{\mathbf{x}}_{i}^{-}\}_{i}^{N_{-}}$ be the sets of feature vectors of positive and negative training data, respectively. The problem of learning $h_{\mathbf{w}}$ through maximizing its empirical AUC on the training data can be formulated as

where $\mathbf{1}(\cdot)$ is the indicator function which equals one if the inequality inside the parentheses holds and equals zero, otherwise. According to the introduction, pAUC can be a better performance measure of $h_{{\mathbf{w}}}$ than AUC. Consider two FPRs $\alpha$ and $\beta$ with $0\leq\alpha<\beta\leq 1$. For simplicity of exposition, we assume $N_{-}\alpha$ and $N_{-}\beta$ are both integers. Let $m=N_{-}\alpha$ and $n=N_{-}\beta$. The problem of maximizing the empirical pAUC with FPR between $\alpha$ and $\beta$ can be formulated as

where $[j]$ denotes the index of the $j$th largest coordinate in vector $(h_{{\mathbf{w}}}({\mathbf{x}}_{j}^{-}))_{j=1}^{N_{-}}$ with ties broken arbitrarily. Note that $N_{+}(n-m)$ in (2) is a normalizer that makes the objective value between zero and one. Solving (2) is challenging due to discontinuity. Let $\ell:\mathbb{R}\rightarrow\mathbb{R}$ be a differential non-increasing loss function. Problem (2) can be approximated by the loss minimization problem

To facilitate the discussion, we first introduce a few notations. Given a vector $S=\left(s_{i}\right)_{i=1}^{N}\in\mathbb{R}^{N}$ and an integer $l$ with $0\leq l\leq N$, the sum of the top-$l$ values in $S$ is

where $[j]$ denotes the index of the $j$th largest coordinate in $S$ with ties broken arbitrarily. For integers $l_{1}$ and $l_{2}$ with $0\leq l_{1}<l_{2}\leq N$, $\phi_{l_{2}}(S)-\phi_{l_{1}}(S)$ is the sum from the $(l_{1}+1)$th to the $l_{2}$th (inclusive) largest coordinates of $S$, also called a sum of ranked range (SoRR). In addition, we define vectors

for $i=1,\dots,N_{+}$, where $s_{ij}({\mathbf{w}}):=\ell(h_{{\mathbf{w}}}({\mathbf{x}}_{i}^{+})-h_{{\mathbf{w}}}({\mathbf{x}}_{j}^{-}))$ for $i=1,\dots,N_{+}$ and $j=1,\dots,N_{-}$. Since $\ell$ is non-increasing, the $j$th largest coordinate of $S_{i}({\mathbf{w}})$ is $\ell(h_{{\mathbf{w}}}({\mathbf{x}}_{i}^{+})-h_{{\mathbf{w}}}({\mathbf{x}}_{[j]}^{-}))$. As a result, we have, for $i=1,\dots,N_{+}$,

Hence, after dropping the normalizer, (3) can be equivalently written as

where

Next, we introduce an interesting special case of (5), namely, the problem of minimizing SoRR loss. We still consider a supervised learning problem but the target $y\in\mathbb{R}$ does not need to be binary. We want to predict $y$ based on a feature vector ${\mathbf{x}}\in\mathbb{R}^{p}$ using $h_{\mathbf{w}}({\mathbf{x}})$. With a little abuse of notation, we measure the discrepancy between $h_{\mathbf{w}}({\mathbf{x}})$ and $y$ by $\ell(h_{\mathbf{w}}({\mathbf{x}}),y)$, where $\ell:\mathbb{R}^{2}\rightarrow\mathbb{R}_{+}$ is a loss function. We consider learning the model’s parameter ${\mathbf{w}}$ from a training set $\mathcal{D}=\{({\mathbf{x}}_{j},y_{j})\}_{j=1}^{N}$, where ${\mathbf{x}}_{j}\in\mathbb{R}^{p}$ and $y_{j}\in\mathbb{R}$ for $j=1,\dots,N$, by minimizing the SoRR loss. More specifically, we define vector

where $s_{j}({\mathbf{w}}):=\ell(h_{\mathbf{w}}({\mathbf{x}}_{j}),y_{j}),~{}j=1,\dots,N$. Recall (4). For any integers $m$ and $n$ with $0\leq m<n\leq N$, the problem of minimizing the SoRR loss with a range from $m+1$ to $n$ is formulated as $\min_{{\mathbf{w}}}\left\{\phi_{n}(S({\mathbf{w}}))-\phi_{m}(S({\mathbf{w}}))\right\}$, which is an instance of (5) with

If we view $S_{i}({\mathbf{w}})$ and $S({\mathbf{w}})$ only as vector-value functions of ${\mathbf{w}}$ but ignore how they are formulated using data, (7) is a special case of (6) with $N_{+}=1$ and $N_{-}=N$.

We first develop a stochastic algorithm for (5) with $f^{l}$ defined in (6). To do so, we make the following assumptions, which are satisfied by many smooth $h_{{\mathbf{w}}}$’s and $\ell$’s.

Given $f:\mathbb{R}^{d}\rightarrow\mathbb{R}\cup\{+\infty\}$, the subdifferential of $f$ is

where each element in $\partial f({\mathbf{w}})$ is called a subgradient of $f$ at ${\mathbf{w}}$. We say $f$ is $\rho$-weakly convex for some $\rho\geq 0$ if $f({\mathbf{v}})\geq f({\mathbf{w}})+\left\langle{\boldsymbol{\xi}},{\mathbf{v}}-{\mathbf{w}}\right\rangle-\frac{\rho}{2}\|{\mathbf{v}}-{\mathbf{w}}\|^{2}$ for any ${\mathbf{v}}$ and ${\mathbf{w}}$ and ${\boldsymbol{\xi}}\in\partial f({\mathbf{w}})$ and say $f$ is $\rho$-strongly convex for some $\rho\geq 0$ if $f({\mathbf{v}})\geq f({\mathbf{w}})+\left\langle{\boldsymbol{\xi}},{\mathbf{v}}-{\mathbf{w}}\right\rangle+\frac{\rho}{2}\|{\mathbf{v}}-{\mathbf{w}}\|^{2}$ for any ${\mathbf{v}}$ and ${\mathbf{w}}$ and ${\boldsymbol{\xi}}\in\partial f({\mathbf{w}})$. It is known that, if $f$ is $\rho$-weakly convex, then $f({\mathbf{w}})+\frac{1}{2\mu}\|{\mathbf{w}}\|^{2}$ is a $(\mu^{-1}-\rho)$-strongly convex function when $\mu^{-1}>\rho$.

Under Assumption 1, $\phi_{l}(S_{i}({\mathbf{w}}))$ is a composite of the closed convex function $\phi_{l}$ and the smooth map $S_{i}({\mathbf{w}})$. According to Lemma 4.2 in Drusvyatskiy and Paquette (2019), we have the following lemma.

To solve (5) numerically, we need to overcome the following challenges. (i) $F({\mathbf{w}})$ is non-convex even if each $s_{ij}({\mathbf{w}})$ is convex. In fact, $F({\mathbf{w}})$ is a DC function because, by Lemma 1, we can represent $F({\mathbf{w}})$ as the difference of the convex functions $f^{n}({\mathbf{w}})+\frac{1}{2\mu}\|{\mathbf{w}}\|^{2}$ and $f^{m}({\mathbf{w}})+\frac{1}{2\mu}\|{\mathbf{w}}\|^{2}$ with $\mu^{-1}>\rho$. (ii) $F({\mathbf{w}})$ is non-smooth due to $\phi_{l}$ so that finding an approximate critical point (defined below) of $F({\mathbf{w}})$ is difficult. (iii) Computing the exact subgradient of $f^{l}({\mathbf{w}})$ for $l=m,n$ requires processing $N_{+}N_{-}$ data pairs, which is computationally expensive for a large data set.

Because of challenges (i) and (ii), we have to consider a reasonable goal when solving (5). We say ${\mathbf{w}}\in\mathbb{R}^{d}$ is a critical point of $\eqref{eqn:pAUC_raw}$ if $\mathbf{0}\in\partial f^{n}({\mathbf{w}})-\partial f^{m}({\mathbf{w}})$. Given $\epsilon>0$, we say ${\mathbf{w}}\in\mathbb{R}^{d}$ is an $\epsilon$-critical point of $\eqref{eqn:pAUC_raw}$ if there exists ${\boldsymbol{\xi}}\in\partial f^{n}({\mathbf{w}})-\partial f^{m}({\mathbf{w}})$ such that $\|{\boldsymbol{\xi}}\|\leq\epsilon$. A critical point can only be achieved asymptotically in general.³³3A stronger notion than criticality is (directional) stationarity, which can also be achieved asymptotically (Pang et al., 2017). Within finitely many iterations, there also exists no algorithm that can find an $\epsilon$-critical point unless at least one of $f^{m}$ and $f^{n}$ is smooth, e.g., Xu et al. (2019). Since $f^{m}$ and $f^{n}$ are both non-smooth, we have to consider a weaker but achievable target, which is a nearly $\epsilon$-critical point defined below.

Definition 2 is reduced to the $\epsilon$-stationary point defined by Sun and Sun (2021); Moudafi (2021) when ${\mathbf{w}}$ equals ${\mathbf{w}}^{\prime}$ or ${\mathbf{w}}^{\prime\prime}$. However, obtaining their $\epsilon$-stationary point requires exactly solving the proximal mapping of $f^{m}$ or $f^{n}$ while finding a nearly $\epsilon$-critical point requires only solving the proximal mapping inexactly. When ${\mathbf{w}}$ is generated by a stochastic algorithm, we also call ${\mathbf{w}}$ a nearly $\epsilon$-critical point if it satisfies Definition 2 with each $\|\cdot\|$ replaced by $\mathbb{E}\|\cdot\|$.

Motivated by Sun and Sun (2021) and Moudafi (2021), we approximate non-smooth $F({\mathbf{w}})$ by a smooth function using the Moreau envelopes. Given a proper, $\rho$-weakly convex and closed function $f$ on $\mathbb{R}^{d}$, the Moreau envelope of $f$ with the smoothing parameter $\mu\in(0,\rho^{-1})$ is defined as

and the proximal mapping of $f$ is defined as

Note that the ${\mathbf{v}}_{\mu f}({\mathbf{w}})$ is unique because the minimization above is strongly convex. Standard results show that $f_{\mu}({\mathbf{w}})$ is smooth with $\nabla f_{\mu}({\mathbf{w}})=\mu^{-1}({\mathbf{w}}-{\mathbf{v}}_{\mu f}({\mathbf{w}}))$ and ${\mathbf{v}}_{\mu f}({\mathbf{w}})$ is $(1-\mu\rho)^{-1}$-Lipschitz continuous. See Proposition 13.37 in Rockafellar and Wets (2009) and Proposition 1 in Sun and Sun (2021). Hence, using the Moreau envelope, we can construct a smooth approximation of (5) as follows

Function $F^{\mu}$ has the following properties. The first property is shown in Sun and Sun (2021). We give the proof for the second in Appendix B.

Since $F^{\mu}$ is smooth, we can directly apply a first-order method for smooth non-convex optimization to (10). To do so, we need to evaluate $\nabla F^{\mu}({\mathbf{w}})$, which requires computing ${\mathbf{v}}_{\mu f^{m}}({\mathbf{w}})$ and ${\mathbf{v}}_{\mu f^{n}}({\mathbf{w}})$, i.e., exactly solving (9) with $f=f^{m}$ and $f=f^{n}$, respectively. Computing the subgradients of $f^{m}$ and $f^{n}$ require processing $N_{+}N_{-}$ data pairs which is costly. Unfortunately, the standard approach of sampling over data pairs does not produce unbiased stochastic subgradients of $f^{m}$ and $f^{n}$ due to the composite structure $\phi_{l}(S_{i}({\mathbf{w}}))$. In the next section, we will discuss a solution to overcome this challenge and approximate ${\mathbf{v}}_{\mu f^{m}}({\mathbf{w}})$ and ${\mathbf{v}}_{\mu f^{n}}({\mathbf{w}})$, which leads to an efficient algorithm for (10).

Consider (10) with $f^{l}$ defined in (6) for $l=m$ and $n$. To avoid of processing $N_{+}N_{-}$ data points, one method is to introduce dual variables ${\mathbf{p}}_{i}=(p_{ij})_{j=1}^{N_{-}}$ for $i=1,\dots,N_{+}$ and formulate $f^{l}$ as

where $\mathcal{P}^{l}=\{{\mathbf{p}}\in\mathbb{R}^{N_{-}}|\sum_{j=1}^{N_{-}}p_{j}=l,~{}p_{j}\in[0,1]\}$. Then (10) can be reformulated as a min-max problem and solved by a primal-dual stochastic gradient method (e.g. Rafique et al. (2021)). However, the maximization in (11) involves $N_{+}N_{-}$ decision variables and equality constraints, so the per-iteration cost is still $O(N_{+}N_{-})$ even after using stochastic gradients.

To further reduce the per-iteration cost, we take the dual form of the maximization in (11) (see Lemma 10 in Appendix B) and formulate $f^{l}$ as

where ${\boldsymbol{\lambda}}=(\lambda_{1},\dots,\lambda_{N_{+}})$. Hence, (9) with $f=f^{l}$ for $l=m$ and $n$ can be reformulated as

Note that $g^{l}({\mathbf{v}},{\boldsymbol{\lambda}})$ is jointly convex in ${\mathbf{v}}$ and ${\boldsymbol{\lambda}}$ when $\mu^{-1}>\rho=N_{+}N_{-}L$ (see Lemma 9 in Appendix B). Thanks to formulation (13), we can construct stochastic subgradient of $g^{l}$ and apply coordinate update to ${\boldsymbol{\lambda}}$ by sampling indexes $i$’s and $j$’s, which significantly reduce the computational cost when $N_{+}$ and $N_{-}$ are both large. We present this standard stochastic block coordinate descent (SBCD) method for solving (13) in Algorithm 1 and present its convergence property as follows.

Using Algorithm 1 to compute an approximation of ${\mathbf{v}}_{\mu f^{l}}({\mathbf{w}})$ for $l=m$ and $n$ and thus, an approximation of $\nabla F^{\mu}({\mathbf{w}})$, we can apply an approximate gradient descent (AGD) method to (10) and find a nearly $\epsilon$-critical point of (5) according to Lemma 3. We present the AGD method in Algorithm 2 and its convergence property as follows.

According to Theorem 5, to find a nearly $\epsilon$-critical point of (5), we need $K=\tilde{O}(\epsilon^{-2})$ iterations in Algorithm 2 and $\sum_{k=0}^{K-1}T_{k}=O(K^{3})=\tilde{O}(\epsilon^{-6})$ iterations of Algorithm 1 in total across all calls.

The technique in the previous sections can be directly applied to minimize the SoRR loss, which is formulated as (5) but with $f^{l}$ defined in (7). Due to the limit of space, we present the algorithm for minimizing the SoRR loss and its convergence result in Appendix A.

In this section, we demonstrate the effectiveness of our algorithm AGD-SBCD for pAUC maximization and SoRR loss minimization problems (see Appendix E.1 for details). All experiments are conducted in Python and Matlab on a computer with the CPU 2GHz Quad-Core Intel Core i5 and the GPU NVIDIA GeForce RTX 2080 Ti. All datasets we used are publicly available and contain no personally identifiable information and offensive contents.

Most existing methods for optimizing pAUC are deterministic and only have an asymptotic convergence property. We formulate pAUC optimization as a non-smooth DC program and develop a stochastic subgradient method based on the Moreau envelope smoothing technique. We show that our method finds a nearly $\epsilon$-critical point in $\tilde{O}(\epsilon^{-6})$ iterations and demonstrate its performance numerically. A limitation of this paper is the smoothness assumption on $s_{ij}({\mathbf{w}})$, which does not hold for some models, e.g., neural networks using ReLU activation functions. It is a future work to extend our results for non-smooth models.

This work was jointly supported by the University of Iowa Jumpstarting Tomorrow Program and NSF award 2147253. T. Yang was also supported by NSF awards 2110545 and 1844403, and Amazon research award. We thank Zhishuai Guo, Zhuoning Yuan and Qi Qi for discussing about processing the image dataset.