Yuanyu Wan

Runang He, Tongya Zheng, Huiling Peng et al.

Ever-evolving transaction patterns have significantly hindered anomaly detection on emerging cryptocurrency blockchains due to the vast number of addresses and diverse anomalous behaviors. Recently, advanced Graph Anomaly Detection (GAD) approaches applied to blockchains have faced two critical challenges: \textit{adversarial pattern evolution by malicious actors} and \textit{the out-of-distribution (OOD) problem caused by varied transaction semantics on blockchains}. To address these challenges, we propose a novel framework termed \textbf{TE}mporal \textbf{M}otif-aware \textbf{G}raph \textbf{T}est-\textbf{T}ime \textbf{A}daptation (\textbf{TEMG-TTA}). First, we comprehensively capture the 3-node temporal motif distribution of each active address using an efficient computational mechanism, enabling downstream temporal motif-aware graph learning. Second, we design a simple yet effective test-time adaptation strategy to facilitate the sharing of common patterns between training and testing graphs. Extensive experiments on 5 real-world datasets demonstrate that our proposed \textbf{TEMG-TTA} outperforms \textit{state-of-the-art} GAD approaches by an average of 54.88\%. A further case study on interpretable motif patterns reveals that \textbf{TEMG-TTA} explicitly characterizes the complex transaction patterns of anomalous addresses, thereby verifying the effectiveness of our technical designs. Our code will be made publicly available https://github.com/LuoXishuang0712/TEMG-TTA/.

Dingzhi Yu, Hongyi Tao, Yuanyu Wan et al.

While adaptive gradient methods are the workhorse of modern machine learning, sign-based optimization algorithms such as Lion and Muon have recently demonstrated superior empirical performance over AdamW in training large language models (LLM). However, a theoretical understanding of why sign-based updates outperform variance-adapted methods remains elusive. In this paper, we aim to bridge the gap between theory and practice through the lens of heavy-tailed gradient noise, a phenomenon frequently observed in language modeling tasks. Theoretically, we introduce a novel generalized heavy-tailed noise condition that captures the behavior of LLMs more accurately than standard finite variance assumptions. Under this noise model, we establish sharp convergence rates of SignSGD and Lion for generalized smooth function classes, matching or surpassing previous best-known bounds. Furthermore, we extend our analysis to Muon and Muonlight, providing what is, to our knowledge, the first rigorous analysis of matrix optimization under heavy-tailed stochasticity. These results offer a strong theoretical justification for the empirical superiority of sign-based optimizers, showcasing that they are naturally suited to handle the noisy gradients associated with heavy tails. Empirically, LLM pretraining experiments validate our theoretical insights and confirm that our proposed noise models are well-aligned with practice.

Yuechen Xie, Xiaoyan Zhang, Yicheng Shan et al.

Vision-Language Models (VLMs) have been increasingly applied in real-world scenarios due to their outstanding understanding and reasoning capabilities. Although VLMs have already demonstrated impressive capabilities in common visual question answering and logical reasoning, they still lack the ability to make reasonable decisions in complex real-world environments. We define this ability as spatial logical reasoning, which not only requires understanding the spatial relationships among objects in complex scenes, but also the logical dependencies between steps in multi-step tasks. To bridge this gap, we introduce Spatial Logical Question Answering (SpatiaLQA), a benchmark designed to evaluate the spatial logical reasoning capabilities of VLMs. SpatiaLQA consists of 9,605 question answer pairs derived from 241 real-world indoor scenes. We conduct extensive experiments on 41 mainstream VLMs, and the results show that even the most advanced models still struggle with spatial logical reasoning. To address this issue, we propose a method called recursive scene graph assisted reasoning, which leverages visual foundation models to progressively decompose complex scenes into task-relevant scene graphs, thereby enhancing the spatial logical reasoning ability of VLMs, outperforming all previous methods. Code and dataset are available at https://github.com/xieyc99/SpatiaLQA.

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.

Yuanyu Wan, Yibo Wang, Chang Yao et al.

Projection-free online learning, which eschews the projection operation via less expensive computations such as linear optimization (LO), has received much interest recently due to its efficiency in handling high-dimensional problems with complex constraints. However, previous studies assume that any queried gradient is revealed immediately, which may not hold in practice and limits their applications. To address this limitation, we generalize the online Frank-Wolfe (OFW) algorithm and the online smooth projection-free (OSPF) algorithm, which are state-of-the-art LO-based projection-free online algorithms for non-smooth and smooth functions respectively, into a delayed setting where queried gradients can be delayed by arbitrary rounds. Specifically, the main idea of our generalized OFW is to perform an update similar to the original OFW after receiving any delayed gradient, and play the latest decision for each round. Moreover, the essential change on OSPF is to replace the sum of queried gradients, which is originally utilized in each update, with the sum of available gradients. Despite their simplicities, our novel analysis shows that under a relatively large amount of delay, the generalized OFW and OSPF enjoy the same regret bound as OFW and OSPF in the non-delayed setting, respectively.

Yuwen Wang, Shunyu Liu, Kaixuan Chen et al.

Graph Lottery Ticket (GLT), a combination of core subgraph and sparse subnetwork, has been proposed to mitigate the computational cost of deep Graph Neural Networks (GNNs) on large input graphs while preserving original performance. However, the winning GLTs in exisiting studies are obtained by applying iterative magnitude-based pruning (IMP) without re-evaluating and re-considering the pruned information, which disregards the dynamic changes in the significance of edges/weights during graph/model structure pruning, and thus limits the appeal of the winning tickets. In this paper, we formulate a conjecture, i.e., existing overlooked valuable information in the pruned graph connections and model parameters which can be re-grouped into GLT to enhance the final performance. Specifically, we propose an adversarial complementary erasing (ACE) framework to explore the valuable information from the pruned components, thereby developing a more powerful GLT, referred to as the ACE-GLT. The main idea is to mine valuable information from pruned edges/weights after each round of IMP, and employ the ACE technique to refine the GLT processing. Finally, experimental results demonstrate that our ACE-GLT outperforms existing methods for searching GLT in diverse tasks. Our code will be made publicly available.

Mao-Lin Luo, Zi-Hao Zhou, Yi-Lin Zhang et al.

Continual learning for pre-trained vision-language models requires balancing three competing objectives: retaining pre-trained knowledge, preserving knowledge from a sequence of learned tasks, and maintaining the plasticity to acquire new knowledge. This paper presents a simple but effective approach called KeepLoRA to effectively balance these objectives. We first analyze the knowledge retention mechanism within the model parameter space and find that general knowledge is mainly encoded in the principal subspace, while task-specific knowledge is encoded in the residual subspace. Motivated by this finding, KeepLoRA learns new tasks by restricting LoRA parameter updates in the residual subspace to prevent interfering with previously learned capabilities. Specifically, we infuse knowledge for a new task by projecting its gradient onto a subspace orthogonal to both the principal subspace of pre-trained model and the dominant directions of previous task features. Our theoretical and empirical analyses confirm that KeepLoRA balances the three objectives and achieves state-of-the-art performance. The implementation code is available at https://github.com/MaolinLuo/KeepLoRA.

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.

Wenhao Yang, Yu Xia, Jinlong Huang et al.

Recent advances in large Vision-Language Models (VLMs) have exhibited strong reasoning capabilities on complex visual tasks by thinking with images in their Chain-of-Thought (CoT), which is achieved by actively invoking tools to analyze visual inputs rather than merely perceiving them. However, existing models often struggle to reflect on and correct themselves when attempting incorrect reasoning trajectories. To address this limitation, we propose DRIM, a model that enables deep but reliable multi-turn reasoning when thinking with images in its multimodal CoT. Our pipeline comprises three stages: data construction, cold-start SFT and RL. Based on a high-resolution image dataset, we construct high-difficulty and verifiable visual question-answer pairs, where solving each task requires multi-turn tool calls to reach the correct answer. In the SFT stage, we collect tool trajectories as cold-start data, guiding a multi-turn reasoning pattern. In the RL stage, we introduce redundancy-penalized policy optimization, which incentivizes the model to develop a self-reflective reasoning pattern. The basic idea is to impose judgment on reasoning trajectories and penalize those that produce incorrect answers without sufficient multi-scale exploration. Extensive experiments demonstrate that DRIM achieves superior performance on visual understanding benchmarks.

Wenhao Yang, Yu Xia, Jinlong Huang et al.

Recent advancements in Multimodal Large Language Models (MLLMs) have incentivized models to ``think with images'' by actively invoking visual tools during multi-turn reasoning. The common Reinforcement Learning (RL) practice of relying on outcome-based rewards ignores the fact that textual plausibility often masks executive failure, meaning that models may exhibit intuitive textual reasoning while executing imprecise or irrelevant visual actions within their agentic reasoning trajectories. This reasoning-action discrepancy introduces noise that accumulates throughout the multi-turn reasoning process, severely degrading the model's multimodal reasoning capabilities and potentially leading to training collapse. In this paper, we introduce Multimodal Agentic Policy Optimization (MAPO), bridging the gap between textual reasoning and visual actions generated by models within their Multimodal Chain-of-Thought (MCoT). Specifically, MAPO mandates the model to generate explicit textual descriptions for the visual content obtained via tool usage. We then employ a novel advantage estimation that couples the semantic alignment between these descriptions and the actual observations with the task reward. Theoretical findings are provided to justify the rationale behind MAPO, which inherently reduces the variance of gradients, and extensive experiments demonstrate that our method achieves superior performance across multiple visual reasoning benchmarks.

Yuanyu Wan, Yu Shen, Dingzhi Yu et al.

To expand the applicability of decentralized online learning, previous studies have proposed several algorithms for decentralized online continuous submodular maximization (D-OCSM) -- a non-convex/non-concave setting with continuous DR-submodular reward functions. However, there exist large gaps between their approximate regret bounds and the regret bounds achieved in the convex setting. Moreover, if focusing on projection-free algorithms, which can efficiently handle complex decision sets, they cannot even recover the approximate regret bounds achieved in the centralized setting. In this paper, we first demonstrate that for D-OCSM over general convex decision sets, these two issues can be addressed simultaneously. Furthermore, for D-OCSM over downward-closed decision sets, we show that the second issue can be addressed while significantly alleviating the first issue. Our key techniques are two reductions from D-OCSM to decentralized online convex optimization (D-OCO), which can exploit D-OCO algorithms to improve the approximate regret of D-OCSM in these two cases, respectively.

Bo Xue, Yuanyu Wan, Zhichao Lu et al.

In multi-objective decision-making with hierarchical preferences, lexicographic bandits provide a natural framework for optimizing multiple objectives in a prioritized order. In this setting, a learner repeatedly selects arms and observes reward vectors, aiming to maximize the reward for the highest-priority objective, then the next, and so on. While previous studies have primarily focused on regret minimization, this work bridges the gap between \textit{regret minimization} and \textit{best arm identification} under lexicographic preferences. We propose two elimination-based algorithms to address this joint objective. The first algorithm eliminates suboptimal arms sequentially, layer by layer, in accordance with the objective priorities, and achieves sample complexity and regret bounds comparable to those of the best single-objective algorithms. The second algorithm simultaneously leverages reward information from all objectives in each round, effectively exploiting cross-objective dependencies. Remarkably, it outperforms the known lower bound for the single-objective bandit problem, highlighting the benefit of cross-objective information sharing in the multi-objective setting. Empirical results further validate their superior performance over baselines.

Yuechen Xie, Jie Song, Yicheng Shan et al.

High-quality open-source datasets have emerged as a pivotal catalyst driving the swift advancement of deep learning, while facing the looming threat of potential exploitation. Protecting these datasets is of paramount importance for the interests of their owners. The verification of dataset ownership has evolved into a crucial approach in this domain; however, existing verification techniques are predominantly tailored to supervised models and contrastive pre-trained models, rendering them ill-suited for direct application to the increasingly prevalent masked models. In this work, we introduce the inaugural methodology addressing this critical, yet unresolved challenge, termed Dataset Ownership Verification for Masked Modeling (DOV4MM). The central objective is to ascertain whether a suspicious black-box model has been pre-trained on a particular unlabeled dataset, thereby assisting dataset owners in safeguarding their rights. DOV4MM is grounded in our empirical observation that when a model is pre-trained on the target dataset, the difficulty of reconstructing masked information within the embedding space exhibits a marked contrast to models not pre-trained on that dataset. We validated the efficacy of DOV4MM through ten masked image models on ImageNet-1K and four masked language models on WikiText-103. The results demonstrate that DOV4MM rejects the null hypothesis, with a $p$-value considerably below 0.05, surpassing all prior approaches. Code is available at https://github.com/xieyc99/DOV4MM.

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.

Dingzhi Yu, Wei Jiang, Yuanyu Wan et al.

Smoothness is crucial for attaining fast rates in first-order optimization. However, many optimization problems in modern machine learning involve non-smooth objectives. Recent studies relax the smoothness assumption by allowing the Lipschitz constant of the gradient to grow with respect to the gradient norm, which accommodates a broad range of objectives in practice. Despite this progress, existing generalizations of smoothness are restricted to Euclidean geometry with $\ell_2$-norm and only have theoretical guarantees for optimization in the Euclidean space. In this paper, we address this limitation by introducing a new $\ell*$-smoothness concept that measures the norm of Hessians in terms of a general norm and its dual, and establish convergence for mirror-descent-type algorithms, matching the rates under the classic smoothness. Notably, we propose a generalized self-bounding property that facilitates bounding the gradients via controlling suboptimality gaps, serving as a principal component for convergence analysis. Beyond deterministic optimization, we establish an anytime convergence for stochastic mirror descent based on a new bounded noise condition that encompasses the widely adopted bounded or affine noise assumptions.

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.

Yuanyu Wan, Tong Wei, Bo Xue et al.

We investigate decentralized online convex optimization (D-OCO), in which a set of local learners are required to minimize a sequence of global loss functions using only local computations and communications. Previous studies have established $O(n^{5/4}ρ^{-1/2}\sqrt{T})$ and ${O}(n^{3/2}ρ^{-1}\log T)$ regret bounds for convex and strongly convex functions respectively, where $n$ is the number of local learners, $ρ<1$ is the spectral gap of the communication matrix, and $T$ is the time horizon. However, there exist large gaps from the existing lower bounds, i.e., $Ω(n\sqrt{T})$ for convex functions and $Ω(n)$ for strongly convex functions. To fill these gaps, in this paper, we first develop a novel D-OCO algorithm that can respectively reduce the regret bounds for convex and strongly convex functions to $\tilde{O}(nρ^{-1/4}\sqrt{T})$ and $\tilde{O}(nρ^{-1/2}\log T)$. The primary technique is to design an online accelerated gossip strategy that enjoys a faster average consensus among local learners. Furthermore, by carefully exploiting spectral properties of a specific network topology, we enhance the lower bounds for convex and strongly convex functions to $Ω(nρ^{-1/4}\sqrt{T})$ and $Ω(nρ^{-1/2}\log T)$, respectively. These results suggest that the regret of our algorithm is nearly optimal in terms of $T$, $n$, and $ρ$ for both convex and strongly convex functions. Finally, we propose a projection-free variant of our algorithm to efficiently handle practical applications with complex constraints. Our analysis reveals that the projection-free variant can achieve ${O}(nT^{3/4})$ and ${O}(nT^{2/3}(\log T)^{1/3})$ regret bounds for convex and strongly convex functions with nearly optimal $\tilde{O}(ρ^{-1/2}\sqrt{T})$ and $\tilde{O}(ρ^{-1/2}T^{1/3}(\log T)^{2/3})$ communication rounds, respectively.

Sifan Yang, Yuanyu Wan, Lijun Zhang

We investigate the online nonsubmodular optimization with delayed feedback in the bandit setting, where the loss function is $α$-weakly DR-submodular and $β$-weakly DR-supermodular. Previous work has established an $(α,β)$-regret bound of $\mathcal{O}(nd^{1/3}T^{2/3})$, where $n$ is the dimensionality and $d$ is the maximum delay. However, its regret bound relies on the maximum delay and is thus sensitive to irregular delays. Additionally, it couples the effects of delays and bandit feedback as its bound is the product of the delay term and the $\mathcal{O}(nT^{2/3})$ regret bound in the bandit setting without delayed feedback. In this paper, we develop two algorithms to address these limitations, respectively. Firstly, we propose a novel method, namely DBGD-NF, which employs the one-point gradient estimator and utilizes all the available estimated gradients in each round to update the decision. It achieves a better $\mathcal{O}(n\bar{d}^{1/3}T^{2/3})$ regret bound, which is relevant to the average delay $\bar{d} = \frac{1}{T}\sum_{t=1}^T d_t\leq d$. Secondly, we extend DBGD-NF by employing a blocking update mechanism to decouple the joint effect of the delays and bandit feedback, which enjoys an $\mathcal{O}(n(T^{2/3} + \sqrt{dT}))$ regret bound. When $d = \mathcal{O}(T^{1/3})$, our regret bound matches the $\mathcal{O}(nT^{2/3})$ bound in the bandit setting without delayed feedback. Compared to our first $\mathcal{O}(n\bar{d}^{1/3}T^{2/3})$ bound, it is more advantageous when the maximum delay $d = o(\bar{d}^{2/3}T^{1/3})$. Finally, we conduct experiments on structured sparse learning to demonstrate the superiority of our methods.

Qian Zeng, Jie Song, Yuanyu Wan et al.

Diffusion models have recently emerged as the dominant approach in visual generation tasks. However, the lengthy denoising chains and the computationally intensive noise estimation networks hinder their applicability in low-latency and resource-limited environments. Previous research has endeavored to address these limitations in a decoupled manner, utilizing either advanced samplers or efficient model quantization techniques. In this study, we uncover that quantization-induced noise disrupts directional estimation at each sampling step, further distorting the precise directional estimations of higher-order samplers when solving the sampling equations through discretized numerical methods, thereby altering the optimal sampling trajectory. To attain dual acceleration with high fidelity, we propose a sampling-aware quantization strategy, wherein a Mixed-Order Trajectory Alignment technique is devised to impose a more stringent constraint on the error bounds at each sampling step, facilitating a more linear probability flow. Extensive experiments on sparse-step fast sampling across multiple datasets demonstrate that our approach preserves the rapid convergence characteristics of high-speed samplers while maintaining superior generation quality. Code will be made publicly available soon.

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.

Lingchan Bao, Tong Wei, Yuanyu Wan

We revisit multi-agent asynchronous online optimization with delays, where only one of the agents becomes active for making the decision at each round, and the corresponding feedback is received by all the agents after unknown delays. Although previous studies have established an $O(\sqrt{dT})$ regret bound for this problem, they assume that the maximum delay $d$ is knowable or the arrival order of feedback satisfies a special property, which may not hold in practice. In this paper, we surprisingly find that when the loss functions are strongly convex, these assumptions can be eliminated, and the existing regret bound can be significantly improved to $O(d\log T)$ meanwhile. Specifically, to exploit the strong convexity of functions, we first propose a delayed variant of the classical follow-the-leader algorithm, namely FTDL, which is very simple but requires the full information of functions as feedback. Moreover, to handle the more general case with only the gradient feedback, we develop an approximate variant of FTDL by combining it with surrogate loss functions. Experimental results show that the approximate FTDL outperforms the existing algorithm in the strongly convex case.

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.

Yucheng Liao, Yuanyu Wan, Chang Yao et al.

We investigate the problem of online learning with monotone and continuous DR-submodular reward functions, which has received great attention recently. To efficiently handle this problem, especially in the case with complicated decision sets, previous studies have proposed an efficient projection-free algorithm called Mono-Frank-Wolfe (Mono-FW) using $O(T)$ gradient evaluations and linear optimization steps in total. However, it only attains a $(1-1/e)$-regret bound of $O(T^{4/5})$. In this paper, we propose an improved projection-free algorithm, namely POBGA, which reduces the regret bound to $O(T^{3/4})$ while keeping the same computational complexity as Mono-FW. Instead of modifying Mono-FW, our key idea is to make a novel combination of a projection-based algorithm called online boosting gradient ascent, an infeasible projection technique, and a blocking technique. Furthermore, we consider the decentralized setting and develop a variant of POBGA, which not only reduces the current best regret bound of efficient projection-free algorithms for this setting from $O(T^{4/5})$ to $O(T^{3/4})$, but also reduces the total communication complexity from $O(T)$ to $O(\sqrt{T})$.

Yuanyu Wan, Chang Yao, Yitao Ma et al.

Although online convex optimization (OCO) under arbitrary delays has received increasing attention recently, previous studies focus on stationary environments with the goal of minimizing static regret. In this paper, we investigate the delayed OCO in non-stationary environments, and choose dynamic regret with respect to any sequence of comparators as the performance metric. To this end, we first propose an algorithm called Mild-OGD for the full-information case, where delayed gradients are available. The basic idea is to maintain multiple experts in parallel, each performing a gradient descent step with different learning rates for every delayed gradient according to their arrival order, and utilize a meta-algorithm to track the best one based on their delayed performance. Despite the simplicity of this idea, our novel analysis shows that the dynamic regret of Mild-OGD can be automatically bounded by $O(\sqrt{\bar{d}T(P_T+1)})$ under the in-order assumption and $O(\sqrt{dT(P_T+1)})$ in the worst case, where $\bar{d}$ and $d$ denote the average and maximum delay respectively, $T$ is the time horizon, and $P_T$ is the path-length of comparators. Moreover, we demonstrate that the result in the worst case is optimal by deriving a matching lower bound. Finally, we develop a bandit variant of Mild-OGD for a more challenging case with only delayed loss values. Interestingly, we prove that under a relatively large amount of delay, our bandit algorithm even enjoys the best dynamic regret bound of existing non-delayed bandit algorithms.

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.

Zhilin Zhao, Longbing Cao, Yuanyu Wan

We consider a general and realistic scenario involving non-stationary time series, consisting of several offline intervals with different distributions within a fixed offline time horizon, and an online interval that continuously receives new samples. For non-stationary time series, the data distribution in the current online interval may have appeared in previous offline intervals. We theoretically explore the feasibility of applying knowledge from offline intervals to the current online interval. To this end, we propose the Mixture of Online and Offline Experts (MOOE). MOOE learns static offline experts from offline intervals and maintains a dynamic online expert for the current online interval. It then adaptively combines the offline and online experts using a meta expert to make predictions for the samples received in the online interval. Specifically, we focus on theoretical analysis, deriving parameter convergence, regret bounds, and generalization error bounds to prove the effectiveness of the algorithm.

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.

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.

Yuanyu Wan, Lijun Zhang

Approximate matrix multiplication with limited space has received ever-increasing attention due to the emergence of large-scale applications. Recently, based on a popular matrix sketching algorithm -- frequent directions, previous work has introduced co-occuring directions (COD) to reduce the approximation error for this problem. Although it enjoys the space complexity of $O((m_x+m_y)\ell)$ for two input matrices $X\in\mathbb{R}^{m_x\times n}$ and $Y\in\mathbb{R}^{m_y\times n}$ where $\ell$ is the sketch size, its time complexity is $O\left(n(m_x+m_y+\ell)\ell\right)$, which is still very high for large input matrices. In this paper, we propose to reduce the time complexity by exploiting the sparsity of the input matrices. The key idea is to employ an approximate singular value decomposition (SVD) method which can utilize the sparsity, to reduce the number of QR decompositions required by COD. In this way, we develop sparse co-occuring directions, which reduces the time complexity to $\widetilde{O}\left((\nnz(X)+\nnz(Y))\ell+n\ell^2\right)$ in expectation while keeps the same space complexity as $O((m_x+m_y)\ell)$, where $\nnz(X)$ denotes the number of non-zero entries in $X$ and the $\widetilde{O}$ notation hides constant factors as well as polylogarithmic factors. Theoretical analysis reveals that the approximation error of our algorithm is almost the same as that of COD. Furthermore, we empirically verify the efficiency and effectiveness of our algorithm.

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.