In this paper, we revisit the online non-monotone continuous DR-submodular maximization problem over a down-closed convex set, which finds wide real-world applications in the domain of machine learning, economics, and operations research. At first, we present the Meta-MFW algorithm achieving a $1/e$-regret of $O(\sqrt{T})$ at the cost of $T^{3/2}$ stochastic gradient evaluations per round. As far as we know, Meta-MFW is the first algorithm to obtain $1/e$-regret of $O(\sqrt{T})$ for the online non-monotone continuous DR-submodular maximization problem over a down-closed convex set. Furthermore, in sharp contrast with ODC algorithm (Thang $&$ Srivastav, 2021), Meta-MFW relies on the simple online linear oracle without discretization, lifting, or rounding operations. Considering the practical restrictions, we then propose the Mono-MFW algorithm, which reduces the per-function stochastic gradient evaluations from $T^{3/2}$ to 1 and achieves a $1/e$-regret bound of $O(T^{4/5})$. Next, we extend Mono-MFW to the bandit setting and propose the Bandit-MFW algorithm which attains a $1/e$-regret bound of $O(T^{8/9})$. To the best of our knowledge, Mono-MFW and Bandit-MFW are the first sublinear-regret algorithms to explore the one-shot and bandit setting for online non-monotone continuous DR-submodular maximization problem over a down-closed convex set, respectively. Finally, we conduct numerical experiments on both synthetic and real-world datasets to verify the effectiveness of our methods.

Maximizing a monotone submodular function is a fundamental task in data mining, machine learning, economics, and statistics. In this paper, we present two communication-efficient decentralized online algorithms for the monotone continuous DR-submodular maximization problem, both of which reduce the number of per-function gradient evaluations and per-round communication complexity from T3/2 to 1. The first one, One-shot Decentralized Meta-Frank-Wolfe~(Mono-DMFW), achieves a (1-1/e)-regret bound of O(T4/5). As far as we know, this is the first one-shot and projection-free decentralized online algorithm for monotone continuous DR-submodular maximization. Next, inspired by the non-oblivious boosting function[29], we propose the Decentralized Online Boosting Gradient Ascent (DOBGA) algorithm, which attains a (1-1/e)-regret of O(√T). To the best of our knowledge, this is the first result to obtain the optimal O(√T) against a (1-1/e)-approximation with only one gradient inquiry for each local objective function per step. Finally, various experimental results confirm the effectiveness of the proposed methods.

This paper studies the online stochastic resource allocation problem (RAP) with chance constraints. The online RAP is a 0-1 integer linear programming problem where the resource consumption coefficients are revealed column by column along with the corresponding revenue coefficients. When a column is revealed, the corresponding decision variables are determined instantaneously without future information. Moreover, in online applications, the resource consumption coefficients are often obtained by prediction. To model their uncertainties, we take the chance constraints into the consideration. To the best of our knowledge, this is the first time chance constraints are introduced in the online RAP problem. Assuming that the uncertain variables have known Gaussian distributions, the stochastic RAP can be transformed into a deterministic but nonlinear problem with integer second-order cone constraints. Next, we linearize this nonlinear problem and analyze the performance of vanilla online primal-dual algorithm for solving the linearized stochastic RAP. Under mild technical assumptions, the optimality gap and constraint violation are both on the order of $\sqrt n $ . Then, to further improve the performance of the algorithm, several modified online primal-dual algorithms with heuristic corrections are proposed. Finally, extensive numerical experiments on both synthetic and real data demonstrate the applicability and effectiveness of our methods.

In this paper, we investigate the online allocation problem of maximizing the overall revenue subject to both lower and upper bound constraints. Compared to the extensively studied online problems with only resource upper bounds, the two-sided constraints affect the prospects of resource consumption more severely. As a result, only limited violations of constraints or pessimistic competitive bounds could be guaranteed. To tackle the challenge, we define a measure of feasibility $\xi^*$ to evaluate the hardness of this problem, and estimate this measurement by an optimization routine with theoretical guarantees. We propose an online algorithm adopting a constructive framework, where we initialize a threshold price vector using the estimation, then dynamically update the price vector and use it for decision-making at each step. It can be shown that the proposed algorithm is $\big(1-O(\frac{\varepsilon}{\xi^*-\varepsilon})\big)$ or $\big(1-O(\frac{\varepsilon}{\xi^*-\sqrt{\varepsilon}})\big)$ competitive with high probability for $\xi^*$ known or unknown respectively. To the best of our knowledge, this is the first result establishing a nearly optimal competitive algorithm for solving two-sided constrained online allocation problems with a high probability of feasibility.

In this paper, we revisit Stochastic Continuous Submodular Maximization in both offline and online settings, which can benefit wide applications in machine learning and operations research areas. We present a boosting framework covering gradient ascent and online gradient ascent. The fundamental ingredient of our methods is a novel non-oblivious function $F$ derived from a factor-revealing optimization problem, whose any stationary point provides a $(1-e^{-\gamma})$-approximation to the global maximum of the $\gamma$-weakly DR-submodular objective function $f\in C^{1,1}_L(\mathcal{X})$. Under the offline scenario, we propose a boosting gradient ascent method achieving $(1-e^{-\gamma}-\epsilon^{2})$-approximation after $O(1/\epsilon^2)$ iterations, which improves the $(\frac{\gamma^2}{1+\gamma^2})$ approximation ratio of the classical gradient ascent algorithm. In the online setting, for the first time we consider the adversarial delays for stochastic gradient feedback, under which we propose a boosting online gradient algorithm with the same non-oblivious function $F$. Meanwhile, we verify that this boosting online algorithm achieves a regret of $O(\sqrt{D})$ against a $(1-e^{-\gamma})$-approximation to the best feasible solution in hindsight, where $D$ is the sum of delays of gradient feedback. To the best of our knowledge, this is the first result to obtain $O(\sqrt{T})$ regret against a $(1-e^{-\gamma})$-approximation with $O(1)$ gradient inquiry at each time step, when no delay exists, i.e., $D=T$. Finally, numerical experiments demonstrate the effectiveness of our boosting methods.

Training accurate deep neural networks (DNNs) in the presence of noisy labels is an important and challenging task. Though a number of approaches have been proposed for learning with noisy labels, many open issues remain. In this paper, we show that DNN learning with Cross Entropy (CE) exhibits overfitting to noisy labels on some classes ("easy" classes), but more surprisingly, it also suffers from significant under learning on some other classes ("hard" classes). Intuitively, CE requires an extra term to facilitate learning of hard classes, and more importantly, this term should be noise tolerant, so as to avoid overfitting to noisy labels. Inspired by the symmetric KL-divergence, we propose the approach of Symmetric cross entropy Learning (SL), boosting CE symmetrically with a noise robust counterpart Reverse Cross Entropy (RCE). Our proposed SL approach simultaneously addresses both the under learning and overfitting problem of CE in the presence of noisy labels. We provide a theoretical analysis of SL and also empirically show, on a range of benchmark and real-world datasets, that SL outperforms state-of-the-art methods. We also show that SL can be easily incorporated into existing methods in order to further enhance their performance.

An important class of non-convex objectives that has wide applications in machine learning consists of a sum of $n$ smooth functions and a non-smooth convex function. Tremendous studies have been devoted to conquering these problems by leveraging one of the two types of variance reduction techniques, i.e., SVRG-type that computes a full gradient occasionally and SAGA-type that maintains $n$ stochastic gradients at every iteration. In practice, SVRG-type is preferred to SAGA-type due to its potentially less memory costs. An interesting question that has been largely ignored is how to improve the complexity of variance reduction methods for problems with a large condition number that measures the degree to which the objective is close to a convex function. In this paper, we present a simple but non-trivial boosting of a state-of-the-art SVRG-type method for convex problems (namely Katyusha) to enjoy an improved complexity for solving non-convex problems with a large condition number (that is close to a convex function). To the best of our knowledge, its complexity has the best dependence on $n$ and the degree of non-convexity, and also matches that of a recent SAGA-type accelerated stochastic algorithm for a constrained non-convex smooth optimization problem.

Although the convergence rates of existing variants of ADAGRAD have a better dependence on the number of iterations under the strong convexity condition, their iteration complexities have a explicitly linear dependence on the dimensionality of the problem. To alleviate this bad dependence, we propose a simple yet novel variant of ADAGRAD for stochastic (weakly) strongly convex optimization. Different from existing variants, the proposed variant (referred to as SADAGRAD) uses an adaptive restarting scheme in which (i) ADAGRAD serves as a sub-routine and is restarted periodically; (ii) the number of iterations for restarting ADAGRAD depends on the history of learning that incorporates knowledge of the geometry of the data. In addition to the adaptive proximal functions and adaptive number of iterations for restarting, we also develop a variant that is adaptive to the (implicit) strong convexity from the data, which together makes the proposed algorithm strongly adaptive. In terms of iteration complexity, in the worst case SADAGRAD has an O(1/\epsilon) for finding an \epsilon-optimal solution similar to other variants. However, it could enjoy faster convergence and much better dependence on the problem’s dimensionality when stochastic gradients are sparse. Extensive experiments on large-scale data sets demonstrate the efficiency of the proposed algorithms in comparison with several variants of ADAGRAD and stochastic gradient method.

In this paper, we consider statistical learning with AUC (area under ROC curve) maximization in the classical stochastic setting where one random data drawn from an unknown distribution is revealed at each iteration for updating the model. Although consistent convex surrogate losses for AUC maximization have been proposed to make the problem tractable, it remains an challenging problem to design fast optimization algorithms in the classical stochastic setting due to that the convex surrogate loss depends on random pairs of examples from positive and negative classes. Building on a saddle point formulation for a consistent square loss, this paper proposes a novel stochastic algorithm to improve the standard $O(1/\sqrt{n})$ convergence rate to $\widetilde O(1/n)$ convergence rate without strong convexity assumption or any favorable statistical assumptions (e.g., low noise), where $n$ is the number of random samples. To the best of our knowledge, this is the first stochastic algorithm for AUC maximization with a statistical convergence rate as fast as $O(1/n)$ up to a logarithmic factor. Extensive experiments on eight large-scale benchmark data sets demonstrate the superior performance of the proposed algorithm comparing with existing stochastic or online algorithms for AUC maximization.

Matrix completion is the task of recovering a data matrix from a sample of entries, and has received significant attention in theory and practice. Normally, matrix completion considers a single matrix, which can be a noisy image or a rating matrix in recommendation. In practice however, data is often obtained from multiple domains rather than a single domain. For example, in recommendation, multiple matrices may exist as user x movie and user x book, while correlations among the multiple domains can be reasonably exploited to improve the quality of matrix completion. In this paper, we consider the problem of aligned matrix completion, where multiple matrices are recovered that correspond to different representations of the same group of objects. In the proposed model, we maintain consistency of multiple domains with a shared latent structure, while allowing independent patterns for each separate domain. In addition, we impose the low-rank structure of a matrix with a novel regularizer which provides better approximation than the standard nuclear norm relaxation.

With the explosive growth of social media, it has gained significantly increasing attention from both journalists and their readership in recent years by enhancing the reading experience with its timeliness, high participation, interactivity, etc. On the other hand, the popularity of social media services such as Twitter also leads to the challenge of information overload by generating thousands of responses (tweets) for each article of hot news, which will be overwhelming for readers. In this paper, we address the problem of selecting a representative subset of responses to news in order to deliver the most important information. We consider different criteria regarding the importance of the selected subset, and treat the problem from the data reconstruction perspective with concerns for both quality and generalizability of the selection. The intuition behind our work is that a good selection should be relevant from two levels: i) at the message level, it brings readers new information as much as possible or generalizes other people's opinions comprehensively; ii) at the text level, it is able to reconstruct the corpus. Specifically, the task of selecting responses to news can be formulated as a convex optimization problem where sparse non-negative weights are introduced for all the responses indicating whether they are selected or not. Several gradient based optimization and step size selection methods are also investigated in this paper to achieve a faster rate of convergence. More importantly, the proposed framework evaluates the utility of a set of responses jointly and therefore is able to reduce redundancy of the selected responses. We evaluate our approach on real-world data obtained from Twitter, and the results demonstrate superior performance over the state of the art in both accuracy and generalizability.