Lijun Zhang

Feng Lu, Lijun Zhang, Shuting Dong et al.

Visual place recognition (VPR) is one of the research hotspots in robotics, which uses visual information to locate robots. Recently, the hierarchical two-stage VPR methods have become popular in this field due to the trade-off between accuracy and efficiency. These methods retrieve the top-k candidate images using the global features in the first stage, then re-rank the candidates by matching the local features in the second stage. However, they usually require additional algorithms (e.g. RANSAC) for geometric consistency verification in re-ranking, which is time-consuming. Here we propose a Dynamically Aligning Local Features (DALF) algorithm to align the local features under spatial constraints. It is significantly more efficient than the methods that need geometric consistency verification. We present a unified network capable of extracting global features for retrieving candidates via an aggregation module and aligning local features for re-ranking via the DALF alignment module. We call this network AANet. Meanwhile, many works use the simplest positive samples in triplet for weakly supervised training, which limits the ability of the network to recognize harder positive pairs. To address this issue, we propose a Semi-hard Positive Sample Mining (ShPSM) strategy to select appropriate hard positive images for training more robust VPR networks. Extensive experiments on four benchmark VPR datasets show that the proposed AANet can outperform several state-of-the-art methods with less time consumption. The code is released at https://github.com/Lu-Feng/AANet.

Lijun Zhang, Wei Jiang, Jinfeng Yi et al.

In this paper, we investigate an online prediction strategy named as Discounted-Normal-Predictor (Kapralov and Panigrahy, 2010) for smoothed online convex optimization (SOCO), in which the learner needs to minimize not only the hitting cost but also the switching cost. In the setting of learning with expert advice, Daniely and Mansour (2019) demonstrate that Discounted-Normal-Predictor can be utilized to yield nearly optimal regret bounds over any interval, even in the presence of switching costs. Inspired by their results, we develop a simple algorithm for SOCO: Combining online gradient descent (OGD) with different step sizes sequentially by Discounted-Normal-Predictor. Despite its simplicity, we prove that it is able to minimize the adaptive regret with switching cost, i.e., attaining nearly optimal regret with switching cost on every interval. By exploiting the theoretical guarantee of OGD for dynamic regret, we further show that the proposed algorithm can minimize the dynamic regret with switching cost in every interval.

Sijia Chen, Yu-Jie Zhang, Wei-Wei Tu et al.

Stochastically Extended Adversarial (SEA) model is introduced by Sachs et al. [2022] as an interpolation between stochastic and adversarial online convex optimization. Under the smoothness condition, they demonstrate that the expected regret of optimistic follow-the-regularized-leader (FTRL) depends on the cumulative stochastic variance $σ_{1:T}^2$ and the cumulative adversarial variation $Σ_{1:T}^2$ for convex functions. They also provide a slightly weaker bound based on the maximal stochastic variance $σ_{\max}^2$ and the maximal adversarial variation $Σ_{\max}^2$ for strongly convex functions. Inspired by their work, we investigate the theoretical guarantees of optimistic online mirror descent (OMD) for the SEA model. For convex and smooth functions, we obtain the same $\mathcal{O}(\sqrt{σ_{1:T}^2}+\sqrt{Σ_{1:T}^2})$ regret bound, without the convexity requirement of individual functions. For strongly convex and smooth functions, we establish an $\mathcal{O}((σ_{\max}^2 + Σ_{\max}^2) \log (σ_{1:T}^2+Σ_{1:T}^2))$ bound, better than their $\mathcal{O}((σ_{\max}^2 + Σ_{\max}^2) \log T)$ result. For exp-concave and smooth functions, we achieve a new $\mathcal{O}(d\log(σ_{1:T}^2+Σ_{1:T}^2))$ bound. Owing to the OMD framework, we broaden our work to study dynamic regret minimization and scenarios where the online functions are non-smooth. We establish the first dynamic regret guarantee for the SEA model with convex and smooth functions, which is more favorable than static regret bounds in non-stationary scenarios. Furthermore, to deal with non-smooth and convex functions in the SEA model, we propose novel algorithms building on optimistic OMD with an implicit update, which provably attain static regret and dynamic regret guarantees without smoothness conditions.

Peng Zhao, Yan-Feng Xie, Lijun Zhang et al.

Non-stationary online learning has drawn much attention in recent years. In particular, dynamic regret and adaptive regret are proposed as two principled performance measures for online convex optimization in non-stationary environments. To optimize them, a two-layer online ensemble is usually deployed due to the inherent uncertainty of non-stationarity, in which multiple base-learners are maintained and a meta-algorithm is employed to track the best one on the fly. However, the two-layer structure raises concerns about computational complexity -- such methods typically maintain $O(\log T)$ base-learners simultaneously for a $T$-round online game and thus perform multiple projections onto the feasible domain per round, which becomes the computational bottleneck when the domain is complicated. In this paper, we present efficient methods for optimizing dynamic regret and adaptive regret that reduce the number of projections per round from $O(\log T)$ to $1$. The proposed algorithms require only one gradient query and one function evaluation at each round. Our technique hinges on the reduction mechanism developed in parameter-free online learning and requires non-trivial modifications for non-stationary online methods. Furthermore, we study an even stronger measure, namely "interval dynamic regret", and reduce the number of projections per round from $O(\log^2 T)$ to $1$ for minimizing it. Our reduction demonstrates broad generality and applies to two important applications: online stochastic control and online principal component analysis, resulting in methods that are both efficient and optimal. Finally, empirical studies verify our theoretical findings.

Yuanyu Wan, Lijun Zhang, Mingli Song

To deal with non-stationary online problems with complex constraints, we investigate the dynamic regret of online Frank-Wolfe (OFW), which is an efficient projection-free algorithm for online convex optimization. It is well-known that in the setting of offline optimization, the smoothness of functions and the strong convexity of functions accompanying specific properties of constraint sets can be utilized to achieve fast convergence rates for the Frank-Wolfe (FW) algorithm. However, for OFW, previous studies only establish a dynamic regret bound of $O(\sqrt{T}(V_T+\sqrt{D_T}+1))$ by utilizing the convexity of problems, where $T$ is the number of rounds, $V_T$ is the function variation, and $D_T$ is the gradient variation. In this paper, we derive improved dynamic regret bounds for OFW by extending the fast convergence rates of FW from offline optimization to online optimization. The key technique for this extension is to set the step size of OFW with a line search rule. In this way, we first show that the dynamic regret bound of OFW can be improved to $O(\sqrt{T(V_T+1)})$ for smooth functions. Second, we achieve a better dynamic regret bound of $O(T^{1/3}(V_T+1)^{2/3})$ when functions are smooth and strongly convex, and the constraint set is strongly convex. Finally, for smooth and strongly convex functions with minimizers in the interior of the constraint set, we demonstrate that the dynamic regret of OFW reduces to $O(V_T+1)$, and can be further strengthened to $O(\min\{P_T^\ast,S_T^\ast,V_T\}+1)$ by performing a constant number of FW iterations per round, where $P_T^\ast$ and $S_T^\ast$ denote the path length and squared path length of minimizers, respectively.

Siddhant Garg, Lijun Zhang, Hui Guan · amazon-science

Although multi-task deep neural network (DNN) models have computation and storage benefits over individual single-task DNN models, they can be further optimized via model compression. Numerous structured pruning methods are already developed that can readily achieve speedups in single-task models, but the pruning of multi-task networks has not yet been extensively studied. In this work, we investigate the effectiveness of structured pruning on multi-task models. We use an existing single-task filter pruning criterion and also introduce an MTL-based filter pruning criterion for estimating the filter importance scores. We prune the model using an iterative pruning strategy with both pruning methods. We show that, with careful hyper-parameter tuning, architectures obtained from different pruning methods do not have significant differences in their performances across tasks when the number of parameters is similar. We also show that iterative structure pruning may not be the best way to achieve a well-performing pruned model because, at extreme pruning levels, there is a high drop in performance across all tasks. But when the same models are randomly initialized and re-trained, they show better results.

Bo Xue, Yimu Wang, Yuanyu Wan et al.

This paper investigates the problem of generalized linear bandits with heavy-tailed rewards, whose $(1+ε)$-th moment is bounded for some $ε\in (0,1]$. Although there exist methods for generalized linear bandits, most of them focus on bounded or sub-Gaussian rewards and are not well-suited for many real-world scenarios, such as financial markets and web-advertising. To address this issue, we propose two novel algorithms based on truncation and mean of medians. These algorithms achieve an almost optimal regret bound of $\widetilde{O}(dT^{\frac{1}{1+ε}})$, where $d$ is the dimension of contextual information and $T$ is the time horizon. Our truncation-based algorithm supports online learning, distinguishing it from existing truncation-based approaches. Additionally, our mean-of-medians-based algorithm requires only $O(\log T)$ rewards and one estimator per epoch, making it more practical. Moreover, our algorithms improve the regret bounds by a logarithmic factor compared to existing algorithms when $ε=1$. Numerical experimental results confirm the merits of our algorithms.

Ye Han, Lijun Zhang, Dejian Meng et al.

The exploration-exploitation trade-off constitutes one of the fundamental challenges in reinforcement learning (RL), which is exacerbated in multi-agent reinforcement learning (MARL) due to the exponential growth of joint state-action spaces. This paper proposes a topology-enhanced MARL (TPE-MARL) method for optimizing cooperative decision-making of connected and autonomous vehicles (CAVs) in mixed traffic. This work presents two primary contributions: First, we construct a game topology tensor for dynamic traffic flow, effectively compressing high-dimensional traffic state information and decrease the search space for MARL algorithms. Second, building upon the designed game topology tensor and using QMIX as the backbone RL algorithm, we establish a topology-enhanced MARL framework incorporating visit counts and agent mutual information. Extensive simulations across varying traffic densities and CAV penetration rates demonstrate the effectiveness of TPE-MARL. Evaluations encompassing training dynamics, exploration patterns, macroscopic traffic performance metrics, and microscopic vehicle behaviors reveal that TPE-MARL successfully balances exploration and exploitation. Consequently, it exhibits superior performance in terms of traffic efficiency, safety, decision smoothness, and task completion. Furthermore, the algorithm demonstrates decision-making rationality comparable to or exceeding that of human drivers in both mixed-autonomy and fully autonomous traffic scenarios. Code of our work is available at \href{https://github.com/leoPub/tpemarl}{https://github.com/leoPub/tpemarl}.

Qinkai Yu, Jianyang Xie, Yitian Zhao et al.

Multimodal ophthalmic imaging-based diagnosis integrates color fundus image with optical coherence tomography (OCT) to provide a comprehensive view of ocular pathologies. However, the uneven global distribution of healthcare resources often results in real-world clinical scenarios encountering incomplete multimodal data, which significantly compromises diagnostic accuracy. Existing commonly used pipelines, such as modality imputation and distillation methods, face notable limitations: 1)Imputation methods struggle with accurately reconstructing key lesion features, since OCT lesions are localized, while fundus images vary in style. 2)distillation methods rely heavily on fully paired multimodal training data. To address these challenges, we propose a novel multimodal alignment and fusion framework capable of robustly handling missing modalities in the task of ophthalmic diagnostics. By considering the distinctive feature characteristics of OCT and fundus images, we emphasize the alignment of semantic features within the same category and explicitly learn soft matching between modalities, allowing the missing modality to utilize existing modality information, achieving robust cross-modal feature alignment under the missing modality. Specifically, we leverage the Optimal Transport for multi-scale modality feature alignment: class-wise alignment through predicted class prototypes and feature-wise alignment via cross-modal shared feature transport. Furthermore, we propose an asymmetric fusion strategy that effectively exploits the distinct characteristics of OCT and fundus modalities. Extensive evaluations on three large ophthalmic multimodal datasets demonstrate our model's superior performance under various modality-incomplete scenarios, achieving Sota performance in both complete modality and inter-modality incompleteness conditions. Code is available at https://github.com/Qinkaiyu/RIMA

Xiao Liu, Lijun Zhang, Deepak Ganesan et al.

Transformer models power many AI applications but suffer from high inference latency, limiting their use in real-time settings. Multi-device inference can reduce latency by parallelizing computation. Yet, existing methods require high inter-device bandwidth, making them impractical for bandwidth-constrained environments. We propose ASTRA, a communication-efficient framework that accelerates Transformer inference through a novel integration of sequence parallelism and a Mixed-Precision Attention mechanism designed to minimize inter-device communication. ASTRA compresses non-local token embeddings via vector quantization and preserves task accuracy through two optimizations, Noise-Augmented Quantization and Distributed Class Tokens. Experiments on ViT and GPT2 across vision and NLP tasks show that ASTRA achieves up to 2.64X speedups over single-device inference and up to 15.25X speedups over state-of-the-art multi-device inferences, while operating under bandwidths as low as 10 Mbps. ASTRA is open-sourced at https://github.com/xl1990/Astra.

Zhuoning Yuan, Yuexin Wu, Zi-Hao Qiu et al.

In this paper, we study contrastive learning from an optimization perspective, aiming to analyze and address a fundamental issue of existing contrastive learning methods that either rely on a large batch size or a large dictionary of feature vectors. We consider a global objective for contrastive learning, which contrasts each positive pair with all negative pairs for an anchor point. From the optimization perspective, we explain why existing methods such as SimCLR require a large batch size in order to achieve a satisfactory result. In order to remove such requirement, we propose a memory-efficient Stochastic Optimization algorithm for solving the Global objective of Contrastive Learning of Representations, named SogCLR. We show that its optimization error is negligible under a reasonable condition after a sufficient number of iterations or is diminishing for a slightly different global contrastive objective. Empirically, we demonstrate that SogCLR with small batch size (e.g., 256) can achieve similar performance as SimCLR with large batch size (e.g., 8192) on self-supervised learning task on ImageNet-1K. We also attempt to show that the proposed optimization technique is generic and can be applied to solving other contrastive losses, e.g., two-way contrastive losses for bimodal contrastive learning. The proposed method is implemented in our open-sourced library LibAUC (www.libauc.org).

Lijun Zhang, Xiao Liu, Hui Guan

Multi-task learning (MTL) jointly learns a set of tasks by sharing parameters among tasks. It is a promising approach for reducing storage costs while improving task accuracy for many computer vision tasks. The effective adoption of MTL faces two main challenges. The first challenge is to determine what parameters to share across tasks to optimize for both memory efficiency and task accuracy. The second challenge is to automatically apply MTL algorithms to an arbitrary CNN backbone without requiring time-consuming manual re-implementation and significant domain expertise. This paper addresses the challenges by developing the first programming framework AutoMTL that automates efficient MTL model development for vision tasks. AutoMTL takes as inputs an arbitrary backbone convolutional neural network (CNN) and a set of tasks to learn, and automatically produces a multi-task model that achieves high accuracy and small memory footprint simultaneously. Experiments on three popular MTL benchmarks (CityScapes, NYUv2, Tiny-Taskonomy) demonstrate the effectiveness of AutoMTL over state-of-the-art approaches as well as the generalizability of AutoMTL across CNNs. AutoMTL is open-sourced and available at https://github.com/zhanglijun95/AutoMTL.

Yuanyu Wan, Chang Yao, Mingli Song et al.

We investigate bandit convex optimization (BCO) with delayed feedback, where only the loss value of the action is revealed under an arbitrary delay. Let $n,T,\bar{d}$ denote the dimensionality, time horizon, and average delay, respectively. Previous studies have achieved an $O(\sqrt{n}T^{3/4}+(n\bar{d})^{1/3}T^{2/3})$ regret bound for this problem, whose delay-independent part matches the regret of the classical non-delayed bandit gradient descent algorithm. However, there is a large gap between its delay-dependent part, i.e., $O((n\bar{d})^{1/3}T^{2/3})$, and an existing $Ω(\sqrt{\bar{d}T})$ lower bound. In this paper, we illustrate that this gap can be filled in the worst case, where $\bar{d}$ is very close to the maximum delay $d$. Specifically, we first develop a novel algorithm, and prove that it enjoys a regret bound of $O(\sqrt{n}T^{3/4}+\sqrt{dT})$ in general. Compared with the previous result, our regret bound is better for $d=O((n\bar{d})^{2/3}T^{1/3})$, and the delay-dependent part is tight in the worst case. The primary idea is to decouple the joint effect of the delays and the bandit feedback on the regret by carefully incorporating the delayed bandit feedback with a blocking update mechanism. Furthermore, we show that the proposed algorithm can improve the regret bound to $O((nT)^{2/3}\log^{1/3}T+d\log T)$ for strongly convex functions. Finally, if the action sets are unconstrained, we demonstrate that it can be simply extended to achieve an $O(n\sqrt{T\log T}+d\log T)$ regret bound for strongly convex and smooth functions.

Yibo Wang, Yuanyu Wan, Lijun Zhang

We investigate constrained online convex optimization, in which decisions must belong to a fixed and typically complicated domain, and are required to approximately satisfy additional time-varying constraints over the long term. In this setting, the commonly used projection operations are often computationally expensive or even intractable. To avoid the time-consuming operation, several projection-free methods have been proposed with an $\mathcal{O}(T^{3/4} \sqrt{\log T})$ regret bound and an $\mathcal{O}(T^{7/8})$ cumulative constraint violation (CCV) bound for general convex losses. In this paper, we improve this result and further establish \textit{novel} regret and CCV bounds when loss functions are strongly convex. The primary idea is to first construct a composite surrogate loss, involving the original loss and constraint functions, by utilizing the Lyapunov-based technique. Then, we propose a parameter-free variant of the classical projection-free method, namely online Frank-Wolfe (OFW), and run this new extension over the online-generated surrogate loss. Theoretically, for general convex losses, we achieve an $\mathcal{O}(T^{3/4})$ regret bound and an $\mathcal{O}(T^{3/4} \log T)$ CCV bound, both of which are order-wise tighter than existing results. For strongly convex losses, we establish new guarantees of an $\mathcal{O}(T^{2/3})$ regret bound and an $\mathcal{O}(T^{5/6})$ CCV bound. Moreover, we also extend our methods to a more challenging setting with bandit feedback, obtaining similar theoretical findings. Empirically, experiments on real-world datasets have demonstrated the effectiveness of our methods.

Wei Jiang, Dingzhi Yu, Sifan Yang et al.

In this paper, we present enhanced analysis for sign-based optimization algorithms with momentum updates. Traditional sign-based methods, under the separable smoothness assumption, guarantee a convergence rate of $\mathcal{O}(T^{-1/4})$, but they either require large batch sizes or assume unimodal symmetric stochastic noise. To address these limitations, we demonstrate that signSGD with momentum can achieve the same convergence rate using constant batch sizes without additional assumptions. Our analysis, under the standard $l_2$-smoothness condition, improves upon the result of the prior momentum-based signSGD method by a factor of $\mathcal{O}(d^{1/2})$, where $d$ is the problem dimension. Furthermore, we explore sign-based methods with majority vote in distributed settings and show that the proposed momentum-based method yields convergence rates of $\mathcal{O}\left( d^{1/2}T^{-1/2} + dn^{-1/2} \right)$ and $\mathcal{O}\left( \max \{ d^{1/4}T^{-1/4}, d^{1/10}T^{-1/5} \} \right)$, which outperform the previous results of $\mathcal{O}\left( dT^{-1/4} + dn^{-1/2} \right)$ and $\mathcal{O}\left( d^{3/8}T^{-1/8} \right)$, respectively. Numerical experiments further validate the effectiveness of the proposed methods.

Quan Cheng, Yuanyu Wan, Lingyu Wu et al.

Continual learning aims to learn multiple tasks sequentially while preserving prior knowledge, but faces the challenge of catastrophic forgetting when adapting to new tasks. Recently, approaches leveraging pre-trained models have gained increasing popularity in mitigating this issue, due to the strong generalization ability of foundation models. To adjust pre-trained models for new tasks, existing methods usually employ low-rank adaptation, which restricts parameter updates to a fixed low-rank subspace. However, constraining the optimization space inherently compromises the model's learning capacity, resulting in inferior performance. To address this limitation, we propose Continuous Subspace Optimization for Continual Learning (CoSO) to fine-tune the model in a series of subspaces rather than a single one. These sequential subspaces are dynamically determined through the singular value decomposition of the gradients. CoSO updates the model by projecting gradients onto these subspaces, ensuring memory-efficient optimization. To mitigate forgetting, the optimization subspace of each task is constrained to be orthogonal to the historical task subspace. During task learning, CoSO maintains a task-specific component that captures the critical update directions for the current task. Upon completing a task, this component is used to update the historical task subspace, laying the groundwork for subsequent learning. Extensive experiments on multiple datasets demonstrate that CoSO significantly outperforms state-of-the-art methods, especially in challenging scenarios with long task sequences.

Lijun Zhang, Yifan Zhang, Weicheng Tang et al.

Micro-expression recognition (MER) presents a significant challenge due to the transient and subtle nature of the motion changes involved. In recent years, deep learning methods based on attention mechanisms have made some breakthroughs in MER. However, these methods still suffer from the limitations of insufficient feature capture and poor dynamic adaptation when coping with the instantaneous subtle movement changes of micro-expressions. Therefore, in this paper, we design an Adaptive Hierarchical Multi-Scale Attention Network (AHMSA-Net) for MER. Specifically, we first utilize the onset and apex frames of the micro-expression sequence to extract three-dimensional (3D) optical flow maps, including horizontal optical flow, vertical optical flow, and optical flow strain. Subsequently, the optical flow feature maps are inputted into AHMSA-Net, which consists of two parts: an adaptive hierarchical framework and a multi-scale attention mechanism. Based on the adaptive downsampling hierarchical attention framework, AHMSA-Net captures the subtle changes of micro-expressions from different granularities (fine and coarse) by dynamically adjusting the size of the optical flow feature map at each layer. Based on the multi-scale attention mechanism, AHMSA-Net learns micro-expression action information by fusing features from different scales (channel and spatial). These two modules work together to comprehensively improve the accuracy of MER. Additionally, rigorous experiments demonstrate that the proposed method achieves competitive results on major micro-expression databases, with AHMSA-Net achieving recognition accuracy of up to 78.21% on composite databases (SMIC, SAMM, CASMEII) and 77.08% on the CASME^{}3 database.

Xiao Liu, Lijun Zhang, Deepak Ganesan et al.

Vision Language Models (VLMs) are central to Visual Question Answering (VQA) systems and are typically deployed in the cloud due to their high computational demands. However, this cloud-only approach underutilizes edge computational resources and requires significant bandwidth for transmitting raw images. In this paper, we introduce an edge-cloud collaborative VQA system, called LLaVA-AlignedVQ, which features a novel Aligned Vector Quantization algorithm (AlignedVQ) that efficiently compress intermediate features without compromising accuracy to support partitioned execution. Our experiments demonstrate that LLaVA-AlignedVQ achieves approximately 1365x compression rate of intermediate features, reducing data transmission overhead by 96.8% compared to transmitting JPEG90-compressed images to the cloud. LLaVA-AlignedVQ achieves an inference speedup of 2-15x while maintaining high accuracy, remaining within -2.23% to +1.6% of the original model's accuracy performance across eight VQA datasets, compared to the cloud-only solution.

Wei Jiang, Sifan Yang, Wenhao Yang et al.

This paper investigates projection-free algorithms for stochastic constrained multi-level optimization. In this context, the objective function is a nested composition of several smooth functions, and the decision set is closed and convex. Existing projection-free algorithms for solving this problem suffer from two limitations: 1) they solely focus on the gradient mapping criterion and fail to match the optimal sample complexities in unconstrained settings; 2) their analysis is exclusively applicable to non-convex functions, without considering convex and strongly convex objectives. To address these issues, we introduce novel projection-free variance reduction algorithms and analyze their complexities under different criteria. For gradient mapping, our complexities improve existing results and match the optimal rates for unconstrained problems. For the widely-used Frank-Wolfe gap criterion, we provide theoretical guarantees that align with those for single-level problems. Additionally, by using a stage-wise adaptation, we further obtain complexities for convex and strongly convex functions. Finally, numerical experiments on different tasks demonstrate the effectiveness of our methods.

Wei Jiang, Sifan Yang, Wenhao Yang et al.

Sign stochastic gradient descent (signSGD) is a communication-efficient method that transmits only the sign of stochastic gradients for parameter updating. Existing literature has demonstrated that signSGD can achieve a convergence rate of $\mathcal{O}(d^{1/2}T^{-1/4})$, where $d$ represents the dimension and $T$ is the iteration number. In this paper, we improve this convergence rate to $\mathcal{O}(d^{1/2}T^{-1/3})$ by introducing the Sign-based Stochastic Variance Reduction (SSVR) method, which employs variance reduction estimators to track gradients and leverages their signs to update. For finite-sum problems, our method can be further enhanced to achieve a convergence rate of $\mathcal{O}(m^{1/4}d^{1/2}T^{-1/2})$, where $m$ denotes the number of component functions. Furthermore, we investigate the heterogeneous majority vote in distributed settings and introduce two novel algorithms that attain improved convergence rates of $\mathcal{O}(d^{1/2}T^{-1/2} + dn^{-1/2})$ and $\mathcal{O}(d^{1/4}T^{-1/4})$ respectively, outperforming the previous results of $\mathcal{O}(dT^{-1/4} + dn^{-1/2})$ and $\mathcal{O}(d^{3/8}T^{-1/8})$, where $n$ represents the number of nodes. Numerical experiments across different tasks validate the effectiveness of our proposed methods.

Yibo Wang, Wenhao Yang, Wei Jiang et al.

Projection-free online learning has drawn increasing interest due to its efficiency in solving high-dimensional problems with complicated constraints. However, most existing projection-free online methods focus on minimizing the static regret, which unfortunately fails to capture the challenge of changing environments. In this paper, we investigate non-stationary projection-free online learning, and choose dynamic regret and adaptive regret to measure the performance. Specifically, we first provide a novel dynamic regret analysis for an existing projection-free method named $\text{BOGD}_\text{IP}$, and establish an $\mathcal{O}(T^{3/4}(1+P_T))$ dynamic regret bound, where $P_T$ denotes the path-length of the comparator sequence. Then, we improve the upper bound to $\mathcal{O}(T^{3/4}(1+P_T)^{1/4})$ by running multiple $\text{BOGD}_\text{IP}$ algorithms with different step sizes in parallel, and tracking the best one on the fly. Our results are the first general-case dynamic regret bounds for projection-free online learning, and can recover the existing $\mathcal{O}(T^{3/4})$ static regret by setting $P_T = 0$. Furthermore, we propose a projection-free method to attain an $\tilde{\mathcal{O}}(τ^{3/4})$ adaptive regret bound for any interval with length $τ$, which nearly matches the static regret over that interval. The essential idea is to maintain a set of $\text{BOGD}_\text{IP}$ algorithms dynamically, and combine them by a meta algorithm. Moreover, we demonstrate that it is also equipped with an $\mathcal{O}(T^{3/4}(1+P_T)^{1/4})$ dynamic regret bound. Finally, empirical studies verify our theoretical findings.

Lijun Zhang, Yibo Wang, Guanghui Wang et al.

Recently, several universal methods have been proposed for online convex optimization, and attain minimax rates for multiple types of convex functions simultaneously. However, they need to design and optimize one surrogate loss for each type of functions, making it difficult to exploit the structure of the problem and utilize existing algorithms. In this paper, we propose a simple strategy for universal online convex optimization, which avoids these limitations. The key idea is to construct a set of experts to process the original online functions, and deploy a meta-algorithm over the linearized losses to aggregate predictions from experts. Specifically, the meta-algorithm is required to yield a second-order bound with excess losses, so that it can leverage strong convexity and exponential concavity to control the meta-regret. In this way, our strategy inherits the theoretical guarantee of any expert designed for strongly convex functions and exponentially concave functions, up to a double logarithmic factor. As a result, we can plug in off-the-shelf online solvers as black-box experts to deliver problem-dependent regret bounds. For general convex functions, it maintains the minimax optimality and also achieves a small-loss bound. Furthermore, we extend our universal strategy to online composite optimization, where the loss function comprises a time-varying function and a fixed regularizer. To deal with the composite loss functions, we employ a meta-algorithm based on the optimistic online learning framework, which not only possesses a second-order bound, but also can utilize estimations for upcoming loss functions. With appropriate configurations, we demonstrate that the additional regularizer does not contribute to the meta-regret, thus maintaining the universality in the composite setting.

Zhishuai Guo, Quanqi Hu, Lijun Zhang et al.

In this paper, we consider non-convex stochastic bilevel optimization (SBO) problems that have many applications in machine learning. Although numerous studies have proposed stochastic algorithms for solving these problems, they are limited in two perspectives: (i) their sample complexities are high, which do not match the state-of-the-art result for non-convex stochastic optimization; (ii) their algorithms are tailored to problems with only one lower-level problem. When there are many lower-level problems, it could be prohibitive to process all these lower-level problems at each iteration. To address these limitations, this paper proposes fast randomized stochastic algorithms for non-convex SBO problems. First, we present a stochastic method for non-convex SBO with only one lower problem and establish its sample complexity of $O(1/ε^3)$ for finding an $ε$-stationary point under Lipschitz continuous conditions of stochastic oracles, matching the lower bound for stochastic smooth non-convex optimization. Second, we present a randomized stochastic method for non-convex SBO with $m>1$ lower level problems (multi-task SBO) by processing a constant number of lower problems at each iteration, and establish its sample complexity no worse than $O(m/ε^3)$, which could be a better complexity than that of simply processing all $m$ lower problems at each iteration. Lastly, we establish even faster convergence results for gradient-dominant functions. To the best of our knowledge, this is the first work considering multi-task SBO and developing state-of-the-art sample complexity results.

Guanghui Wang, Yuanyu Wan, Tianbao Yang et al.

In many sequential decision making applications, the change of decision would bring an additional cost, such as the wear-and-tear cost associated with changing server status. To control the switching cost, we introduce the problem of online convex optimization with continuous switching constraint, where the goal is to achieve a small regret given a budget on the \emph{overall} switching cost. We first investigate the hardness of the problem, and provide a lower bound of order $Ω(\sqrt{T})$ when the switching cost budget $S=Ω(\sqrt{T})$, and $Ω(\min\{\frac{T}{S},T\})$ when $S=O(\sqrt{T})$, where $T$ is the time horizon. The essential idea is to carefully design an adaptive adversary, who can adjust the loss function according to the cumulative switching cost of the player incurred so far based on the orthogonal technique. We then develop a simple gradient-based algorithm which enjoys the minimax optimal regret bound. Finally, we show that, for strongly convex functions, the regret bound can be improved to $O(\log T)$ for $S=Ω(\log T)$, and $O(\min\{T/\exp(S)+S,T\})$ for $S=O(\log T)$.

Yuanyu Wan, Wei-Wei Tu, Lijun Zhang

We investigate the problem of online convex optimization with unknown delays, in which the feedback of a decision arrives with an arbitrary delay. Previous studies have presented a delayed variant of online gradient descent (OGD), and achieved the regret bound of $O(\sqrt{T+D})$ by only utilizing the convexity condition, where $D$ is the sum of delays over $T$ rounds. In this paper, we further exploit the strong convexity to improve the regret bound. Specifically, we first extend the delayed variant of OGD for strongly convex functions, and establish a better regret bound of $O(d\log T)$, where $d$ is the maximum delay. The essential idea is to let the learning rate decay with the total number of received feedback linearly. Furthermore, we consider the more challenging bandit setting, and obtain similar theoretical guarantees by incorporating the classical multi-point gradient estimator into our extended method. To the best of our knowledge, this is the first work that solves online strongly convex optimization under the general delayed setting.

Yuanyu Wan, Guanghui Wang, Wei-Wei Tu et al.

To deal with complicated constraints via locally light computations in distributed online learning, a recent study has presented a projection-free algorithm called distributed online conditional gradient (D-OCG), and achieved an $O(T^{3/4})$ regret bound for convex losses, where $T$ is the number of total rounds. However, it requires $T$ communication rounds, and cannot utilize the strong convexity of losses. In this paper, we propose an improved variant of D-OCG, namely D-BOCG, which can attain the same $O(T^{3/4})$ regret bound with only $O(\sqrt{T})$ communication rounds for convex losses, and a better regret bound of $O(T^{2/3}(\log T)^{1/3})$ with fewer $O(T^{1/3}(\log T)^{2/3})$ communication rounds for strongly convex losses. The key idea is to adopt a delayed update mechanism that reduces the communication complexity, and redefine the surrogate loss function in D-OCG for exploiting the strong convexity. Furthermore, we provide lower bounds to demonstrate that the $O(\sqrt{T})$ communication rounds required by D-BOCG are optimal (in terms of $T$) for achieving the $O(T^{3/4})$ regret with convex losses, and the $O(T^{1/3}(\log T)^{2/3})$ communication rounds required by D-BOCG are near-optimal (in terms of $T$) for achieving the $O(T^{2/3}(\log T)^{1/3})$ regret with strongly convex losses up to polylogarithmic factors. Finally, to handle the more challenging bandit setting, in which only the loss value is available, we incorporate the classical one-point gradient estimator into D-BOCG, and obtain similar theoretical guarantees.

Peng Zhao, Lijun Zhang, Yuan Jiang et al.

This paper investigates the problem of non-stationary linear bandits, where the unknown regression parameter is evolving over time. Existing studies develop various algorithms and show that they enjoy an $\widetilde{\mathcal{O}}(T^{2/3}P_T^{1/3})$ dynamic regret, where $T$ is the time horizon and $P_T$ is the path-length that measures the fluctuation of the evolving unknown parameter. In this paper, we discover that a serious technical flaw makes their results ungrounded, and then present a fix, which gives an $\widetilde{\mathcal{O}}(T^{3/4}P_T^{1/4})$ dynamic regret without modifying original algorithms. Furthermore, we demonstrate that instead of using sophisticated mechanisms, such as sliding window or weighted penalty, a simple restarted strategy is sufficient to attain the same regret guarantee. Specifically, we design an UCB-type algorithm to balance exploitation and exploration, and restart it periodically to handle the drift of unknown parameters. Our approach enjoys an $\widetilde{\mathcal{O}}(T^{3/4}P_T^{1/4})$ dynamic regret. Note that to achieve this bound, the algorithm requires an oracle knowledge of the path-length $P_T$. Combining the bandits-over-bandits mechanism by treating our algorithm as the base learner, we can further achieve the same regret bound in a parameter-free way. Empirical studies also validate the effectiveness of our approach.

Lijun Zhang, Wei Jiang, Shiyin Lu et al.

In this paper, we revisit the problem of smoothed online learning, in which the online learner suffers both a hitting cost and a switching cost, and target two performance metrics: competitive ratio and dynamic regret with switching cost. To bound the competitive ratio, we assume the hitting cost is known to the learner in each round, and investigate the simple idea of balancing the two costs by an optimization problem. Surprisingly, we find that minimizing the hitting cost alone is $\max(1, \frac{2}α)$-competitive for $α$-polyhedral functions and $1 + \frac{4}λ$-competitive for $λ$-quadratic growth functions, both of which improve state-of-the-art results significantly. Moreover, when the hitting cost is both convex and $λ$-quadratic growth, we reduce the competitive ratio to $1 + \frac{2}{\sqrtλ}$ by minimizing the weighted sum of the hitting cost and the switching cost. To bound the dynamic regret with switching cost, we follow the standard setting of online convex optimization, in which the hitting cost is convex but hidden from the learner before making predictions. We modify Ader, an existing algorithm designed for dynamic regret, slightly to take into account the switching cost when measuring the performance. The proposed algorithm, named as Smoothed Ader, attains an optimal $O(\sqrt{T(1+P_T)})$ bound for dynamic regret with switching cost, where $P_T$ is the path-length of the comparator sequence. Furthermore, if the hitting cost is accessible in the beginning of each round, we obtain a similar guarantee without the bounded gradient condition, and establish an $Ω(\sqrt{T(1+P_T)})$ lower bound to confirm the optimality.

Yuanyu Wan, Lijun Zhang

To efficiently solve online problems with complicated constraints, projection-free algorithms including online frank-wolfe (OFW) and its variants have received significant interest recently. However, in the general case, existing efficient projection-free algorithms only achieved the regret bound of $O(T^{3/4})$, which is worse than the regret of projection-based algorithms, where $T$ is the number of decision rounds. In this paper, we study the special case of online learning over strongly convex sets, for which we first prove that OFW can enjoy a better regret bound of $O(T^{2/3})$ for general convex losses. The key idea is to refine the decaying step-size in the original OFW by a simple line search rule. Furthermore, for strongly convex losses, we propose a strongly convex variant of OFW by redefining the surrogate loss function in OFW. We show that it achieves a regret bound of $O(T^{2/3})$ over general convex sets and a better regret bound of $O(\sqrt{T})$ over strongly convex sets.

Peng Zhao, Yu-Jie Zhang, Lijun Zhang et al.

We investigate online convex optimization in non-stationary environments and choose the dynamic regret as the performance measure, defined as the difference between cumulative loss incurred by the online algorithm and that of any feasible comparator sequence. Let $T$ be the time horizon and $P_T$ be the path-length that essentially reflects the non-stationarity of environments, the state-of-the-art dynamic regret is $\mathcal{O}(\sqrt{T(1+P_T)})$. Although this bound is proved to be minimax optimal for convex functions, in this paper, we demonstrate that it is possible to further enhance the dynamic regret by exploiting the smoothness condition. Specifically, we propose novel online algorithms that are capable of leveraging smoothness and replace the dependence on $T$ in the dynamic regret by problem-dependent quantities: the variation in gradients of loss functions, the cumulative loss of the comparator sequence, and the minimum of the previous two terms. These quantities are at most $\mathcal{O}(T)$ while could be much smaller in benign environments. Therefore, our results are adaptive to the intrinsic difficulty of the problem, since the bounds are tighter than existing results for easy problems and meanwhile guarantee the same rate in the worst case.

Peng Zhao, Lijun Zhang

In this paper, we present an improved analysis for dynamic regret of strongly convex and smooth functions. Specifically, we investigate the Online Multiple Gradient Descent (OMGD) algorithm proposed by Zhang et al. (2017). The original analysis shows that the dynamic regret of OMGD is at most $\mathcal{O}(\min\{\mathcal{P}_T,\mathcal{S}_T\})$, where $\mathcal{P}_T$ and $\mathcal{S}_T$ are path-length and squared path-length that measures the cumulative movement of minimizers of the online functions. We demonstrate that by an improved analysis, the dynamic regret of OMGD can be improved to $\mathcal{O}(\min\{\mathcal{P}_T,\mathcal{S}_T,\mathcal{V}_T\})$, where $\mathcal{V}_T$ is the function variation of the online functions. Note that the quantities of $\mathcal{P}_T, \mathcal{S}_T, \mathcal{V}_T$ essentially reflect different aspects of environmental non-stationarity -- they are not comparable in general and are favored in different scenarios. Therefore, the dynamic regret presented in this paper actually achieves a \emph{best-of-three-worlds} guarantee and is strictly tighter than previous results.

Bo Xue, Guanghui Wang, Yimu Wang et al.

In this paper, we study the problem of stochastic linear bandits with finite action sets. Most of existing work assume the payoffs are bounded or sub-Gaussian, which may be violated in some scenarios such as financial markets. To settle this issue, we analyze the linear bandits with heavy-tailed payoffs, where the payoffs admit finite $1+ε$ moments for some $ε\in(0,1]$. Through median of means and dynamic truncation, we propose two novel algorithms which enjoy a sublinear regret bound of $\widetilde{O}(d^{\frac{1}{2}}T^{\frac{1}{1+ε}})$, where $d$ is the dimension of contextual information and $T$ is the time horizon. Meanwhile, we provide an $Ω(d^{\fracε{1+ε}}T^{\frac{1}{1+ε}})$ lower bound, which implies our upper bound matches the lower bound up to polylogarithmic factors in the order of $d$ and $T$ when $ε=1$. Finally, we conduct numerical experiments to demonstrate the effectiveness of our algorithms and the empirical results strongly support our theoretical guarantees.

Lijun Zhang, Shiyin Lu, Tianbao Yang

Regret minimization is treated as the golden rule in the traditional study of online learning. However, regret minimization algorithms tend to converge to the static optimum, thus being suboptimal for changing environments. To address this limitation, new performance measures, including dynamic regret and adaptive regret have been proposed to guide the design of online algorithms. The former one aims to minimize the global regret with respect to a sequence of changing comparators, and the latter one attempts to minimize every local regret with respect to a fixed comparator. Existing algorithms for dynamic regret and adaptive regret are developed independently, and only target one performance measure. In this paper, we bridge this gap by proposing novel online algorithms that are able to minimize the dynamic regret and adaptive regret simultaneously. In fact, our theoretical guarantee is even stronger in the sense that one algorithm is able to minimize the dynamic regret over any interval.

Shiyin Lu, Lijun Zhang

In this paper, we consider the problem of prediction with expert advice in dynamic environments. We choose tracking regret as the performance metric and develop two adaptive and efficient algorithms with data-dependent tracking regret bounds. The first algorithm achieves a second-order tracking regret bound, which improves existing first-order bounds. The second algorithm enjoys a path-length bound, which is generally not comparable to the second-order bound but offers advantages in slowly moving environments. Both algorithms are developed under the online mirror descent framework and draw inspiration from existing algorithms that attain data-dependent bounds of static regret. The key idea is to use a clipped simplex in the updating step of online mirror descent. Finally, we extend our algorithms and analysis to online matrix prediction and provide the first data-dependent tracking regret bound for this problem.

Lijun Zhang, Guanghui Wang, Wei-Wei Tu et al.

To deal with changing environments, a new performance measure -- adaptive regret, defined as the maximum static regret over any interval, was proposed in online learning. Under the setting of online convex optimization, several algorithms have been successfully developed to minimize the adaptive regret. However, existing algorithms lack universality in the sense that they can only handle one type of convex functions and need apriori knowledge of parameters. By contrast, there exist universal algorithms, such as MetaGrad, that attain optimal static regret for multiple types of convex functions simultaneously. Along this line of research, this paper presents the first universal algorithm for minimizing the adaptive regret of convex functions. Specifically, we borrow the idea of maintaining multiple learning rates in MetaGrad to handle the uncertainty of functions, and utilize the technique of sleeping experts to capture changing environments. In this way, our algorithm automatically adapts to the property of functions (convex, exponentially concave, or strongly convex), as well as the nature of environments (stationary or changing). As a by product, it also allows the type of functions to switch between rounds.

Shiyin Lu, Guanghui Wang, Yao Hu et al.

In this paper, we study the multi-objective bandits (MOB) problem, where a learner repeatedly selects one arm to play and then receives a reward vector consisting of multiple objectives. MOB has found many real-world applications as varied as online recommendation and network routing. On the other hand, these applications typically contain contextual information that can guide the learning process which, however, is ignored by most of existing work. To utilize this information, we associate each arm with a context vector and assume the reward follows the generalized linear model (GLM). We adopt the notion of Pareto regret to evaluate the learner's performance and develop a novel algorithm for minimizing it. The essential idea is to apply a variant of the online Newton step to estimate model parameters, based on which we utilize the upper confidence bound (UCB) policy to construct an approximation of the Pareto front, and then uniformly at random choose one arm from the approximate Pareto front. Theoretical analysis shows that the proposed algorithm achieves an $\tilde O(d\sqrt{T})$ Pareto regret, where $T$ is the time horizon and $d$ is the dimension of contexts, which matches the optimal result for single objective contextual bandits problem. Numerical experiments demonstrate the effectiveness of our method.

Guanghui Wang, Shiyin Lu, Lijun Zhang

In this paper, we study adaptive online convex optimization, and aim to design a universal algorithm that achieves optimal regret bounds for multiple common types of loss functions. Existing universal methods are limited in the sense that they are optimal for only a subclass of loss functions. To address this limitation, we propose a novel online method, namely Maler, which enjoys the optimal $O(\sqrt{T})$, $O(d\log T)$ and $O(\log T)$ regret bounds for general convex, exponentially concave, and strongly convex functions respectively. The essential idea is to run multiple types of learning algorithms with different learning rates in parallel, and utilize a meta algorithm to track the best one on the fly. Empirical results demonstrate the effectiveness of our method.

Guanghui Wang, Shiyin Lu, Weiwei Tu et al.

The Adam algorithm has become extremely popular for large-scale machine learning. Under convexity condition, it has been proved to enjoy a data-dependant $O(\sqrt{T})$ regret bound where $T$ is the time horizon. However, whether strong convexity can be utilized to further improve the performance remains an open problem. In this paper, we give an affirmative answer by developing a variant of Adam (referred to as SAdam) which achieves a data-dependant $O(\log T)$ regret bound for strongly convex functions. The essential idea is to maintain a faster decaying yet under controlled step size for exploiting strong convexity. In addition, under a special configuration of hyperparameters, our SAdam reduces to SC-RMSprop, a recently proposed variant of RMSprop for strongly convex functions, for which we provide the first data-dependent logarithmic regret bound. Empirical results on optimizing strongly convex functions and training deep networks demonstrate the effectiveness of our method.

Bo-Jian Hou, Lijun Zhang, Zhi-Hua Zhou

Learning with feature evolution studies the scenario where the features of the data streams can evolve, i.e., old features vanish and new features emerge. Its goal is to keep the model always performing well even when the features happen to evolve. To tackle this problem, canonical methods assume that the old features will vanish simultaneously and the new features themselves will emerge simultaneously as well. They also assume there is an overlapping period where old and new features both exist when the feature space starts to change. However, in reality, the feature evolution could be unpredictable, which means the features can vanish or emerge arbitrarily, causing the overlapping period incomplete. In this paper, we propose a novel paradigm: Prediction with Unpredictable Feature Evolution (PUFE) where the feature evolution is unpredictable. To address this problem, we fill the incomplete overlapping period and formulate it as a new matrix completion problem. We give a theoretical bound on the least number of observed entries to make the overlapping period intact. With this intact overlapping period, we leverage an ensemble method to take the advantage of both the old and new feature spaces without manually deciding which base models should be incorporated. Theoretical and experimental results validate that our method can always follow the best base models and thus realize the goal of learning with feature evolution.

Lijun Zhang, Tie-Yan Liu, Zhi-Hua Zhou

We investigate online convex optimization in changing environments, and choose the adaptive regret as the performance measure. The goal is to achieve a small regret over every interval so that the comparator is allowed to change over time. Different from previous works that only utilize the convexity condition, this paper further exploits smoothness to improve the adaptive regret. To this end, we develop novel adaptive algorithms for convex and smooth functions, and establish problem-dependent regret bounds over any interval. Our regret bounds are comparable to existing results in the worst case, and become much tighter when the comparator has a small loss.

Yan Yan, Yi Xu, Qihang Lin et al.

Previous studies on stochastic primal-dual algorithms for solving min-max problems with faster convergence heavily rely on the bilinear structure of the problem, which restricts their applicability to a narrowed range of problems. The main contribution of this paper is the design and analysis of new stochastic primal-dual algorithms that use a mixture of stochastic gradient updates and a logarithmic number of deterministic dual updates for solving a family of convex-concave problems with no bilinear structure assumed. Faster convergence rates than $O(1/\sqrt{T})$ with $T$ being the number of stochastic gradient updates are established under some mild conditions of involved functions on the primal and the dual variable. For example, for a family of problems that enjoy a weak strong convexity in terms of the primal variable and has a strongly concave function of the dual variable, the convergence rate of the proposed algorithm is $O(1/T)$. We also investigate the effectiveness of the proposed algorithms for learning robust models and empirical AUC maximization.

Yuanyu Wan, Jinfeng Yi, Lijun Zhang

In this paper, we consider matrix completion from non-uniformly sampled entries including fully observed and partially observed columns. Specifically, we assume that a small number of columns are randomly selected and fully observed, and each remaining column is partially observed with uniform sampling. To recover the unknown matrix, we first recover its column space from the fully observed columns. Then, for each partially observed column, we recover it by finding a vector which lies in the recovered column space and consists of the observed entries. When the unknown $m\times n$ matrix is low-rank, we show that our algorithm can exactly recover it from merely $Ω(rn\ln n)$ entries, where $r$ is the rank of the matrix. Furthermore, for a noisy low-rank matrix, our algorithm computes a low-rank approximation of the unknown matrix and enjoys an additive error bound measured by Frobenius norm. Experimental results on synthetic datasets verify our theoretical claims and demonstrate the effectiveness of our proposed algorithm.

Mingrui Liu, Xiaoxuan Zhang, Lijun Zhang et al.

Error bound conditions (EBC) are properties that characterize the growth of an objective function when a point is moved away from the optimal set. They have recently received increasing attention in the field of optimization for developing optimization algorithms with fast convergence. However, the studies of EBC in statistical learning are hitherto still limited. The main contributions of this paper are two-fold. First, we develop fast and intermediate rates of empirical risk minimization (ERM) under EBC for risk minimization with Lipschitz continuous, and smooth convex random functions. Second, we establish fast and intermediate rates of an efficient stochastic approximation (SA) algorithm for risk minimization with Lipschitz continuous random functions, which requires only one pass of $n$ samples and adapts to EBC. For both approaches, the convergence rates span a full spectrum between $\widetilde O(1/\sqrt{n})$ and $\widetilde O(1/n)$ depending on the power constant in EBC, and could be even faster than $O(1/n)$ in special cases for ERM. Moreover, these convergence rates are automatically adaptive without using any knowledge of EBC. Overall, this work not only strengthens the understanding of ERM for statistical learning but also brings new fast stochastic algorithms for solving a broad range of statistical learning problems.

Lijun Zhang, Zhi-Hua Zhou

In this paper, we consider the problem of linear regression with heavy-tailed distributions. Different from previous studies that use the squared loss to measure the performance, we choose the absolute loss, which is capable of estimating the conditional median. To address the challenge that both the input and output could be heavy-tailed, we propose a truncated minimization problem, and demonstrate that it enjoys an $\widetilde{O}(\sqrt{d/n})$ excess risk, where $d$ is the dimensionality and $n$ is the number of samples. Compared with traditional work on $\ell_1$-regression, the main advantage of our result is that we achieve a high-probability risk bound without exponential moment conditions on the input and output. Furthermore, if the input is bounded, we show that the classical empirical risk minimization is competent for $\ell_1$-regression even when the output is heavy-tailed.

Fanhua Shang, Kaiwen Zhou, Hongying Liu et al.

In this paper, we propose a simple variant of the original SVRG, called variance reduced stochastic gradient descent (VR-SGD). Unlike the choices of snapshot and starting points in SVRG and its proximal variant, Prox-SVRG, the two vectors of VR-SGD are set to the average and last iterate of the previous epoch, respectively. The settings allow us to use much larger learning rates, and also make our convergence analysis more challenging. We also design two different update rules for smooth and non-smooth objective functions, respectively, which means that VR-SGD can tackle non-smooth and/or non-strongly convex problems directly without any reduction techniques. Moreover, we analyze the convergence properties of VR-SGD for strongly convex problems, which show that VR-SGD attains linear convergence. Different from its counterparts that have no convergence guarantees for non-strongly convex problems, we also provide the convergence guarantees of VR-SGD for this case, and empirically verify that VR-SGD with varying learning rates achieves similar performance to its momentum accelerated variant that has the optimal convergence rate $\mathcal{O}(1/T^2)$. Finally, we apply VR-SGD to solve various machine learning problems, such as convex and non-convex empirical risk minimization, and leading eigenvalue computation. Experimental results show that VR-SGD converges significantly faster than SVRG and Prox-SVRG, and usually outperforms state-of-the-art accelerated methods, e.g., Katyusha.

Tianbao Yang, Zhe Li, Lijun Zhang

In this paper, we present a simple analysis of {\bf fast rates} with {\it high probability} of {\bf empirical minimization} for {\it stochastic composite optimization} over a finite-dimensional bounded convex set with exponential concave loss functions and an arbitrary convex regularization. To the best of our knowledge, this result is the first of its kind. As a byproduct, we can directly obtain the fast rate with {\it high probability} for exponential concave empirical risk minimization with and without any convex regularization, which not only extends existing results of empirical risk minimization but also provides a unified framework for analyzing exponential concave empirical risk minimization with and without {\it any} convex regularization. Our proof is very simple only exploiting the covering number of a finite-dimensional bounded set and a concentration inequality of random vectors.

Bo-Jian Hou, Lijun Zhang, Zhi-Hua Zhou

Learning with streaming data has attracted much attention during the past few years. Though most studies consider data stream with fixed features, in real practice the features may be evolvable. For example, features of data gathered by limited-lifespan sensors will change when these sensors are substituted by new ones. In this paper, we propose a novel learning paradigm: \emph{Feature Evolvable Streaming Learning} where old features would vanish and new features would occur. Rather than relying on only the current features, we attempt to recover the vanished features and exploit it to improve performance. Specifically, we learn two models from the recovered features and the current features, respectively. To benefit from the recovered features, we develop two ensemble methods. In the first method, we combine the predictions from two models and theoretically show that with the assistance of old features, the performance on new features can be improved. In the second approach, we dynamically select the best single prediction and establish a better performance guarantee when the best model switches. Experiments on both synthetic and real data validate the effectiveness of our proposal.

Jinfeng Yi, Cho-Jui Hsieh, Kush Varshney et al.

Recommendation for e-commerce with a mix of durable and nondurable goods has characteristics that distinguish it from the well-studied media recommendation problem. The demand for items is a combined effect of form utility and time utility, i.e., a product must both be intrinsically appealing to a consumer and the time must be right for purchase. In particular for durable goods, time utility is a function of inter-purchase duration within product category because consumers are unlikely to purchase two items in the same category in close temporal succession. Moreover, purchase data, in contrast to ratings data, is implicit with non-purchases not necessarily indicating dislike. Together, these issues give rise to the positive-unlabeled demand-aware recommendation problem that we pose via joint low-rank tensor completion and product category inter-purchase duration vector estimation. We further relax this problem and propose a highly scalable alternating minimization approach with which we can solve problems with millions of users and millions of items in a single thread. We also show superior prediction accuracies on multiple real-world data sets.

Lijun Zhang, Tianbao Yang, Rong Jin

Although there exist plentiful theories of empirical risk minimization (ERM) for supervised learning, current theoretical understandings of ERM for a related problem---stochastic convex optimization (SCO), are limited. In this work, we strengthen the realm of ERM for SCO by exploiting smoothness and strong convexity conditions to improve the risk bounds. First, we establish an $\widetilde{O}(d/n + \sqrt{F_*/n})$ risk bound when the random function is nonnegative, convex and smooth, and the expected function is Lipschitz continuous, where $d$ is the dimensionality of the problem, $n$ is the number of samples, and $F_*$ is the minimal risk. Thus, when $F_*$ is small we obtain an $\widetilde{O}(d/n)$ risk bound, which is analogous to the $\widetilde{O}(1/n)$ optimistic rate of ERM for supervised learning. Second, if the objective function is also $λ$-strongly convex, we prove an $\widetilde{O}(d/n + κF_*/n )$ risk bound where $κ$ is the condition number, and improve it to $O(1/[λn^2] + κF_*/n)$ when $n=\widetildeΩ(κd)$. As a result, we obtain an $O(κ/n^2)$ risk bound under the condition that $n$ is large and $F_*$ is small, which to the best of our knowledge, is the first $O(1/n^2)$-type of risk bound of ERM. Third, we stress that the above results are established in a unified framework, which allows us to derive new risk bounds under weaker conditions, e.g., without convexity of the random function and Lipschitz continuity of the expected function. Finally, we demonstrate that to achieve an $O(1/[λn^2] + κF_*/n)$ risk bound for supervised learning, the $\widetildeΩ(κd)$ requirement on $n$ can be replaced with $Ω(κ^2)$, which is dimensionality-independent.

Lijun Zhang, Tianbao Yang, Rong Jin et al.

To cope with changing environments, recent developments in online learning have introduced the concepts of adaptive regret and dynamic regret independently. In this paper, we illustrate an intrinsic connection between these two concepts by showing that the dynamic regret can be expressed in terms of the adaptive regret and the functional variation. This observation implies that strongly adaptive algorithms can be directly leveraged to minimize the dynamic regret. As a result, we present a series of strongly adaptive algorithms that have small dynamic regrets for convex functions, exponentially concave functions, and strongly convex functions, respectively. To the best of our knowledge, this is the first time that exponential concavity is utilized to upper bound the dynamic regret. Moreover, all of those adaptive algorithms do not need any prior knowledge of the functional variation, which is a significant advantage over previous specialized methods for minimizing dynamic regret.