Understanding the generalization and optimization of neural networks is a longstanding problem in modern learning theory. The prior analysis often leads to risk bounds of order $1/\sqrt{n}$ for ReLU networks, where $n$ is the sample size. In this paper, we present a general optimization and generalization analysis for gradient descent applied to shallow ReLU networks. We develop convergence rates of the order $1/T$ for gradient descent with $T$ iterations, and show that the gradient descent iterates fall inside local balls around either an initialization point or a reference point. Then we develop improved Rademacher complexity estimates by using the activation pattern of the ReLU function in these local balls. We apply our general result to NTK-separable data with a margin $\gamma$, and develop an almost optimal risk bound of the order $1/(n\gamma^2)$ for the ReLU network with a polylogarithmic width.

Recent developments of stochastic optimization often suggest biased gradient estimators to improve either the robustness, communication efficiency or computational speed. Representative biased stochastic gradient methods (BSGMs) include Zeroth-order stochastic gradient descent (SGD), Clipped-SGD and SGD with delayed gradients. The practical success of BSGMs motivates a lot of convergence analysis to explain their impressive training behaviour. As a comparison, there is far less work on their generalization analysis, which is a central topic in modern machine learning. In this paper, we present the first framework to study the stability and generalization of BSGMs for convex and smooth problems. We introduce a generalized Lipschitz-type condition on gradient estimators and bias, under which we develop a rather general stability bound to show how the bias and the gradient estimators affect the stability. We apply our general result to develop the first stability bound for Zeroth-order SGD with reasonable step size sequences, and the first stability bound for Clipped-SGD. While our stability analysis is developed for general BSGMs, the resulting stability bounds for both Zeroth-order SGD and Clipped-SGD match those of SGD under appropriate smoothing/clipping parameters. We combine the stability and convergence analysis together, and derive excess risk bounds of order $O(1/\sqrt{n})$ for both Zeroth-order SGD and Clipped-SGD, where $n$ is the sample size.

Stochastic optimization is the workhorse behind the success of many machine learning algorithms. The existing theoretical analysis of stochastic optimization mainly focuses on the behavior on the training dataset or requires a convexity assumption. In this paper, we provide a comprehensive analysis on the generalization behavior of stochastic optimization with nonconvex problems. We first present both upper and lower bounds on the uniform convergence of gradients. Our analysis outperforms existing results by incorporating the 2nd moment of the gradient at a single model into the upper bound. Based on this uniform convergence, we provide a high-probability bound on the gradient norm of population risks for stochastic gradient descent (SGD), which significantly improves the existing results. We show that better bounds can be achieved under further assumptions such as quasi-convexity or Polyak-Łojasiewicz condition. Our analysis shows the computation cost can be further decreased by taking the variance-reduction trick. Finally, we study the utility guarantee of SGD under a privacy constraint. Our results show a linear speed up with respect to the batch size, which shows the benefit of computing gradients in a distributed manner.

Algorithmic stability is a fundamental concept in learning theory for studying the generalization guarantees of learning algorithms. A notable limitation of classical stability analyses is that they often require convexity assumptions to obtain nontrivial bounds. In this paper, we investigate the stability and generalization properties of learning algorithms in nonconvex settings. We introduce an algorithm-dependent quantity that depends only on the training dataset and the algorithm's output. Under a mild differentiability assumption, we establish stability and generalization bounds that apply to almost any algorithm. Our bounds explicitly involve the optimization error and the algorithm-dependent quantity, thereby capturing the local curvature of the objective function around the learned model. A key feature of our analysis is that it remains valid even when the algorithm does not converge to a global or local minimizer. We further apply our general framework to gradient descent and demonstrate its implications for both linear models and shallow neural networks. Empirical studies verify the effectiveness of our stability analyses.

In this paper, we present generalization bounds for the unsupervised risk in the Deep Contrastive Representation Learning framework, which employs deep neural networks as representation functions. We approach this problem from two angles. On the one hand, we derive a parameter-counting bound that scales with the overall size of the neural networks. On the other hand, we provide a norm-based bound that scales with the norms of neural networks' weight matrices. Ignoring logarithmic factors, the bounds are independent of the size of the tuples provided for contrastive learning. To the best of our knowledge, this property is only shared by one other work, which employed a different proof strategy and suffers from very strong exponential dependence on the depth of the network which is due to a use of the peeling technique. Our results circumvent this by leveraging powerful results on covering numbers with respect to uniform norms over samples. In addition, we utilize loss augmentation techniques to further reduce the dependency on matrix norms and the implicit dependence on network depth. In fact, our techniques allow us to produce many bounds for the contrastive learning setting with similar architectural dependencies as in the study of the sample complexity of ordinary loss functions, thereby bridging the gap between the learning theories of contrastive learning and DNNs.

Pairwise learning includes various machine learning tasks, with ranking and metric learning serving as the primary representatives. While randomized coordinate descent (RCD) is popular in various problems, there is much less theoretical analysis on the generalization behavior of models trained by RCD, especially under the pairwise learning framework. In this paper, we consider the generalization of RCD for pairwise learning. We measure the on-average argument stability for both convex and strongly convex objective functions, based on which we develop generalization bounds in expectation. The early-stopping strategy is adopted to quantify the balance between estimation and optimization. Our analysis further incorporates the low-noise setting into the excess risk bounds to achieve the optimistic bound as O(1/n), where n is the sample size.

Deep learning algorithms excel at extracting fine-grained patterns from data to enable accurate predictions. However, this capability can conflict with the goal of protecting the privacy of individuals. This paper addresses both the practical and theoretical challenges of developing privacy-preserving deep learning algorithms that maintain strong predictive performance. Specifically, we propose a differentially private GD algorithm for three-layer neural networks with gradient perturbation. Both privacy and utility guarantees of the proposed method are presented, attaining-up to constants-an optimal excess population risk of order$\mathcal{O}\left(\frac{1}{\sqrt{n}}+\frac{\sqrt{d \log (1 / \delta)}}{n \epsilon}\right)$, where$s$is the data dimension,$\epsilon$is the privacy budget, and$\delta$is the failure probability. To our knowledge, this is the first utility analysis achieving optimal rates, on par with their counterparts in the convex setting, for differentially private GD algorithms in multi-laver neural networks.

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.

Decentralized SGD (D-SGD) is a popular optimization method to train large-scale machine learning models. In this paper, we study the generalization behavior of D-SGD for both smooth and nonsmooth problems by leveraging the algorithm stability. For convex and smooth problems, we develop stability bounds involving the training errors to show the benefit of optimization in generalization. This improves the existing results by removing the Lipschitzness assumption and implying fast rates in a low-noise condition. We also develop the first optimal stability-based generalization bounds for D-SGD applied to nonsmooth problems. We further develop optimization error bounds which imply minimax optimal excess risk rates. Our novelty in the analysis consists of an error decomposition to use the co-coercivity of functions as well as the control of a neighboring-consensus error.

It has been recently observed in much of the literature that neural networks exhibit a bottleneck rank property: for larger depths, the activation and weights of neural networks trained with gradient-based methods tend to be of approximately low rank. In fact, the rank of the activations of each layer converges to a fixed value referred to as the ``bottleneck rank", which is the minimum rank required to represent the training data. This perspective is in line with the observation that regularizing linear networks (without activations) with weight decay is equivalent to minimizing the Schatten $p$ quasi norm of the neural network. In this paper we investigate the implications of this phenomenon for generalization. More specifically, we prove generalization bounds for neural networks which exploit the approximate low rank structure of the weight matrices if present. The final results rely on the Schatten $p$ quasi norms of the weight matrices: for small p, the bounds exhibit a sample complexity $ \widetilde{O}(WrL^2)$ where $W$ and $L$ are the width and depth of the neural network respectively and where $r$ is the rank of the weight matrices. As $p$ increases, the bound behaves more like a norm-based bound instead. The proof techniques involve a careful interpolation between the parametric and norm based regimes. We also demonstrate in experiments that this bound outperforms both classic parameter counting and norm based bounds in the typical overparametrized regime.

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 $\gamma$, we prove an excess risk rate of $\widetilde{O}(L^4 (1 + \gamma L^2) / (n \gamma^2))$, which aligns with the optimal SVM-type rate $\widetilde{O}(1 / (n \gamma^2))$ 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.

We consider a PAC-Bayes analysis of stochastic optimization algorithms, and devise a new SGDA algorithm inspired from our bounds. Our algorithm learns a data-dependent sampling scheme along with model parameters, which may be seen as assigning a probability to each training point. We demonstrate that learning the sampling scheme increases robustness against misleading training points, as our algorithm learns to avoid bad examples during training. We conduct experiments in both standard and adversarial learning problems on several benchmark datasets, and demonstrate various applications including interpretability upon visual inspection, and robustness to the ill effects of bad training points. We also extend our analysis to pairwise SGD to demonstrate the generalizability of our methodology.

Significant progress has been made recently 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. Here, network scaling corresponds to the normalization of the layers. In this article, we greatly extend the previous work (Lei et al., 2022; Richards & Kuzborskij, 2021) by conducting a comprehensive stability and generalization analysis of GD for two-layer and three-layer NNs. For two-layer NNs, our results are established under general network scaling, 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 overparameterization. As a direct application of our general findings, we derive the excess risk rate of O(1/n) for GD in both two-layer and three-layer NNs. This sheds light on sufficient or necessary conditions for underparameterized and overparameterized NNs trained by GD to attain the desired risk rate of O(1/n). Moreover, we demonstrate that as the scaling factor increases or the network complexity decreases, less overparameterization 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.

In this article, we present a family of adaptive stochastic optimization methods, which are associated with mirror maps that are widely used to capture the geometry properties of optimization problems during iteration processes. The well-known adaptive moment estimation (Adam)-type algorithm falls into the family when the mirror maps take the form of temporal adaptation. In the context of convex objective functions, we show that with proper step sizes and hyperparameters, the average regret can achieve the convergence rate ${\mathcal { O}}(T^{-(1/2)})$ after T iterations under some standard assumptions. We further improve it to $O(T^{-1}\log T)$ when the objective functions are strongly convex. In the context of smooth objective functions (not necessarily convex), based on properties of the strongly convex differentiable mirror map, our algorithms achieve convergence rates of order ${\mathcal { O}}(T^{-(1/2)})$ up to a logarithmic term, requiring large or increasing hyperparameters that are coincident with practical usage of Adam-type algorithms. Thus, our work gives explanations for the selection of the hyperparameters in Adam-type algorithms' implementation.

The Alternating Direction Method of Multipliers (ADMM) has gained significant attention across a broad spectrum of machine learning applications. Incorporating the over-relaxation technique shows potential for enhancing the convergence rate of ADMM. However, determining optimal algorithmic parameters, including both the associated penalty and relaxation parameters, often relies on empirical approaches tailored to specific problem domains and contextual scenarios. Incorrect parameter selection can significantly hinder ADMM's convergence rate. To address this challenge, in this paper we first propose a general approach to optimize the value of penalty parameter, followed by a novel closed-form formula to compute the optimal relaxation parameter in the context of linear quadratic problems (LQPs). We then experimentally validate our parameter selection methods through random instantiations and diverse imaging applications, encompassing diffeomorphic image registration, image deblurring, and MRI reconstruction.

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.

In recent years, few-shot learning has received increasing attention in the Internet of Things areas. Few-shot learning aims to distinguish unseen classes with a few labeled samples from each class. Most recently transductive few-shot studies highly rely on the static geometry distributions generated on the feature space during the label propagation process between unseen class instances. However, these recent methods fail to guarantee that the generated graph structure preserves the true distributions between data properly. In this article, we propose a novel dynamic graph structure preserving (DGSP) model for few-shot learning. Specifically, we formulate the objective function of DGSP by simultaneously considering the data correlations from the feature space and the label space to update the generated graph structure, which can reasonably revise the inappropriate or mistaken local geometry relationships. Then, we design an efficient alternating optimization algorithm to jointly learn the label prediction matrix and the optimal graph structure, the latter of which can be formulated as a linear programming problem. Moreover, our proposed DGSP can be easily combined with any backbone networks during the learning process. We conduct extensive experimental results across different benchmarks, backbones, and task settings, and our method achieves state-of-the-art performance compared with methods based on transductive few-shot learning.

We study inductive matrix completion (matrix completion with side information) under an i.i.d. subgaussian noise assumption at a low noise regime, with uniform sampling of the entries. We obtain for the first time generalization bounds with the following three properties: (1) they scale like the standard deviation of the noise and in particular approach zero in the exact recovery case; (2) even in the presence of noise, they converge to zero when the sample size approaches infinity; and (3) for a fixed dimension of the side information, they only have a logarithmic dependence on the size of the matrix. Differently from many works in approximate recovery, we present results both for bounded Lipschitz losses and for the absolute loss, with the latter relying on Talagrand-type inequalities. The proofs create a bridge between two approaches to the theoretical analysis of matrix completion, since they consist in a combination of techniques from both the exact recovery literature and the approximate recovery literature.

Stochastic optimization has found wide applications in minimizing objective functions in machine learning, which motivates a lot of theoretical studies to understand its practical success. Most of existing studies focus on the convergence of optimization errors, while the generalization analysis of stochastic optimization is much lagging behind. This is especially the case for nonconvex and nonsmooth problems often encountered in practice. In this paper, we initialize a systematic stability and generalization analysis of stochastic optimization on nonconvex and nonsmooth problems. We introduce novel algorithmic stability measures and establish their quantitative connection on the gap between population gradients and empirical gradients, which is then further extended to study the gap between the Moreau envelope of the empirical risk and that of the population risk. To our knowledge, these quantitative connection between stability and generalization in terms of either gradients or Moreau envelopes have not been studied in the literature. We introduce a class of sampling-determined algorithms, for which we develop bounds for three stability measures. Finally, we apply these results to derive error bounds for stochastic gradient descent and its adaptive variant, where we show how to achieve an implicit regularization by tuning the step sizes and the number of iterations.

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.

We give sharper bounds for uniformly stable randomized algorithms in a PAC-Bayesian framework, which improve the existing results by up to a factor of $\sqrt{n}$ (ignoring a log factor), where $n$ is the sample size. The key idea is to bound the moment generating function of the generalization gap using concentration of weakly dependent random variables due to Bousquet et al (2020). We introduce an assumption of sub-exponential stability parameter, which allows a general treatment that we instantiate in two applications: stochastic gradient descent and randomized coordinate descent. Our results eliminate the requirement of strong convexity from previous results, and hold for non-smooth convex problems.

Gradient descent methods are simple and efficient optimization algorithms with widespread applications. To handle high-dimensional problems, we study compressed stochastic gradient descent (SGD) with low-dimensional gradient updates. We provide a detailed analysis in terms of both optimization rates and generalization rates. To this end, we develop uniform stability bounds for CompSGD for both smooth and nonsmooth problems, based on which we develop almost optimal population risk bounds. Then we extend our analysis to two variants of SGD: batch and mini-batch gradient descent. Furthermore, we show that these variants achieve almost optimal rates compared to their high-dimensional gradient setting. Thus, our results provide a way to reduce the dimension of gradient updates without affecting the convergence rate in the generalization analysis. Moreover, we show that the same result also holds in the differentially private setting, which allows us to reduce the dimension of added noise with "almost free" cost.

Many recent studies have highlighted the susceptibility of virtually all machine-learning models to adversarial attacks. Adversarial attacks are imperceptible changes to an input example of a given prediction model. Such changes are carefully designed to alter the otherwise correct prediction of the model. In this paper, we study the generalization properties of adversarial learning. In particular, we derive high-probability generalization bounds on the adversarial risk in terms of the empirical adversarial risk, the complexity of the function class and the adversarial noise set. Our bounds are generally applicable to many models, losses, and adversaries. We showcase its applicability by deriving adversarial generalization bounds for the multi-class classification setting and various prediction models (including linear models and Deep Neural Networks). We also derive optimistic adversarial generalization bounds for the case of smooth losses. These are the first fast-rate bounds valid for adversarial deep learning to the best of our knowledge.

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.

In distributed training of deep neural networks, people usually run Stochastic Gradient Descent (SGD) or its variants on each machine and communicate with other machines periodically. However, SGD might converge slowly in training some deep neural networks (e.g., RNN, LSTM) because of the exploding gradient issue. Gradient clipping is usually employed to address this issue in the single machine setting, but exploring this technique in the distributed setting is still in its infancy: it remains mysterious whether the gradient clipping scheme can take advantage of multiple machines to enjoy parallel speedup. The main technical difficulty lies in dealing with nonconvex loss function, non-Lipschitz continuous gradient, and skipping communication rounds simultaneously. In this paper, we explore a relaxed-smoothness assumption of the loss landscape which LSTM was shown to satisfy in previous works, and design a communication-efficient gradient clipping algorithm. This algorithm can be run on multiple machines, where each machine employs a gradient clipping scheme and communicate with other machines after multiple steps of gradient-based updates. Our algorithm is proved to have $O\left(\frac{1}{N\epsilon^4}\right)$ iteration complexity and $O(\frac{1}{\epsilon^3})$ communication complexity for finding an $\epsilon$-stationary point in the homogeneous data setting, where $N$ is the number of machines. This indicates that our algorithm enjoys linear speedup and reduced communication rounds. Our proof relies on novel analysis techniques of estimating truncated random variables, which we believe are of independent interest. Our experiments on several benchmark datasets and various scenarios demonstrate that our algorithm indeed exhibits fast convergence speed in practice and thus validates our theory.

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 (Lei et al., 2021). 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.

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.

In this paper, we investigate the early stopping strategy for the iterative regularization technique, which is based on gradient descent of convex loss functions in reproducing kernel Hilbert spaces without an explicit regularization term. This work shows that projecting the last iterate of the stopping time produces an estimator that can improve the generalization ability. Using the upper bound of the generalization errors, we establish a close link between the iterative regularization and Tikhonov regularization scheme and explain theoretically why the two schemes have similar regularization paths in the existing numerical simulations. We introduce a data-dependent way based on cross-validation to select the stopping time. We prove that the a-posteriori selection way can retain the comparable generalization errors to those obtained by our stopping rules with a-prior parameters.

We show generalisation error bounds for deep learning with two main improvements over the state of the art. (1) Our bounds have no explicit dependence on the number of classes except for logarithmic factors. This holds even when formulating the bounds in terms of the Frobenius-norm of the weight matrices, where previous bounds exhibit at least a square-root dependence on the number of classes. (2) We adapt the classic Rademacher analysis of DNNs to incorporate weight sharing---a task of fundamental theoretical importance which was previously attempted only under very restrictive assumptions. In our results, each convolutional filter contributes only once to the bound, regardless of how many times it is applied. Further improvements exploiting pooling and sparse connections are provided. The presented bounds scale as the norms of the parameter matrices, rather than the number of parameters. In particular, contrary to bounds based on parameter counting, they are asymptotically tight (up to log factors) when the weights approach initialisation, making them suitable as a basic ingredient in bounds sensitive to the optimisation procedure. We also show how to adapt the recent technique of loss function augmentation to replace spectral norms by empirical analogues whilst maintaining the advantages of our approach.

Many fundamental machine learning tasks can be formulated as a problem of learning with vector-valued functions, where we learn multiple scalar-valued functions together. Although there is some generalization analysis on different specific algorithms under the empirical risk minimization principle, a unifying analysis of vector-valued learning under a regularization framework is still lacking. In this paper, we initiate the generalization analysis of regularized vector-valued learning algorithms by presenting bounds with a mild dependency on the output dimension and a fast rate on the sample size. Our discussions relax the existing assumptions on the restrictive constraint of hypothesis spaces, smoothness of loss functions and low-noise condition. To understand the interaction between optimization and learning, we further use our results to derive the first generalization bounds for stochastic gradient descent with vector-valued functions. We apply our general results to multi-class classification and multi-label classification, which yield the first bounds with a logarithmic dependency on the output dimension for extreme multi-label classification with the Frobenius regularization. As a byproduct, we derive a Rademacher complexity bound for loss function classes defined in terms of a general strongly convex function.

Pairwise learning has recently received increasing attention since it subsumes many important machine learning tasks (e.g. AUC maximization and metric learning) into a unifying framework. In this paper, we give the first-ever-known stability and generalization analysis of stochastic gradient descent (SGD) for pairwise learning with non-smooth loss functions, which are widely used (e.g. Ranking SVM with the hinge loss). We introduce a novel decomposition in its stability analysis to decouple the pairwisely dependent random variables, and derive generalization bounds consistent with pointwise learning. Furthermore, we apply our stability analysis to develop differentially private SGD for pairwise learning, for which our utility bounds match with the state-of-the-art output perturbation method (Huai et al., 2020) with smooth losses. Finally, we illustrate the results using specific examples of AUC maximization and similarity metric learning. As a byproduct, we provide an affirmative solution to an open question on the advantage of the nuclear-norm constraint over Frobenius norm constraint in similarity metric learning.

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 understanding 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. We report preliminary experimental results to verify our theory.

In machine learning we often encounter structured output prediction problems (SOPPs), i.e. problems where the output space admits a rich internal structure. Application domains where SOPPs naturally occur include natural language processing, speech recognition, and computer vision. Typical SOPPs have an extremely large label set, which grows exponentially as a function of the size of the output. Existing generalization analysis implies generalization bounds with at least a square-root dependency on the cardinality d of the label set, which can be vacuous in practice. In this paper, we significantly improve the state of the art by developing novel high-probability bounds with a logarithmic dependency on d. Furthermore, we leverage the lens of algorithmic stability to develop generalization bounds in expectation without any dependency on d. Our results therefore build a solid theoretical foundation for learning in large-scale SOPPs. Furthermore, we extend our results to learning with weakly dependent data.

We propose a novel training methodology---Concept Group Learning (CGL)---that encourages training of interpretable CNN filters by partitioning filters in each layer into \emph{concept groups}, each of which is trained to learn a single visual concept. We achieve this through a novel regularization strategy that forces filters in the same group to be active in similar image regions for a given layer. We additionally use a regularizer to encourage a sparse weighting of the concept groups in each layer so that a few concept groups can have greater importance than others. We quantitatively evaluate CGL's model interpretability using standard interpretability evaluation techniques and find that our method increases interpretability scores in most cases. Qualitatively we compare the image regions which are most active under filters learned using CGL versus filters learned without CGL and find that CGL activation regions more strongly concentrate around semantically relevant features.

Randomized coordinate descent (RCD) is a popular optimization algorithm with wide applications in various machine learning problems, which motivates a lot of theoretical analysis on its convergence behavior. As a comparison, there is no work studying how the models trained by RCD would generalize to test examples. In this paper, we initialize the generalization analysis of RCD by leveraging the powerful tool of algorithmic stability. We establish argument stability bounds of RCD for both convex and strongly convex objectives, from which we develop optimal generalization bounds by showing how to early-stop the algorithm to tradeoff the estimation and optimization. Our analysis shows that RCD enjoys better stability as compared to stochastic gradient descent.

In this paper, we bridge the gap between the state-of-the-art theoretical results for matrix completion with the nuclear norm and their equivalent in \textit{inductive matrix completion}: (1) In the distribution-free setting, we prove bounds improving the previously best scaling of $O(rd^2)$ to $\widetilde{O}(d^{3/2}\sqrt{r})$, where $d$ is the dimension of the side information and $r$ is the rank. (2) We introduce the (smoothed) \textit{adjusted trace-norm minimization} strategy, an inductive analogue of the weighted trace norm, for which we show guarantees of the order $\widetilde{O}(dr)$ under arbitrary sampling. In the inductive case, a similar rate was previously achieved only under uniform sampling and for exact recovery. Both our results align with the state of the art in the particular case of standard (non-inductive) matrix completion, where they are known to be tight up to log terms. Experiments further confirm that our strategy outperforms standard inductive matrix completion on various synthetic datasets and real problems, justifying its place as an important tool in the arsenal of methods for matrix completion using side information.

Recently, there is a growing interest in studying pairwise learning since it includes many important machine learning tasks as specific examples, e.g., metric learning, AUC maximization and ranking. While stochastic gradient descent (SGD) is an efficient method, there is a lacking study on its generalization behavior for pairwise learning. In this paper, we present a systematic study on the generalization analysis of SGD for pairwise learning to understand the balance between generalization and optimization. We develop a novel high-probability generalization bound for uniformly-stable algorithms to incorporate the variance information for better generalization, based on which we establish the first nonsmooth learning algorithm to achieve almost optimal high-probability and dimension-independent generalization bounds in linear time. We consider both convex and nonconvex pairwise learning problems. Our stability analysis for convex problems shows how the interpolation can help generalization. We establish a uniform convergence of gradients, and apply it to derive the first generalization bounds on population gradients for nonconvex problems. Finally, we develop better generalization bounds for gradient-dominated problems.

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.

Stochastic gradient descent (SGD) has become the method of choice to tackle large-scale datasets due to its low computational cost and good practical performance. Learning rate analysis, either capacity-independent or capacity-dependent, provides a unifying viewpoint to study the computational and statistical properties of SGD, as well as the implicit regularization by tuning the number of passes. Existing capacity-independent learning rates require a nontrivial bounded subgradient assumption and a smoothness assumption to be optimal. Furthermore, existing capacity-dependent learning rates are only established for the specific least squares loss with a special structure. In this paper, we provide both optimal capacity-independent and capacity-dependent learning rates for SGD with general convex loss functions. Our results require neither bounded subgradient assumptions nor smoothness assumptions, and are stated with high probability. We achieve this improvement by a refined estimate on the norm of SGD iterates based on a careful martingale analysis and concentration inequalities on empirical processes.

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.

Stochastic gradient descent (SGD) has become the method of choice for training highly complex and nonconvex models since it can not only recover good solutions to minimize training errors but also generalize well. Computational and statistical properties are separately studied to understand the behavior of SGD in the literature. However, there is a lacking study to jointly consider the computational and statistical properties in a nonconvex learning setting. In this paper, we develop novel learning rates of SGD for nonconvex learning by presenting high-probability bounds for both computational and statistical errors. We show that the complexity of SGD iterates grows in a controllable manner with respect to the iteration number, which sheds insights on how an implicit regularization can be achieved by tuning the number of passes to balance the computational and statistical errors. As a byproduct, we also slightly refine the existing studies on the uniform convergence of gradients by showing its connection to Rademacher chaos complexities.

Over the last decade, research on automated parameter tuning, often referred to as automatic algorithm configuration (AAC), has made significant progress. Although the usefulness of such tools has been widely recognized in real world applications, the theoretical foundations of AAC are still very weak. This paper addresses this gap by studying the performance estimation problem in AAC. More specifically, this paper first proves the universal best performance estimator in a practical setting, and then establishes theoretical bounds on the estimation error, i.e., the difference between the training performance and the true performance for a parameter configuration, considering finite and infinite configuration spaces respectively. These findings were verified in extensive experiments conducted on four algorithm configuration scenarios involving different problem domains. Moreover, insights for enhancing existing AAC methods are also identified.

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 (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 O(bd) 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.

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, 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 non-smooth loss functions (e.g., hinge loss). Finally, we study learning problems with (strongly) convex objectives but non-convex loss functions.

Pairwise learning refers to learning tasks with loss functions depending on a pair of training examples, which includes ranking and metric learning as specific examples. Recently, there has been an increasing amount of attention on the generalization analysis of pairwise learning to understand its practical behavior. However, the existing stability analysis provides suboptimal high-probability generalization bounds. In this paper, we provide a refined stability analysis by developing generalization bounds which can be $\sqrt{n}$-times faster than the existing results, where $n$ is the sample size. This implies excess risk bounds of the order $O(n^{-1/2})$ (up to a logarithmic factor) for both regularized risk minimization and stochastic gradient descent. We also introduce a new on-average stability measure to develop optimistic bounds in a low noise setting. We apply our results to ranking and metric learning, and clearly show the advantage of our generalization bounds over the existing analysis.

Stochastic gradient descent (SGD) is a popular and efficient method with wide applications in training deep neural nets and other nonconvex models. While the behavior of SGD is well understood in the convex learning setting, the existing theoretical results for SGD applied to nonconvex objective functions are far from mature. For example, existing results require to impose a nontrivial assumption on the uniform boundedness of gradients for all iterates encountered in the learning process, which is hard to verify in practical implementations. In this article, we establish a rigorous theoretical foundation for SGD in nonconvex learning by showing that this boundedness assumption can be removed without affecting convergence rates, and relaxing the standard smoothness assumption to Hölder continuity of gradients. In particular, we establish sufficient conditions for almost sure convergence as well as optimal convergence rates for SGD applied to both general nonconvex and gradient-dominated objective functions. A linear convergence is further derived in the case with zero variances.

Stochastic composite mirror descent (SCMD) is a simple and efficient method able to capture both geometric and composite structures of optimization problems in machine learning. Existing strategies require to take either an average or a random selection of iterates to achieve optimal convergence rates, which, however, can either destroy the sparsity of solutions or slow down the practical training speed. In this paper, we propose a theoretically sound strategy to select an individual iterate of the vanilla SCMD, which is able to achieve optimal rates for both convex and strongly convex problems in a non-smooth learning setting. This strategy of outputting an individual iterate can preserve the sparsity of solutions which is crucial for a proper interpretation in sparse learning problems. We report experimental comparisons with several baseline methods to show the effectiveness of our method in achieving a fast training speed as well as in outputting sparse solutions.

Motivated by the growing popularity of time-variant evolutionary algorithms (EAs) in solving practical problems, this paper uses spectral analyses to study convergence in probability for a general class of time-variant EAs which can be asymptotically described by reducible Markov chains with multiple aperiodic recurrent classes, covering many existing concrete case studies as specific instantiations. We provide a universal yet easily checkable characteristic for time-variant EAs satisfying global convergence, by introducing the asymptotical elitism and asymptotical monotonicity. To illustrate the effectiveness of our result, we consider four specific EAs with distinct asymptotical behavior, and recover, under even mild conditions, the state-of-the-art result as simple applications of our general theorem. Besides, simulation experiments further verify these results.

In this paper, we introduce a learning algorithm, boosted kernel ridge regression (BKRR), that combines $L_2$-Boosting with the kernel ridge regression (KRR). We analyze the learning performance of this algorithm in the framework of learning theory. We show that BKRR provides a new bias-variance trade-off via tuning the number of boosting iterations, which is different from KRR via adjusting the regularization parameter. A (semi-)exponential bias-variance trade-off is derived for BKRR, exhibiting a stable relationship between the generalization error and the number of iterations. Furthermore, an adaptive stopping rule is proposed, with which BKRR achieves the optimal learning rate without saturation.

In this paper, we study data-dependent generalization error bounds that exhibit a mild dependency on the number of classes, making them suitable for multi-class learning with a large number of label classes. The bounds generally hold for empirical multi-class risk minimization algorithms using an arbitrary norm as the regularizer. Key to our analysis is new structural results for multi-class Gaussian complexities and empirical ℓ∞-norm covering numbers, which exploit the Lipschitz continuity of the loss function with respect to the ℓ2- and ℓ∞-norm, respectively. We establish data-dependent error bounds in terms of the complexities of a linear function class defined on a finite set induced by training examples, for which we show tight lower and upper bounds. We apply the results to several prominent multi-class learning machines and show a tighter dependency on the number of classes than the state of the art. For instance, for the multi-class support vector machine of Crammer and Singer (2002), we obtain a data-dependent bound with a logarithmic dependency, which is a significant improvement of the previous square-root dependency. The experimental results are reported to verify the effectiveness of our theoretical findings.

The firefly algorithm (FA) is one of the swarm intelligence algorithms for which opposition-based learning (OBL) is an efficient method for improving performance. In most of the existing OBL schemes, the opposite solution is calculated simultaneously for all dimensions of the original solution. However, the opposite solution does not always offer a better value in every dimension than the original solution. This paper develops a new scheme by utilizing the orthogonal experiment design method to select a subset of elements of the individual to be changed into opposite values by the centroid opposition, while the rest remain unchanged. Useful information about the original individual and its opposite can be found by this method. This new scheme is named orthogonal centroid opposition-based learning (OCOBL) and is incorporated into FA to obtain an orthogonal centroid opposition-based firefly algorithm (OCOFA). OCOFA is tested on the CEC's 2013 benchmark suite and compared with state-of-the-art FA variants. The experimental results demonstrate the effectiveness of OCOBL and an improved performance for the proposed OCOFA.

Pairwise learning refers to learning tasks with the associated loss functions depending on pairs of examples. Recently, pairwise learning has received increasing attention since it covers many machine learning schemes, e.g., metric learning, ranking and AUC maximization, in a unified framework. In this paper, we establish a unified generalization error bound for regularized pairwise learning without either Bernstein conditions or capacity assumptions. We apply this general result to typical learning tasks including distance metric learning and ranking, for each of which our discussion is able to improve the state-of-the-art results.

We study stochastic composite mirror descent, a class of scalable algorithms able to exploit the geometry and composite structure of a problem. We consider both convex and strongly convex objectives with non-smooth loss functions, for each of which we establish high-probability convergence rates optimal up to a logarithmic factor. We apply the derived computational error bounds to study the generalization performance of multi-pass stochastic gradient descent (SGD) in a non-parametric setting. Our high-probability generalization bounds enjoy a logarithmical dependency on the number of passes provided that the step size sequence is square-summable, which improves the existing bounds in expectation with a polynomial dependency and therefore gives a strong justification on the ability of multi-pass SGD to overcome overfitting. Our analysis removes boundedness assumptions on subgradients often imposed in the literature. Numerical results are reported to support our theoretical findings.

We show a Talagrand-type concentration inequality for Multi-Task Learning (MTL), with which we establish sharp excess risk bounds for MTL in terms of the Local Rademacher Complexity (LRC). We also give a new bound on the (LRC) for any norm regularized hypothesis classes, which applies not only to MTL, but also to the standard Single-Task Learning (STL) setting. By combining both results, one can easily derive fast-rate bounds on the excess risk for many prominent MTL methods, including–as we demonstrate–Schatten norm, group norm, and graph regularized MTL. The derived bounds reflect a relationship akin to a conservation law of asymptotic convergence rates. When compared to the rates obtained via a traditional, global Rademacher analysis, this very relationship allows for trading off slower rates with respect to the number of tasks for faster rates with respect to the number of available samples per task.

In this paper we propose an online learning algorithm, a general randomized sparse Kaczmarz method, for generating sparse approximate solutions to linear systems and present learning theory analysis for its convergence. Under a mild assumption covering the case of noisy random measurements in the sampling process or nonlinear regression function, we show that the algorithm converges in expectation if and only if the step size sequence $\{\eta_t\}_{t\in\mathbb{N}}$ satisfies $\lim_{t\to\infty}\eta_t=0$ and $\sum_{t=1}^{\infty}\eta_t=\infty$. Convergence rates are also obtained and linear convergence is shown to be impossible under the assumption of positive variance of the sampling process. A sufficient condition for almost sure convergence is derived with an additional restriction $\sum_{t=1}^{\infty}\eta_t^2 <\infty$. Our novel analysis is performed by interpreting the randomized sparse Kaczmarz method as a special online mirror descent algorithm with a nondifferentiable mirror map and using the Bregman distance. The sufficient and necessary conditions are derived by establishing a restricted variant of strong convexity for the involved generalization error and using the special structures of the soft-thresholding operator.

We study the convergence of the online composite mirror descent algorithm, which involves a mirror map to reflect the geometry of the data and a convex objective function consisting of a loss and a regularizer possibly inducing sparsity. Our error analysis provides convergence rates in terms of properties of the strongly convex differentiable mirror map and the objective function. For a class of objective functions with Hölder continuous gradients, the convergence rates of the excess (regularized) risk under polynomially decaying step sizes have the order [Formula: see text] after [Formula: see text] iterates. Our results improve the existing error analysis for the online composite mirror descent algorithm by avoiding averaging and removing boundedness assumptions, and they sharpen the existing convergence rates of the last iterate for online gradient descent without any boundedness assumptions. Our methodology mainly depends on a novel error decomposition in terms of an excess Bregman distance, refined analysis of self-bounding properties of the objective function, and the resulting one-step progress bounds.

We propose a localized approach to multiple kernel learning that can be formulated as a convex optimization problem over a given cluster structure. For which we obtain generalization error guarantees and derive an optimization algorithm based on the Fenchel dual representation. Experiments on real-world datasets from the application domains of computational biology and computer vision show that convex localized multiple kernel learning can achieve higher prediction accuracies than its global and non-convex local counterparts.

This paper studies the generalization performance of multi-class classification algorithms, for which we obtain, for the first time, a data-dependent generalization error bound with a logarithmic dependence on the class size, substantially improving the state-of-the-art linear dependence in the existing data-dependent generalization analysis. The theoretical analysis motivates us to introduce a new multi-class classification machine based on lp-norm regularization, where the parameter p controls the complexity of the corresponding bounds. We derive an efficient optimization algorithm based on Fenchel duality theory. Benchmarks on several real-world datasets show that the proposed algorithm can achieve significant accuracy gains over the state of the art.

This paper studies the generalization performance of radial basis function (RBF) networks using local Rademacher complexities. We propose a general result on controlling local Rademacher complexities with the L1 -metric capacity. We then apply this result to estimate the RBF networks' complexities, based on which a novel estimation error bound is obtained. An effective approximation error bound is also derived by carefully investigating the Hölder continuity of the lp loss function's derivative. Furthermore, it is demonstrated that the RBF network minimizing an appropriately constructed structural risk admits a significantly better learning rate when compared with the existing results. An empirical study is also performed to justify the application of our structural risk in model selection.

Estimating the Rademacher chaos complexity of order two is important for understanding the performance of multikernel learning (MKL) machines. In this letter, we develop a novel entropy integral for Rademacher chaos complexities. As compared to the previous bounds, our result is much improved in that it introduces an adjustable parameter ε to prohibit the divergence of the involved integral. With the use of the iteration technique in Steinwart and Scovel (2007), we also apply our Rademacher chaos complexity bound to the MKL problems and improve existing learning rates.