Yiming Ying

Shu Hu, Zhenhuan Yang, Xin Wang et al.

Supervised learning models are challenged by the intrinsic complexities of training data such as outliers and minority subpopulations and intentional attacks at inference time with adversarial samples. While traditional robust learning methods and the recent adversarial training approaches are designed to handle each of the two challenges, to date, no work has been done to develop models that are robust with regard to the low-quality training data and the potential adversarial attack at inference time simultaneously. It is for this reason that we introduce Outlier Robust Adversarial Training (ORAT) in this work. ORAT is based on a bi-level optimization formulation of adversarial training with a robust rank-based loss function. Theoretically, we show that the learning objective of ORAT satisfies the $\mathcal{H}$-consistency in binary classification, which establishes it as a proper surrogate to adversarial 0/1 loss. Furthermore, we analyze its generalization ability and provide uniform convergence rates in high probability. ORAT can be optimized with a simple algorithm. Experimental evaluations on three benchmark datasets demonstrate the effectiveness and robustness of ORAT in handling outliers and adversarial attacks. Our code is available at https://github.com/discovershu/ORAT.

Tianbao Yang, Yiming Ying

Area under the ROC curve, a.k.a. AUC, is a measure of choice for assessing the performance of a classifier for imbalanced data. AUC maximization refers to a learning paradigm that learns a predictive model by directly maximizing its AUC score. It has been studied for more than two decades dating back to late 90s and a huge amount of work has been devoted to AUC maximization since then. Recently, stochastic AUC maximization for big data and deep AUC maximization for deep learning have received increasing attention and yielded dramatic impact for solving real-world problems. However, to the best our knowledge there is no comprehensive survey of related works for AUC maximization. This paper aims to address the gap by reviewing the literature in the past two decades. We not only give a holistic view of the literature but also present detailed explanations and comparisons of different papers from formulations to algorithms and theoretical guarantees. We also identify and discuss remaining and emerging issues for deep AUC maximization, and provide suggestions on topics for future work.

Yunwen Lei, Tianbao Yang, Yiming Ying et al.

Recently, contrastive learning has found impressive success in advancing the state of the art in solving various machine learning tasks. However, the existing generalization analysis is very limited or even not meaningful. In particular, the existing generalization error bounds depend linearly on the number $k$ of negative examples while it was widely shown in practice that choosing a large $k$ is necessary to guarantee good generalization of contrastive learning in downstream tasks. In this paper, we establish novel generalization bounds for contrastive learning which do not depend on $k$, up to logarithmic terms. Our analysis uses structural results on empirical covering numbers and Rademacher complexities to exploit the Lipschitz continuity of loss functions. For self-bounding Lipschitz loss functions, we further improve our results by developing optimistic bounds which imply fast rates in a low noise condition. We apply our results to learning with both linear representation and nonlinear representation by deep neural networks, for both of which we derive Rademacher complexity bounds to get improved generalization bounds.

Puyu Wang, Yunwen Lei, Yiming Ying et al.

Recently there is a large amount of work devoted to the study of Markov chain stochastic gradient methods (MC-SGMs) which mainly focus on their convergence analysis for solving minimization problems. In this paper, we provide a comprehensive generalization analysis of MC-SGMs for both minimization and minimax problems through the lens of algorithmic stability in the framework of statistical learning theory. For empirical risk minimization (ERM) problems, we establish the optimal excess population risk bounds for both smooth and non-smooth cases by introducing on-average argument stability. For minimax problems, we develop a quantitative connection between on-average argument stability and generalization error which extends the existing results for uniform stability \cite{lei2021stability}. We further develop the first nearly optimal convergence rates for convex-concave problems both in expectation and with high probability, which, combined with our stability results, show that the optimal generalization bounds can be attained for both smooth and non-smooth cases. To the best of our knowledge, this is the first generalization analysis of SGMs when the gradients are sampled from a Markov process.

Zhenhuan Yang, Yan Lok Ko, Kush R. Varshney et al.

The use of machine learning models in consequential decision making often exacerbates societal inequity, in particular yielding disparate impact on members of marginalized groups defined by race and gender. The area under the ROC curve (AUC) is widely used to evaluate the performance of a scoring function in machine learning, but is studied in algorithmic fairness less than other performance metrics. Due to the pairwise nature of the AUC, defining an AUC-based group fairness metric is pairwise-dependent and may involve both \emph{intra-group} and \emph{inter-group} AUCs. Importantly, considering only one category of AUCs is not sufficient to mitigate unfairness in AUC optimization. In this paper, we propose a minimax learning and bias mitigation framework that incorporates both intra-group and inter-group AUCs while maintaining utility. Based on this Rawlsian framework, we design an efficient stochastic optimization algorithm and prove its convergence to the minimum group-level AUC. We conduct numerical experiments on both synthetic and real-world datasets to validate the effectiveness of the minimax framework and the proposed optimization algorithm.

Yunwen Lei, Rong Jin, Yiming Ying

While significant theoretical progress has been achieved, unveiling the generalization mystery of overparameterized neural networks still remains largely elusive. In this paper, we study the generalization behavior of shallow neural networks (SNNs) by leveraging the concept of algorithmic stability. We consider gradient descent (GD) and stochastic gradient descent (SGD) to train SNNs, for both of which we develop consistent excess risk bounds by balancing the optimization and generalization via early-stopping. As compared to existing analysis on GD, our new analysis requires a relaxed overparameterization assumption and also applies to SGD. The key for the improvement is a better estimation of the smallest eigenvalues of the Hessian matrices of the empirical risks and the loss function along the trajectories of GD and SGD by providing a refined estimation of their iterates.

Puyu Wang, Yunwen Lei, Yiming Ying et al.

Modern machine learning algorithms aim to extract fine-grained information from data to provide accurate predictions, which often conflicts with the goal of privacy protection. This paper addresses the practical and theoretical importance of developing privacy-preserving machine learning algorithms that ensure good performance while preserving privacy. In this paper, we focus on the privacy and utility (measured by excess risk bounds) performances of differentially private stochastic gradient descent (SGD) algorithms in the setting of stochastic convex optimization. Specifically, we examine the pointwise problem in the low-noise setting for which we derive sharper excess risk bounds for the differentially private SGD algorithm. In the pairwise learning setting, we propose a simple differentially private SGD algorithm based on gradient perturbation. Furthermore, we develop novel utility bounds for the proposed algorithm, proving that it achieves optimal excess risk rates even for non-smooth losses. Notably, we establish fast learning rates for privacy-preserving pairwise learning under the low-noise condition, which is the first of its kind.

Ming Yang, Xiyuan Wei, Tianbao Yang et al.

Many machine learning tasks can be formulated as a stochastic compositional optimization (SCO) problem such as reinforcement learning, AUC maximization, and meta-learning, where the objective function involves a nested composition associated with an expectation. While a significant amount of studies has been devoted to studying the convergence behavior of SCO algorithms, there is little work on understanding their generalization, i.e., how these learning algorithms built from training examples would behave on future test examples. In this paper, we provide the stability and generalization analysis of stochastic compositional gradient descent algorithms through the lens of algorithmic stability in the framework of statistical learning theory. Firstly, we introduce a stability concept called compositional uniform stability and establish its quantitative relation with generalization for SCO problems. Then, we establish the compositional uniform stability results for two popular stochastic compositional gradient descent algorithms, namely SCGD and SCSC. Finally, we derive dimension-independent excess risk bounds for SCGD and SCSC by trade-offing their stability results and optimization errors. To the best of our knowledge, these are the first-ever-known results on stability and generalization analysis of stochastic compositional gradient descent algorithms.

Zhenhuan Yang, Yingqiang Ge, Congzhe Su et al.

Recently, there has been an increasing adoption of differential privacy guided algorithms for privacy-preserving machine learning tasks. However, the use of such algorithms comes with trade-offs in terms of algorithmic fairness, which has been widely acknowledged. Specifically, we have empirically observed that the classical collaborative filtering method, trained by differentially private stochastic gradient descent (DP-SGD), results in a disparate impact on user groups with respect to different user engagement levels. This, in turn, causes the original unfair model to become even more biased against inactive users. To address the above issues, we propose \textbf{DP-Fair}, a two-stage framework for collaborative filtering based algorithms. Specifically, it combines differential privacy mechanisms with fairness constraints to protect user privacy while ensuring fair recommendations. The experimental results, based on Amazon datasets, and user history logs collected from Etsy, one of the largest e-commerce platforms, demonstrate that our proposed method exhibits superior performance in terms of both overall accuracy and user group fairness on both shallow and deep recommendation models compared to vanilla DP-SGD.

Yicheng Cao, Zhuo Huang, Yu Yao et al.

Physical simulation predicts future states of objects based on material properties and external loads, enabling blueprints for both Industry and Engineering to conduct risk management. Current 3D reconstruction-based simulators typically rely on explicit, step-wise updates, which are sensitive to step time and suffer from rapid accuracy degradation under complicated scenarios, such as high-stiffness materials or quasi-static movement. To address this, we introduce i-PhysGaussian, a framework that couples 3D Gaussian Splatting (3DGS) with an implicit Material Point Method (MPM) integrator. Unlike explicit methods, our solution obtains an end-of-step state by minimizing a momentum-balance residual through implicit Newton-type optimization with a GMRES solver. This formulation significantly reduces time-step sensitivity and ensures physical consistency. Our results demonstrate that i-PhysGaussian maintains stability at up to 20x larger time steps than explicit baselines, preserving structural coherence and smooth motion even in complex dynamic transitions.

Hanpu Shen, Cheng-Long Wang, Zihang Xiang et al.

This paper focuses on the problem of Differentially Private Stochastic Optimization for (multi-layer) fully connected neural networks with a single output node. In the first part, we examine cases with no hidden nodes, specifically focusing on Generalized Linear Models (GLMs). We investigate the well-specific model where the random noise possesses a zero mean, and the link function is both bounded and Lipschitz continuous. We propose several algorithms and our analysis demonstrates the feasibility of achieving an excess population risk that remains invariant to the data dimension. We also delve into the scenario involving the ReLU link function, and our findings mirror those of the bounded link function. We conclude this section by contrasting well-specified and misspecified models, using ReLU regression as a representative example. In the second part of the paper, we extend our ideas to two-layer neural networks with sigmoid or ReLU activation functions in the well-specified model. In the third part, we study the theoretical guarantees of DP-SGD in Abadi et al. (2016) for fully connected multi-layer neural networks. By utilizing recent advances in Neural Tangent Kernel theory, we provide the first excess population risk when both the sample size and the width of the network are sufficiently large. Additionally, we discuss the role of some parameters in DP-SGD regarding their utility, both theoretically and empirically.

Bokun Wang, Yunwen Lei, Yiming Ying et al.

We study the discriminative probabilistic modeling on a continuous domain for the data prediction task of (multimodal) self-supervised representation learning. To address the challenge of computing the integral in the partition function for each anchor data, we leverage the multiple importance sampling (MIS) technique for robust Monte Carlo integration, which can recover InfoNCE-based contrastive loss as a special case. Within this probabilistic modeling framework, we conduct generalization error analysis to reveal the limitation of current InfoNCE-based contrastive loss for self-supervised representation learning and derive insights for developing better approaches by reducing the error of Monte Carlo integration. To this end, we propose a novel non-parametric method for approximating the sum of conditional probability densities required by MIS through convex optimization, yielding a new contrastive objective for self-supervised representation learning. Moreover, we design an efficient algorithm for solving the proposed objective. We empirically compare our algorithm to representative baselines on the contrastive image-language pretraining task. Experimental results on the CC3M and CC12M datasets demonstrate the superior overall performance of our algorithm. Our code is available at https://github.com/bokun-wang/NUCLR.

Lisha Chen, Heshan Fernando, Yiming Ying et al.

Multi-objective learning (MOL) problems often arise in emerging machine learning problems when there are multiple learning criteria, data modalities, or learning tasks. Different from single-objective learning, one of the critical challenges in MOL is the potential conflict among different objectives during the iterative optimization process. Recent works have developed various dynamic weighting algorithms for MOL such as MGDA and its variants, where the central idea is to find an update direction that avoids conflicts among objectives. Albeit its appealing intuition, empirical studies show that dynamic weighting methods may not always outperform static ones. To understand this theory-practical gap, we focus on a new stochastic variant of MGDA - the Multi-objective gradient with Double sampling (MoDo) algorithm, and study the generalization performance of the dynamic weighting-based MoDo and its interplay with optimization through the lens of algorithm stability. Perhaps surprisingly, we find that the key rationale behind MGDA -- updating along conflict-avoidant direction - may hinder dynamic weighting algorithms from achieving the optimal ${\cal O}(1/\sqrt{n})$ population risk, where $n$ is the number of training samples. We further demonstrate the impact of the variability of dynamic weights on the three-way trade-off among optimization, generalization, and conflict avoidance that is unique in MOL. We showcase the generality of our theoretical framework by analyzing other existing stochastic MOL algorithms under the framework. Experiments on various multi-task learning benchmarks are performed to demonstrate the practical applicability. Code is available at https://github.com/heshandevaka/Trade-Off-MOL.

Dixian Zhu, Yiming Ying, Tianbao Yang

We study a family of loss functions named label-distributionally robust (LDR) losses for multi-class classification that are formulated from distributionally robust optimization (DRO) perspective, where the uncertainty in the given label information are modeled and captured by taking the worse case of distributional weights. The benefits of this perspective are several fold: (i) it provides a unified framework to explain the classical cross-entropy (CE) loss and SVM loss and their variants, (ii) it includes a special family corresponding to the temperature-scaled CE loss, which is widely adopted but poorly understood; (iii) it allows us to achieve adaptivity to the uncertainty degree of label information at an instance level. Our contributions include: (1) we study both consistency and robustness by establishing top-$k$ ($\forall k\geq 1$) consistency of LDR losses for multi-class classification, and a negative result that a top-$1$ consistent and symmetric robust loss cannot achieve top-$k$ consistency simultaneously for all $k\geq 2$; (2) we propose a new adaptive LDR loss that automatically adapts the individualized temperature parameter to the noise degree of class label of each instance; (3) we demonstrate stable and competitive performance for the proposed adaptive LDR loss on 7 benchmark datasets under 6 noisy label and 1 clean settings against 13 loss functions, and on one real-world noisy dataset. The code is open-sourced at \url{https://github.com/Optimization-AI/ICML2023_LDR}.

Bokun Wang, Zhuoning Yuan, Yiming Ying et al.

In recent years, model-agnostic meta-learning (MAML) has become a popular research area. However, the stochastic optimization of MAML is still underdeveloped. Existing MAML algorithms rely on the ``episode'' idea by sampling a few tasks and data points to update the meta-model at each iteration. Nonetheless, these algorithms either fail to guarantee convergence with a constant mini-batch size or require processing a large number of tasks at every iteration, which is unsuitable for continual learning or cross-device federated learning where only a small number of tasks are available per iteration or per round. To address these issues, this paper proposes memory-based stochastic algorithms for MAML that converge with vanishing error. The proposed algorithms require sampling a constant number of tasks and data samples per iteration, making them suitable for the continual learning scenario. Moreover, we introduce a communication-efficient memory-based MAML algorithm for personalized federated learning in cross-device (with client sampling) and cross-silo (without client sampling) settings. Our theoretical analysis improves the optimization theory for MAML, and our empirical results corroborate our theoretical findings. Interested readers can access our code at \url{https://github.com/bokun-wang/moml}.

Zhuoning Yuan, Zhishuai Guo, Yi Xu et al.

Deep AUC (area under the ROC curve) Maximization (DAM) has attracted much attention recently due to its great potential for imbalanced data classification. However, the research on Federated Deep AUC Maximization (FDAM) is still limited. Compared with standard federated learning (FL) approaches that focus on decomposable minimization objectives, FDAM is more complicated due to its minimization objective is non-decomposable over individual examples. In this paper, we propose improved FDAM algorithms for heterogeneous data by solving the popular non-convex strongly-concave min-max formulation of DAM in a distributed fashion, which can also be applied to a class of non-convex strongly-concave min-max problems. A striking result of this paper is that the communication complexity of the proposed algorithm is a constant independent of the number of machines and also independent of the accuracy level, which improves an existing result by orders of magnitude. The experiments have demonstrated the effectiveness of our FDAM algorithm on benchmark datasets, and on medical chest X-ray images from different organizations. Our experiment shows that the performance of FDAM using data from multiple hospitals can improve the AUC score on testing data from a single hospital for detecting life-threatening diseases based on chest radiographs. The proposed method is implemented in our open-sourced library LibAUC (www.libauc.org) whose github address is https://github.com/Optimization-AI/ICML2021_FedDeepAUC_CODASCA.

Yuanfan Li, Xiyuan Wei, Tianbao Yang et al.

Contrastive representation learning (CRL) underpins many modern foundation models. Despite recent theoretical progress, existing analyses suffer from several key limitations: (i) the statistical consistency of CRL remains poorly understood; (ii) available generalization bounds deteriorate as the number of negative samples increases, contradicting the empirical benefits of large negative sets; and (iii) the retrieval performance of CRL has received limited theoretical attention. In this paper, we develop a unified statistical learning theory for CRL. For downstream tasks, we evaluate retrieval quality using an AUC-type population criterion and show that the contrastive loss is \emph{statistically consistent} with optimal ranking. We further establish a \emph{calibration-style inequality} that quantitatively relates excess contrastive risk to excess retrieval suboptimality. For upstream training, we study both supervised and self-supervised contrastive objectives and derive generalization bounds of order $O(1/m + 1/\sqrt{n})$ and $O(1/\sqrt{m} + 1/\sqrt{n})$, respectively, where $m$ denotes the number of negative samples and $n$ the number of anchor points. These bounds not only explain the empirical advantages of large negative sets but also reveal an explicit trade-off between $m$ and $n$. Extensive experiments on large-scale vision--language models corroborate our theoretical predictions.

Yuanfan Li, Yunwen Lei, Zheng-Chu Guo et al.

Recent advances have significantly improved our understanding of the generalization performance of gradient descent (GD) methods in deep neural networks. A natural and fundamental question is whether GD can achieve generalization rates comparable to the minimax optimal rates established in the kernel setting. Existing results either yield suboptimal rates of $O(1/\sqrt{n})$, or focus on networks with smooth activation functions, incurring exponential dependence on network depth $L$. In this work, we establish optimal generalization rates for GD with deep ReLU networks by carefully trading off optimization and generalization errors, achieving only polynomial dependence on depth. Specifically, under the assumption that the data are NTK separable from the margin $γ$, we prove an excess risk rate of $\widetilde{O}(L^4 (1 + γL^2) / (n γ^2))$, which aligns with the optimal SVM-type rate $\widetilde{O}(1 / (n γ^2))$ up to depth-dependent factors. A key technical contribution is our novel control of activation patterns near a reference model, enabling a sharper Rademacher complexity bound for deep ReLU networks trained with gradient descent.

Jun Chen, Hong Chen, Yonghua Yu et al.

In recent years, contrastive learning has achieved state-of-the-art performance in the territory of self-supervised representation learning. Many previous works have attempted to provide the theoretical understanding underlying the success of contrastive learning. Almost all of them rely on a default assumption, i.e., the label consistency assumption, which may not hold in practice (the probability of failure is called labeling error) due to the strength and randomness of common augmentation strategies, such as random resized crop (RRC). This paper investigates the theoretical impact of labeling error on the downstream classification performance of contrastive learning. We first reveal several significant negative impacts of labeling error on downstream classification risk. To mitigate these impacts, data dimensionality reduction method (e.g., singular value decomposition, SVD) is applied on original data to reduce false positive samples, and establish both theoretical and empirical evaluations. Moreover, it is also found that SVD acts as a double-edged sword, which may lead to the deterioration of downstream classification accuracy due to the reduced connectivity of the augmentation graph. Based on the above observations, we give the augmentation suggestion that we should use some moderate embedding dimension (such as $512, 1024$ in our experiments), data inflation, weak augmentation, and SVD to ensure large graph connectivity and small labeling error to improve model performance.

Puyu Wang, Yunwen Lei, Di Wang et al.

Recently, significant progress has been made in understanding the generalization of neural networks (NNs) trained by gradient descent (GD) using the algorithmic stability approach. However, most of the existing research has focused on one-hidden-layer NNs and has not addressed the impact of different network scaling parameters. In this paper, we greatly extend the previous work \cite{lei2022stability,richards2021stability} by conducting a comprehensive stability and generalization analysis of GD for multi-layer NNs. For two-layer NNs, our results are established under general network scaling parameters, relaxing previous conditions. In the case of three-layer NNs, our technical contribution lies in demonstrating its nearly co-coercive property by utilizing a novel induction strategy that thoroughly explores the effects of over-parameterization. As a direct application of our general findings, we derive the excess risk rate of $O(1/\sqrt{n})$ for GD algorithms in both two-layer and three-layer NNs. This sheds light on sufficient or necessary conditions for under-parameterized and over-parameterized NNs trained by GD to attain the desired risk rate of $O(1/\sqrt{n})$. Moreover, we demonstrate that as the scaling parameter increases or the network complexity decreases, less over-parameterization is required for GD to achieve the desired error rates. Additionally, under a low-noise condition, we obtain a fast risk rate of $O(1/n)$ for GD in both two-layer and three-layer NNs.

Zhenhuan Yang, Shu Hu, Yunwen Lei et al.

Stochastic gradient descent ascent (SGDA) and its variants have been the workhorse for solving minimax problems. However, in contrast to the well-studied stochastic gradient descent (SGD) with differential privacy (DP) constraints, there is little work on understanding the generalization (utility) of SGDA with DP constraints. In this paper, we use the algorithmic stability approach to establish the generalization (utility) of DP-SGDA in different settings. In particular, for the convex-concave setting, we prove that the DP-SGDA can achieve an optimal utility rate in terms of the weak primal-dual population risk in both smooth and non-smooth cases. To our best knowledge, this is the first-ever-known result for DP-SGDA in the non-smooth case. We further provide its utility analysis in the nonconvex-strongly-concave setting which is the first-ever-known result in terms of the primal population risk. The convergence and generalization results for this nonconvex setting are new even in the non-private setting. Finally, numerical experiments are conducted to demonstrate the effectiveness of DP-SGDA for both convex and nonconvex cases.

Zhenhuan Yang, Yunwen Lei, Puyu Wang et al.

Pairwise learning refers to learning tasks where the loss function depends on a pair of instances. It instantiates many important machine learning tasks such as bipartite ranking and metric learning. A popular approach to handle streaming data in pairwise learning is an online gradient descent (OGD) algorithm, where one needs to pair the current instance with a buffering set of previous instances with a sufficiently large size and therefore suffers from a scalability issue. In this paper, we propose simple stochastic and online gradient descent methods for pairwise learning. A notable difference from the existing studies is that we only pair the current instance with the previous one in building a gradient direction, which is efficient in both the storage and computational complexity. We develop novel stability results, optimization, and generalization error bounds for both convex and nonconvex as well as both smooth and nonsmooth problems. We introduce novel techniques to decouple the dependency of models and the previous instance in both the optimization and generalization analysis. Our study resolves an open question on developing meaningful generalization bounds for OGD using a buffering set with a very small fixed size. We also extend our algorithms and stability analysis to develop differentially private SGD algorithms for pairwise learning which significantly improves the existing results.

Shu Hu, Yiming Ying, Xin Wang et al.

In forming learning objectives, one oftentimes needs to aggregate a set of individual values to a single output. Such cases occur in the aggregate loss, which combines individual losses of a learning model over each training sample, and in the individual loss for multi-label learning, which combines prediction scores over all class labels. In this work, we introduce the sum of ranked range (SoRR) as a general approach to form learning objectives. A ranked range is a consecutive sequence of sorted values of a set of real numbers. The minimization of SoRR is solved with the difference of convex algorithm (DCA). We explore two applications in machine learning of the minimization of the SoRR framework, namely the AoRR aggregate loss for binary/multi-class classification at the sample level and the TKML individual loss for multi-label/multi-class classification at the label level. A combination loss of AoRR and TKML is proposed as a new learning objective for improving the robustness of multi-label learning in the face of outliers in sample and labels alike. Our empirical results highlight the effectiveness of the proposed optimization frameworks and demonstrate the applicability of proposed losses using synthetic and real data sets.

Yunwen Lei, Zhenhuan Yang, Tianbao Yang et al.

Many machine learning problems can be formulated as minimax problems such as Generative Adversarial Networks (GANs), AUC maximization and robust estimation, to mention but a few. A substantial amount of studies are devoted to studying the convergence behavior of their stochastic gradient-type algorithms. In contrast, there is relatively little work on their generalization, i.e., how the learning models built from training examples would behave on test examples. In this paper, we provide a comprehensive generalization analysis of stochastic gradient methods for minimax problems under both convex-concave and nonconvex-nonconcave cases through the lens of algorithmic stability. We establish a quantitative connection between stability and several generalization measures both in expectation and with high probability. For the convex-concave setting, our stability analysis shows that stochastic gradient descent ascent attains optimal generalization bounds for both smooth and nonsmooth minimax problems. We also establish generalization bounds for both weakly-convex-weakly-concave and gradient-dominated problems.

Puyu Wang, Yunwen Lei, Yiming Ying et al.

In this paper, we are concerned with differentially private {stochastic gradient descent (SGD)} algorithms in the setting of stochastic convex optimization (SCO). Most of the existing work requires the loss to be Lipschitz continuous and strongly smooth, and the model parameter to be uniformly bounded. However, these assumptions are restrictive as many popular losses violate these conditions including the hinge loss for SVM, the absolute loss in robust regression, and even the least square loss in an unbounded domain. We significantly relax these restrictive assumptions and establish privacy and generalization (utility) guarantees for private SGD algorithms using output and gradient perturbations associated with non-smooth convex losses. Specifically, the loss function is relaxed to have an $α$-Hölder continuous gradient (referred to as $α$-Hölder smoothness) which instantiates the Lipschitz continuity ($α=0$) and the strong smoothness ($α=1$). We prove that noisy SGD with $α$-Hölder smooth losses using gradient perturbation can guarantee $(ε,δ)$-differential privacy (DP) and attain optimal excess population risk $\mathcal{O}\Big(\frac{\sqrt{d\log(1/δ)}}{nε}+\frac{1}{\sqrt{n}}\Big)$, up to logarithmic terms, with the gradient complexity $ \mathcal{O}( n^{2-α\over 1+α}+ n).$ This shows an important trade-off between $α$-Hölder smoothness of the loss and the computational complexity for private SGD with statistically optimal performance. In particular, our results indicate that $α$-Hölder smoothness with $α\ge {1/2}$ is sufficient to guarantee $(ε,δ)$-DP of noisy SGD algorithms while achieving optimal excess risk with the linear gradient complexity $\mathcal{O}(n).$

Zhenhuan Yang, Baojian Zhou, Yunwen Lei et al.

In this paper, we aim to develop stochastic hard thresholding algorithms for the important problem of AUC maximization in imbalanced classification. The main challenge is the pairwise loss involved in AUC maximization. We overcome this obstacle by reformulating the U-statistics objective function as an empirical risk minimization (ERM), from which a stochastic hard thresholding algorithm (\texttt{SHT-AUC}) is developed. To our best knowledge, this is the first attempt to provide stochastic hard thresholding algorithms for AUC maximization with a per-iteration cost $Ø(b d)$ where $d$ and $b$ are the dimension of the data and the minibatch size, respectively. We show that the proposed algorithm enjoys the linear convergence rate up to a tolerance error. In particular, we show, if the data is generated from the Gaussian distribution, then its convergence becomes slower as the data gets more imbalanced. We conduct extensive experiments to show the efficiency and effectiveness of the proposed algorithms.

Shu Hu, Yiming Ying, Xin Wang et al.

In forming learning objectives, one oftentimes needs to aggregate a set of individual values to a single output. Such cases occur in the aggregate loss, which combines individual losses of a learning model over each training sample, and in the individual loss for multi-label learning, which combines prediction scores over all class labels. In this work, we introduce the sum of ranked range (SoRR) as a general approach to form learning objectives. A ranked range is a consecutive sequence of sorted values of a set of real numbers. The minimization of SoRR is solved with the difference of convex algorithm (DCA). We explore two applications in machine learning of the minimization of the SoRR framework, namely the AoRR aggregate loss for binary classification and the TKML individual loss for multi-label/multi-class classification. Our empirical results highlight the effectiveness of the proposed optimization framework and demonstrate the applicability of proposed losses using synthetic and real datasets.

Baojian Zhou, Yiming Ying, Steven Skiena

The Area Under the ROC Curve (AUC) is a widely used performance measure for imbalanced classification arising from many application domains where high-dimensional sparse data is abundant. In such cases, each $d$ dimensional sample has only $k$ non-zero features with $k \ll d$, and data arrives sequentially in a streaming form. Current online AUC optimization algorithms have high per-iteration cost $\mathcal{O}(d)$ and usually produce non-sparse solutions in general, and hence are not suitable for handling the data challenge mentioned above. In this paper, we aim to directly optimize the AUC score for high-dimensional sparse datasets under online learning setting and propose a new algorithm, \textsc{FTRL-AUC}. Our proposed algorithm can process data in an online fashion with a much cheaper per-iteration cost $\mathcal{O}(k)$, making it amenable for high-dimensional sparse streaming data analysis. Our new algorithmic design critically depends on a novel reformulation of the U-statistics AUC objective function as the empirical saddle point reformulation, and the innovative introduction of the "lazy update" rule so that the per-iteration complexity is dramatically reduced from $\mathcal{O}(d)$ to $\mathcal{O}(k)$. Furthermore, \textsc{FTRL-AUC} can inherently capture sparsity more effectively by applying a generalized Follow-The-Regularized-Leader (FTRL) framework. Experiments on real-world datasets demonstrate that \textsc{FTRL-AUC} significantly improves both run time and model sparsity while achieving competitive AUC scores compared with the state-of-the-art methods. Comparison with the online learning method for logistic loss demonstrates that \textsc{FTRL-AUC} achieves higher AUC scores especially when datasets are imbalanced.

Yunwen Lei, Yiming Ying

Recently there are a considerable amount of work devoted to the study of the algorithmic stability and generalization for stochastic gradient descent (SGD). However, the existing stability analysis requires to impose restrictive assumptions on the boundedness of gradients, strong smoothness and convexity of loss functions. In this paper, we provide a fine-grained analysis of stability and generalization for SGD by substantially relaxing these assumptions. Firstly, we establish stability and generalization for SGD by removing the existing bounded gradient assumptions. The key idea is the introduction of a new stability measure called on-average model stability, for which we develop novel bounds controlled by the risks of SGD iterates. This yields generalization bounds depending on the behavior of the best model, and leads to the first-ever-known fast bounds in the low-noise setting using stability approach. Secondly, the smoothness assumption is relaxed by considering loss functions with Holder continuous (sub)gradients for which we show that optimal bounds are still achieved by balancing computation and stability. To our best knowledge, this gives the first-ever-known stability and generalization bounds for SGD with even non-differentiable loss functions. Finally, we study learning problems with (strongly) convex objectives but non-convex loss functions.

Mingrui Liu, Zhuoning Yuan, Yiming Ying et al.

Stochastic AUC maximization has garnered an increasing interest due to better fit to imbalanced data classification. However, existing works are limited to stochastic AUC maximization with a linear predictive model, which restricts its predictive power when dealing with extremely complex data. In this paper, we consider stochastic AUC maximization problem with a deep neural network as the predictive model. Building on the saddle point reformulation of a surrogated loss of AUC, the problem can be cast into a {\it non-convex concave} min-max problem. The main contribution made in this paper is to make stochastic AUC maximization more practical for deep neural networks and big data with theoretical insights as well. In particular, we propose to explore Polyak-Łojasiewicz (PL) condition that has been proved and observed in deep learning, which enables us to develop new stochastic algorithms with even faster convergence rate and more practical step size scheme. An AdaGrad-style algorithm is also analyzed under the PL condition with adaptive convergence rate. Our experimental results demonstrate the effectiveness of the proposed algorithms.

Yunwen Lei, Yiming Ying

In this paper we consider the problem of maximizing the Area under the ROC curve (AUC) which is a widely used performance metric in imbalanced classification and anomaly detection. Due to the pairwise nonlinearity of the objective function, classical SGD algorithms do not apply to the task of AUC maximization. We propose a novel stochastic proximal algorithm for AUC maximization which is scalable to large scale streaming data. Our algorithm can accommodate general penalty terms and is easy to implement with favorable $O(d)$ space and per-iteration time complexities. We establish a high-probability convergence rate $O(1/\sqrt{T})$ for the general convex setting, and improve it to a fast convergence rate $O(1/T)$ for the cases of strongly convex regularizers and no regularization term (without strong convexity). Our proof does not need the uniform boundedness assumption on the loss function or the iterates which is more fidelity to the practice. Finally, we perform extensive experiments over various benchmark data sets from real-world application domains which show the superior performance of our algorithm over the existing AUC maximization algorithms.

Baojian Zhou, Feng Chen, Yiming Ying

Online learning algorithms update models via one sample per iteration, thus efficient to process large-scale datasets and useful to detect malicious events for social benefits, such as disease outbreak and traffic congestion on the fly. However, existing algorithms for graph-structured models focused on the offline setting and the least square loss, incapable for online setting, while methods designed for online setting cannot be directly applied to the problem of complex (usually non-convex) graph-structured sparsity model. To address these limitations, in this paper we propose a new algorithm for graph-structured sparsity constraint problems under online setting, which we call \textsc{GraphDA}. The key part in \textsc{GraphDA} is to project both averaging gradient (in dual space) and primal variables (in primal space) onto lower dimensional subspaces, thus capturing the graph-structured sparsity effectively. Furthermore, the objective functions assumed here are generally convex so as to handle different losses for online learning settings. To the best of our knowledge, \textsc{GraphDA} is the first online learning algorithm for graph-structure constrained optimization problems. To validate our method, we conduct extensive experiments on both benchmark graph and real-world graph datasets. Our experiment results show that, compared to other baseline methods, \textsc{GraphDA} not only improves classification performance, but also successfully captures graph-structured features more effectively, hence stronger interpretability.

Baojian Zhou, Feng Chen, Yiming Ying

Stochastic optimization algorithms update models with cheap per-iteration costs sequentially, which makes them amenable for large-scale data analysis. Such algorithms have been widely studied for structured sparse models where the sparsity information is very specific, e.g., convex sparsity-inducing norms or $\ell^0$-norm. However, these norms cannot be directly applied to the problem of complex (non-convex) graph-structured sparsity models, which have important application in disease outbreak and social networks, etc. In this paper, we propose a stochastic gradient-based method for solving graph-structured sparsity constraint problems, not restricted to the least square loss. We prove that our algorithm enjoys a linear convergence up to a constant error, which is competitive with the counterparts in the batch learning setting. We conduct extensive experiments to show the efficiency and effectiveness of the proposed algorithms.

Wei Shen, Zhenhuan Yang, Yiming Ying et al.

In this paper we study the stability and its trade-off with optimization error for stochastic gradient descent (SGD) algorithms in the pairwise learning setting. Pairwise learning refers to a learning task which involves a loss function depending on pairs of instances among which notable examples are bipartite ranking, metric learning, area under ROC (AUC) maximization and minimum error entropy (MEE) principle. Our contribution is twofold. Firstly, we establish the stability results of SGD for pairwise learning in the convex, strongly convex and non-convex settings, from which generalization bounds can be naturally derived. Secondly, we establish the trade-off between stability and optimization error of SGD algorithms for pairwise learning. This is achieved by lower-bounding the sum of stability and optimization error by the minimax statistical error over a prescribed class of pairwise loss functions. From this fundamental trade-off, we obtain lower bounds for the optimization error of SGD algorithms and the excess expected risk over a class of pairwise losses. In addition, we illustrate our stability results by giving some specific examples of AUC maximization, metric learning and MEE.

Siwei Lyu, Yiming Ying

Area under ROC (AUC) is an important metric for binary classification and bipartite ranking problems. However, it is difficult to directly optimizing AUC as a learning objective, so most existing algorithms are based on optimizing a surrogate loss to AUC. One significant drawback of these surrogate losses is that they require pairwise comparisons among training data, which leads to slow running time and increasing local storage for online learning. In this work, we describe a new surrogate loss based on a reformulation of the AUC risk, which does not require pairwise comparison but rankings of the predictions. We further show that the ranking operation can be avoided, and the learning objective obtained based on this surrogate enjoys linear complexity in time and storage. We perform experiments to demonstrate the effectiveness of the online and batch algorithms for AUC optimization based on the proposed surrogate loss.

Yunlong Feng, Yiming Ying

In recent years, correntropy and its applications in machine learning have been drawing continuous attention owing to its merits in dealing with non-Gaussian noise and outliers. However, theoretical understanding of correntropy, especially in the statistical learning context, is still limited. In this study, within the statistical learning framework, we investigate correntropy based regression in the presence of non-Gaussian noise or outliers. Motivated by the practical way of generating non-Gaussian noise or outliers, we introduce mixture of symmetric stable noise, which include Gaussian noise, Cauchy noise, and their mixture as special cases, to model non-Gaussian noise or outliers. We demonstrate that under the mixture of symmetric stable noise assumption, correntropy based regression can learn the conditional mean function or the conditional median function well without resorting to the finite-variance or even the finite first-order moment condition on the noise. In particular, for the above two cases, we establish asymptotic optimal learning rates for correntropy based regression estimators that are asymptotically of type $\mathcal{O}(n^{-1})$. These results justify the effectiveness of the correntropy based regression estimators in dealing with outliers as well as non-Gaussian noise. We believe that the present study completes our understanding towards correntropy based regression from a statistical learning viewpoint, and may also shed some light on robust statistical learning for regression.

Yanbo Fan, Siwei Lyu, Yiming Ying et al.

In this work, we introduce the {\em average top-$k$} (\atk) loss as a new aggregate loss for supervised learning, which is the average over the $k$ largest individual losses over a training dataset. We show that the \atk loss is a natural generalization of the two widely used aggregate losses, namely the average loss and the maximum loss, but can combine their advantages and mitigate their drawbacks to better adapt to different data distributions. Furthermore, it remains a convex function over all individual losses, which can lead to convex optimization problems that can be solved effectively with conventional gradient-based methods. We provide an intuitive interpretation of the \atk loss based on its equivalent effect on the continuous individual loss functions, suggesting that it can reduce the penalty on correctly classified data. We further give a learning theory analysis of \matk learning on the classification calibration of the \atk loss and the error bounds of \atk-SVM. We demonstrate the applicability of minimum average top-$k$ learning for binary classification and regression using synthetic and real datasets.

Yiming Ying, Ding-Xuan Zhou

In this paper, we consider unregularized online learning algorithms in a Reproducing Kernel Hilbert Spaces (RKHS). Firstly, we derive explicit convergence rates of the unregularized online learning algorithms for classification associated with a general gamma-activating loss (see Definition 1 in the paper). Our results extend and refine the results in Ying and Pontil (2008) for the least-square loss and the recent result in Bach and Moulines (2011) for the loss function with a Lipschitz-continuous gradient. Moreover, we establish a very general condition on the step sizes which guarantees the convergence of the last iterate of such algorithms. Secondly, we establish, for the first time, the convergence of the unregularized pairwise learning algorithm with a general loss function and derive explicit rates under the assumption of polynomially decaying step sizes. Concrete examples are used to illustrate our main results. The main techniques are tools from convex analysis, refined inequalities of Gaussian averages, and an induction approach.

Yiming Ying, Ding-Xuan Zhou

Pairwise learning usually refers to a learning task which involves a loss function depending on pairs of examples, among which most notable ones include ranking, metric learning and AUC maximization. In this paper, we study an online algorithm for pairwise learning with a least-square loss function in an unconstrained setting of a reproducing kernel Hilbert space (RKHS), which we refer to as the Online Pairwise lEaRning Algorithm (OPERA). In contrast to existing works \cite{Kar,Wang} which require that the iterates are restricted to a bounded domain or the loss function is strongly-convex, OPERA is associated with a non-strongly convex objective function and learns the target function in an unconstrained RKHS. Specifically, we establish a general theorem which guarantees the almost surely convergence for the last iterate of OPERA without any assumptions on the underlying distribution. Explicit convergence rates are derived under the condition of polynomially decaying step sizes. We also establish an interesting property for a family of widely-used kernels in the setting of pairwise learning and illustrate the above convergence results using such kernels. Our methodology mainly depends on the characterization of RKHSs using its associated integral operators and probability inequalities for random variables with values in a Hilbert space.

Zheng-Chu Guo, Yiming Ying

Learning an appropriate (dis)similarity function from the available data is a central problem in machine learning, since the success of many machine learning algorithms critically depends on the choice of a similarity function to compare examples. Despite many approaches for similarity metric learning have been proposed, there is little theoretical study on the links between similarity met- ric learning and the classification performance of the result classifier. In this paper, we propose a regularized similarity learning formulation associated with general matrix-norms, and establish their generalization bounds. We show that the generalization error of the resulting linear separator can be bounded by the derived generalization bound of similarity learning. This shows that a good gen- eralization of the learnt similarity function guarantees a good classification of the resulting linear classifier. Our results extend and improve those obtained by Bellet at al. [3]. Due to the techniques dependent on the notion of uniform stability [6], the bound obtained there holds true only for the Frobenius matrix- norm regularization. Our techniques using the Rademacher complexity [5] and its related Khinchin-type inequality enable us to establish bounds for regularized similarity learning formulations associated with general matrix-norms including sparse L 1 -norm and mixed (2,1)-norm.

Qiong Cao, Zheng-Chu Guo, Yiming Ying

Recently, metric learning and similarity learning have attracted a large amount of interest. Many models and optimisation algorithms have been proposed. However, there is relatively little work on the generalization analysis of such methods. In this paper, we derive novel generalization bounds of metric and similarity learning. In particular, we first show that the generalization analysis reduces to the estimation of the Rademacher average over "sums-of-i.i.d." sample-blocks related to the specific matrix norm. Then, we derive generalization bounds for metric/similarity learning with different matrix-norm regularisers by estimating their specific Rademacher complexities. Our analysis indicates that sparse metric/similarity learning with $L^1$-norm regularisation could lead to significantly better bounds than those with Frobenius-norm regularisation. Our novel generalization analysis develops and refines the techniques of U-statistics and Rademacher complexity analysis.