Yudong Chen

Brahma S. Pavse, Matthew Zurek, Yudong Chen et al.

In many reinforcement learning (RL) applications, we want policies that reach desired states and then keep the controlled system within an acceptable region around the desired states over an indefinite period of time. This latter objective is called stability and is especially important when the state space is unbounded, such that the states can be arbitrarily far from each other and the agent can drift far away from the desired states. For example, in stochastic queuing networks, where queues of waiting jobs can grow without bound, the desired state is all-zero queue lengths. Here, a stable policy ensures queue lengths are finite while an optimal policy minimizes queue lengths. Since an optimal policy is also stable, one would expect that RL algorithms would implicitly give us stable policies. However, in this work, we find that deep RL algorithms that directly minimize the distance to the desired state during online training often result in unstable policies, i.e., policies that drift far away from the desired state. We attribute this instability to poor credit-assignment for destabilizing actions. We then introduce an approach based on two ideas: 1) a Lyapunov-based cost-shaping technique and 2) state transformations to the unbounded state space. We conduct an empirical study on various queueing networks and traffic signal control problems and find that our approach performs competitively against strong baselines with knowledge of the transition dynamics. Our code is available here: https://github.com/Badger-RL/STOP.

Jianglin Lu, Jie Zhou, Yudong Chen et al.

Thanks to the efficient retrieval speed and low storage consumption, learning to hash has been widely used in visual retrieval tasks. However, existing hashing methods assume that the query and retrieval samples lie in homogeneous feature space within the same domain. As a result, they cannot be directly applied to heterogeneous cross-domain retrieval. In this paper, we propose a Generalized Image Transfer Retrieval (GITR) problem, which encounters two crucial bottlenecks: 1) the query and retrieval samples may come from different domains, leading to an inevitable {domain distribution gap}; 2) the features of the two domains may be heterogeneous or misaligned, bringing up an additional {feature gap}. To address the GITR problem, we propose an Asymmetric Transfer Hashing (ATH) framework with its unsupervised/semi-supervised/supervised realizations. Specifically, ATH characterizes the domain distribution gap by the discrepancy between two asymmetric hash functions, and minimizes the feature gap with the help of a novel adaptive bipartite graph constructed on cross-domain data. By jointly optimizing asymmetric hash functions and the bipartite graph, not only can knowledge transfer be achieved but information loss caused by feature alignment can also be avoided. Meanwhile, to alleviate negative transfer, the intrinsic geometrical structure of single-domain data is preserved by involving a domain affinity graph. Extensive experiments on both single-domain and cross-domain benchmarks under different GITR subtasks indicate the superiority of our ATH method in comparison with the state-of-the-art hashing methods.

Xuwei Xu, Sen Wang, Yudong Chen et al.

Vision Transformers (ViTs) have revolutionized the field of computer vision, yet their deployments on resource-constrained devices remain challenging due to high computational demands. To expedite pre-trained ViTs, token pruning and token merging approaches have been developed, which aim at reducing the number of tokens involved in the computation. However, these methods still have some limitations, such as image information loss from pruned tokens and inefficiency in the token-matching process. In this paper, we introduce a novel Graph-based Token Propagation (GTP) method to resolve the challenge of balancing model efficiency and information preservation for efficient ViTs. Inspired by graph summarization algorithms, GTP meticulously propagates less significant tokens' information to spatially and semantically connected tokens that are of greater importance. Consequently, the remaining few tokens serve as a summarization of the entire token graph, allowing the method to reduce computational complexity while preserving essential information of eliminated tokens. Combined with an innovative token selection strategy, GTP can efficiently identify image tokens to be propagated. Extensive experiments have validated GTP's effectiveness, demonstrating both efficiency and performance improvements. Specifically, GTP decreases the computational complexity of both DeiT-S and DeiT-B by up to 26% with only a minimal 0.3% accuracy drop on ImageNet-1K without finetuning, and remarkably surpasses the state-of-the-art token merging method on various backbones at an even faster inference speed. The source code is available at https://github.com/Ackesnal/GTP-ViT.

Liwei Jiang, Yudong Chen, Lijun Ding

We study the asymmetric matrix factorization problem under a natural nonconvex formulation with arbitrary overparametrization. The model-free setting is considered, with minimal assumption on the rank or singular values of the observed matrix, where the global optima provably overfit. We show that vanilla gradient descent with small random initialization sequentially recovers the principal components of the observed matrix. Consequently, when equipped with proper early stopping, gradient descent produces the best low-rank approximation of the observed matrix without explicit regularization. We provide a sharp characterization of the relationship between the approximation error, iteration complexity, initialization size and stepsize. Our complexity bound is almost dimension-free and depends logarithmically on the approximation error, with significantly more lenient requirements on the stepsize and initialization compared to prior work. Our theoretical results provide accurate prediction for the behavior gradient descent, showing good agreement with numerical experiments.

Dongyan Huo, Yudong Chen, Qiaomin Xie

We consider Linear Stochastic Approximation (LSA) with a constant stepsize and Markovian data. Viewing the joint process of the data and LSA iterate as a time-homogeneous Markov chain, we prove its convergence to a unique limiting and stationary distribution in Wasserstein distance and establish non-asymptotic, geometric convergence rates. Furthermore, we show that the bias vector of this limit admits an infinite series expansion with respect to the stepsize. Consequently, the bias is proportional to the stepsize up to higher order terms. This result stands in contrast with LSA under i.i.d. data, for which the bias vanishes. In the reversible chain setting, we provide a general characterization of the relationship between the bias and the mixing time of the Markovian data, establishing that they are roughly proportional to each other. While Polyak-Ruppert tail-averaging reduces the variance of the LSA iterates, it does not affect the bias. The above characterization allows us to show that the bias can be reduced using Richardson-Romberg extrapolation with $m\ge 2$ stepsizes, which eliminates the $m-1$ leading terms in the bias expansion. This extrapolation scheme leads to an exponentially smaller bias and an improved mean squared error, both in theory and empirically. Our results immediately apply to the Temporal Difference learning algorithm with linear function approximation, Markovian data, and constant stepsizes.

Yudong Chen, Sen Wang, Jiajun Liu et al.

In knowledge distillation, previous feature distillation methods mainly focus on the design of loss functions and the selection of the distilled layers, while the effect of the feature projector between the student and the teacher remains under-explored. In this paper, we first discuss a plausible mechanism of the projector with empirical evidence and then propose a new feature distillation method based on a projector ensemble for further performance improvement. We observe that the student network benefits from a projector even if the feature dimensions of the student and the teacher are the same. Training a student backbone without a projector can be considered as a multi-task learning process, namely achieving discriminative feature extraction for classification and feature matching between the student and the teacher for distillation at the same time. We hypothesize and empirically verify that without a projector, the student network tends to overfit the teacher's feature distributions despite having different architecture and weights initialization. This leads to degradation on the quality of the student's deep features that are eventually used in classification. Adding a projector, on the other hand, disentangles the two learning tasks and helps the student network to focus better on the main feature extraction task while still being able to utilize teacher features as a guidance through the projector. Motivated by the positive effect of the projector in feature distillation, we propose an ensemble of projectors to further improve the quality of student features. Experimental results on different datasets with a series of teacher-student pairs illustrate the effectiveness of the proposed method.

Xuwei Xu, Changlin Li, Yudong Chen et al.

Vision Transformers (ViTs) have demonstrated outstanding performance in computer vision tasks, yet their high computational complexity prevents their deployment in computing resource-constrained environments. Various token pruning techniques have been introduced to alleviate the high computational burden of ViTs by dynamically dropping image tokens. However, some undesirable pruning at early stages may result in permanent loss of image information in subsequent layers, consequently hindering model performance. To address this problem, we propose IdleViT, a dynamic token-idle-based method that achieves an excellent trade-off between performance and efficiency. Specifically, in each layer, IdleViT selects a subset of the image tokens to participate in computations while keeping the rest of the tokens idle and directly passing them to this layer's output. By allowing the idle tokens to be re-selected in the following layers, IdleViT mitigates the negative impact of improper pruning in the early stages. Furthermore, inspired by the normalized graph cut, we devise a token cut loss on the attention map as regularization to improve IdleViT's token selection ability. Our method is simple yet effective and can be extended to pyramid ViTs since no token is completely dropped. Extensive experimental results on various ViT architectures have shown that IdleViT can diminish the complexity of pretrained ViTs by up to 33\% with no more than 0.2\% accuracy decrease on ImageNet, after finetuning for only 30 epochs. Notably, when the keep ratio is 0.5, IdleViT outperforms the state-of-the-art EViT on DeiT-S by 0.5\% higher accuracy and even faster inference speed. The source code is available in the supplementary material.

Tyler Sam, Yudong Chen, Christina Lee Yu

The practicality of reinforcement learning algorithms has been limited due to poor scaling with respect to the problem size, as the sample complexity of learning an $ε$-optimal policy is $\tildeΩ\left(|S||A|H^3 / ε^2\right)$ over worst case instances of an MDP with state space $S$, action space $A$, and horizon $H$. We consider a class of MDPs for which the associated optimal $Q^*$ function is low rank, where the latent features are unknown. While one would hope to achieve linear sample complexity in $|S|$ and $|A|$ due to the low rank structure, we show that without imposing further assumptions beyond low rank of $Q^*$, if one is constrained to estimate the $Q$ function using only observations from a subset of entries, there is a worst case instance in which one must incur a sample complexity exponential in the horizon $H$ to learn a near optimal policy. We subsequently show that under stronger low rank structural assumptions, given access to a generative model, Low Rank Monte Carlo Policy Iteration (LR-MCPI) and Low Rank Empirical Value Iteration (LR-EVI) achieve the desired sample complexity of $\tilde{O}\left((|S|+|A|)\mathrm{poly}(d,H)/ε^2\right)$ for a rank $d$ setting, which is minimax optimal with respect to the scaling of $|S|, |A|$, and $ε$. In contrast to literature on linear and low-rank MDPs, we do not require a known feature mapping, our algorithm is computationally simple, and our results hold for long time horizons. Our results provide insights on the minimal low-rank structural assumptions required on the MDP with respect to the transition kernel versus the optimal action-value function.

Young Wu, Jeremy McMahan, Yiding Chen et al.

We study the game modification problem, where a benevolent game designer or a malevolent adversary modifies the reward function of a zero-sum Markov game so that a target deterministic or stochastic policy profile becomes the unique Markov perfect Nash equilibrium and has a value within a target range, in a way that minimizes the modification cost. We characterize the set of policy profiles that can be installed as the unique equilibrium of a game and establish sufficient and necessary conditions for successful installation. We propose an efficient algorithm that solves a convex optimization problem with linear constraints and then performs random perturbation to obtain a modification plan with a near-optimal cost. The code for our algorithm is available at https://github.com/YoungWu559/game-modification .

Emmanouil-Vasileios Vlatakis-Gkaragkounis, Angeliki Giannou, Yudong Chen et al.

For min-max optimization and variational inequalities problems (VIP) encountered in diverse machine learning tasks, Stochastic Extragradient (SEG) and Stochastic Gradient Descent Ascent (SGDA) have emerged as preeminent algorithms. Constant step-size variants of SEG/SGDA have gained popularity, with appealing benefits such as easy tuning and rapid forgiveness of initial conditions, but their convergence behaviors are more complicated even in rudimentary bilinear models. Our work endeavors to elucidate and quantify the probabilistic structures intrinsic to these algorithms. By recasting the constant step-size SEG/SGDA as time-homogeneous Markov Chains, we establish a first-of-its-kind Law of Large Numbers and a Central Limit Theorem, demonstrating that the average iterate is asymptotically normal with a unique invariant distribution for an extensive range of monotone and non-monotone VIPs. Specializing to convex-concave min-max optimization, we characterize the relationship between the step-size and the induced bias with respect to the Von-Neumann's value. Finally, we establish that Richardson-Romberg extrapolation can improve proximity of the average iterate to the global solution for VIPs. Our probabilistic analysis, underpinned by experiments corroborating our theoretical discoveries, harnesses techniques from optimization, Markov chains, and operator theory.

Zhihao Wu, Gracia Gong, Qinglin Zhu et al.

Watermarking embeds statistical signatures in AI-generated text for detection and attribution. We reveal a fundamental vulnerability: when users access multiple models (today's reality), watermarks trivially fail. Watermarks perturb output distributions away from the original, and in competitive markets, these perturbations are typically independent across providers. We theoretically prove that averaging output probability distributions recovers the unwatermarked distribution with up to a second-order error term. Empirically, simply averaging 3-5 models cancels out these perturbations. We introduce WASH (Watermark Attenuation via Statistical Hybridisation), which solves practical challenges in ensemble generation: vocabulary misalignment and tokenisation differences across heterogeneous models. Experiments across six watermarking schemes and three LLMs show that averaging across 3 models suppresses detection z-scores from 5-300 to below 2 (below the detection threshold of 4) and reduces TPR at 5% FPR to below 50%, while improving quality by 27.5% and running 6 times faster than the best baseline on the long sequence generation. Our results suggest that robust AI-text detection via watermarking requires either accepting this fundamental vulnerability or unprecedented coordination among model providers.

Matthew Zurek, Yudong Chen

We study graph clustering in the Stochastic Block Model (SBM) in the presence of both large clusters and small, unrecoverable clusters. Previous convex relaxation approaches achieving exact recovery do not allow any small clusters of size $o(\sqrt{n})$, or require a size gap between the smallest recovered cluster and the largest non-recovered cluster. We provide an algorithm based on semidefinite programming (SDP) which removes these requirements and provably recovers large clusters regardless of the remaining cluster sizes. Mid-sized clusters pose unique challenges to the analysis, since their proximity to the recovery threshold makes them highly sensitive to small noise perturbations and precludes a closed-form candidate solution. We develop novel techniques, including a leave-one-out-style argument which controls the correlation between SDP solutions and noise vectors even when the removal of one row of noise can drastically change the SDP solution. We also develop improved eigenvalue perturbation bounds of potential independent interest. Our results are robust to certain semirandom settings that are challenging for alternative algorithms. Using our gap-free clustering procedure, we obtain efficient algorithms for the problem of clustering with a faulty oracle with superior query complexities, notably achieving $o(n^2)$ sample complexity even in the presence of a large number of small clusters. Our gap-free clustering procedure also leads to improved algorithms for recursive clustering.

Jeremy McMahan, Young Wu, Yudong Chen et al.

Many real-world games suffer from information asymmetry: one player is only aware of their own payoffs while the other player has the full game information. Examples include the critical domain of security games and adversarial multi-agent reinforcement learning. Information asymmetry renders traditional solution concepts such as Strong Stackelberg Equilibrium (SSE) and Robust-Optimization Equilibrium (ROE) inoperative. We propose a novel solution concept called VISER (Victim Is Secure, Exploiter best-Responds). VISER enables an external observer to predict the outcome of such games. In particular, for security applications, VISER allows the victim to better defend itself while characterizing the most damaging attacks available to the attacker. We show that each player's VISER strategy can be computed independently in polynomial time using linear programming (LP). We also extend VISER to its Markov-perfect counterpart for Markov games, which can be solved efficiently using a series of LPs.

Yudong Chen, Sen Wang, Jiajun Liu et al.

Conventionally, during the knowledge distillation process (e.g. feature distillation), an additional projector is often required to perform feature transformation due to the dimension mismatch between the teacher and the student networks. Interestingly, we discovered that even if the student and the teacher have the same feature dimensions, adding a projector still helps to improve the distillation performance. In addition, projectors even improve logit distillation if we add them to the architecture too. Inspired by these surprising findings and the general lack of understanding of the projectors in the knowledge distillation process from existing literature, this paper investigates the implicit role that projectors play but so far have been overlooked. Our empirical study shows that the student with a projector (1) obtains a better trade-off between the training accuracy and the testing accuracy compared to the student without a projector when it has the same feature dimensions as the teacher, (2) better preserves its similarity to the teacher beyond shallow and numeric resemblance, from the view of Centered Kernel Alignment (CKA), and (3) avoids being over-confident as the teacher does at the testing phase. Motivated by the positive effects of projectors, we propose a projector ensemble-based feature distillation method to further improve distillation performance. Despite the simplicity of the proposed strategy, empirical results from the evaluation of classification tasks on benchmark datasets demonstrate the superior classification performance of our method on a broad range of teacher-student pairs and verify from the aspects of CKA and model calibration that the student's features are of improved quality with the projector ensemble design.

Matthew Zurek, Yudong Chen

We study the sample complexity of learning an $\varepsilon$-optimal policy in an average-reward Markov decision process (MDP) under a generative model. We establish the complexity bound $\widetilde{O}\left(SA\frac{H}{\varepsilon^2} \right)$, where $H$ is the span of the bias function of the optimal policy and $SA$ is the cardinality of the state-action space. Our result is the first that is minimax optimal (up to log factors) in all parameters $S,A,H$ and $\varepsilon$, improving on existing work that either assumes uniformly bounded mixing times for all policies or has suboptimal dependence on the parameters. Our result is based on reducing the average-reward MDP to a discounted MDP. To establish the optimality of this reduction, we develop improved bounds for $γ$-discounted MDPs, showing that $\widetilde{O}\left(SA\frac{H}{(1-γ)^2\varepsilon^2} \right)$ samples suffice to learn a $\varepsilon$-optimal policy in weakly communicating MDPs under the regime that $γ\geq 1 - \frac{1}{H}$, circumventing the well-known lower bound of $\widetildeΩ\left(SA\frac{1}{(1-γ)^3\varepsilon^2} \right)$ for general $γ$-discounted MDPs. Our analysis develops upper bounds on certain instance-dependent variance parameters in terms of the span parameter. These bounds are tighter than those based on the mixing time or diameter of the MDP and may be of broader use.

Qinglin Zhu, Runcong Zhao, Hanqi Yan et al.

Large Language Models (LLMs) struggle with complex reasoning due to limited diversity and inefficient search. We propose Soft Reasoning, an embedding-based search framework that optimises the embedding of the first token to guide generation. It combines (1) embedding perturbation for controlled exploration and (2) Bayesian optimisation to refine embeddings via a verifier-guided objective, balancing exploration and exploitation. This approach improves reasoning accuracy and coherence while avoiding reliance on heuristic search. Experiments demonstrate superior correctness with minimal computation, making it a scalable, model-agnostic solution. The code is released at https://github.com/alickzhu/Soft-Reasoning.

Yuanhe Zhang, Fanghui Liu, Yudong Chen

This paper explores how theory can guide and enhance practical algorithms, using Low-Rank Adaptation (LoRA, Hu et al. 2022) in large language models as a case study. We rigorously prove that, under gradient descent, LoRA adapters align with specific singular subspaces of the one-step full fine-tuning gradient. This result suggests that, by properly initializing the adapters using the one-step full gradient, subspace alignment can be achieved immediately and applicable to both linear and nonlinear models. Building on our theory, we propose a theory-driven algorithm, LoRA-One, where the linear convergence (as well as generalization) is built and incorporating preconditioners theoretically helps mitigate the effects of ill-conditioning. Besides, our theory reveals connections between LoRA-One and other gradient-alignment-based methods, helping to clarify misconceptions in the design of such algorithms. LoRA-One achieves significant empirical improvements over LoRA and its variants across benchmarks in natural language understanding, mathematical reasoning, and code generation. Code is available at: https://github.com/YuanheZ/LoRA-One.

Yudong Chen, Xumei Xi, Christina Lee Yu

Matrix completion tackles the task of predicting missing values in a low-rank matrix based on a sparse set of observed entries. It is often assumed that the observation pattern is generated uniformly at random or has a very specific structure tuned to a given algorithm. There is still a gap in our understanding when it comes to arbitrary sampling patterns. Given an arbitrary sampling pattern, we introduce a matrix completion algorithm based on network flows in the bipartite graph induced by the observation pattern. For additive matrices, the particular flow we used is the electrical flow and we establish error upper bounds customized to each entry as a function of the observation set, along with matching minimax lower bounds. Our results show that the minimax squared error for recovery of a particular entry in the matrix is proportional to the effective resistance of the corresponding edge in the graph. Furthermore, we show that our estimator is equivalent to the least squares estimator. We apply our estimator to the two-way fixed effects model and show that it enables us to accurately infer individual causal effects and the unit-specific and time-specific confounders. For rank-$1$ matrices, we use edge-disjoint paths to form an estimator that achieves minimax optimal estimation when the sampling is sufficiently dense. Our discovery introduces a new family of estimators parametrized by network flows, which provide a fine-grained and intuitive understanding of the impact of the given sampling pattern on the relative difficulty of estimation at an entry-specific level. This graph-based approach allows us to quantify the inherent complexity of matrix completion for individual entries, rather than relying solely on global measures of performance.

Xuwei Xu, Sen Wang, Yudong Chen et al.

Vision Transformers (ViTs) have demonstrated remarkable performance in various computer vision tasks. However, the high computational complexity hinders ViTs' applicability on devices with limited memory and computing resources. Although certain investigations have delved into the fusion of convolutional layers with self-attention mechanisms to enhance the efficiency of ViTs, there remains a knowledge gap in constructing tiny yet effective ViTs solely based on the self-attention mechanism. Furthermore, the straightforward strategy of reducing the feature channels in a large but outperforming ViT often results in significant performance degradation despite improved efficiency. To address these challenges, we propose a novel channel shuffle module to improve tiny-size ViTs, showing the potential of pure self-attention models in environments with constrained computing resources. Inspired by the channel shuffle design in ShuffleNetV2 \cite{ma2018shufflenet}, our module expands the feature channels of a tiny ViT and partitions the channels into two groups: the \textit{Attended} and \textit{Idle} groups. Self-attention computations are exclusively employed on the designated \textit{Attended} group, followed by a channel shuffle operation that facilitates information exchange between the two groups. By incorporating our module into a tiny ViT, we can achieve superior performance while maintaining a comparable computational complexity to the vanilla model. Specifically, our proposed channel shuffle module consistently improves the top-1 accuracy on the ImageNet-1K dataset for various tiny ViT models by up to 2.8\%, with the changes in model complexity being less than 0.03 GMACs.

Xuwei Xu, Yang Li, Yudong Chen et al.

We reveal that feedforward network (FFN) layers, rather than attention layers, are the primary contributors to Vision Transformer (ViT) inference latency, with their impact signifying as model size increases. This finding highlights a critical opportunity for optimizing the efficiency of large-scale ViTs by focusing on FFN layers. In this work, we propose a novel channel idle mechanism that facilitates post-training structural reparameterization for efficient FFN layers during testing. Specifically, a set of feature channels remains idle and bypasses the nonlinear activation function in each FFN layer, thereby forming a linear pathway that enables structural reparameterization during inference. This mechanism results in a family of ReParameterizable Vision Transformers (RePaViTs), which achieve remarkable latency reductions with acceptable sacrifices (sometimes gains) in accuracy across various ViTs. The benefits of our method scale consistently with model sizes, demonstrating greater speed improvements and progressively narrowing accuracy gaps or even higher accuracies on larger models. In particular, RePa-ViT-Large and RePa-ViT-Huge enjoy 66.8% and 68.7% speed-ups with +1.7% and +1.1% higher top-1 accuracies under the same training strategy, respectively. RePaViT is the first to employ structural reparameterization on FFN layers to expedite ViTs to our best knowledge, and we believe that it represents an auspicious direction for efficient ViTs. Source code is available at https://github.com/Ackesnal/RePaViT.

Bo Zheng, Yudong Chen, Zihua Xiong et al.

Tabular data forms the backbone of high-stakes decision systems in finance, healthcare, and beyond. Yet industrial tabular datasets are inherently difficult: high-dimensional, riddled with missing entries, and rarely labeled at scale. While foundation models have revolutionized vision and language, tabular learning still leans on handcrafted features and lacks a general self-supervised framework. We present MaskTab, a unified pre-training framework designed specifically for industrial-scale tabular data. MaskTab encodes missing values via dedicated learnable tokens, enabling the model to distinguish structural absence from random dropout. It jointly optimizes a hybrid supervised pre-training scheme--utilizing a twin-path architecture to reconcile masked reconstruction with task-specific supervision--and an MoE-augmented loss that adaptively routes features through specialized subnetworks. On industrial-scale benchmarks, it achieves +5.04% AUC and +8.28% KS over prior art under rigorous scaling. Moreover, its representations distill effectively into lightweight models, yielding +2.55% AUC and +4.85% KS under strict latency and interpretability constraints, while improving robustness to distribution shifts. Our work demonstrates that tabular data admits a foundation-model treatment--when its structural idiosyncrasies are respected.

Guy Zamir, Matthew Zurek, Yudong Chen

Online reinforcement learning in infinite-horizon Markov decision processes (MDPs) remains less theoretically and algorithmically developed than its episodic counterpart, with many algorithms suffering from high ``burn-in'' costs and failing to adapt to benign instance-specific complexity. In this work, we address these shortcomings for two infinite-horizon objectives: the classical average-reward regret and the $γ$-regret. We develop a single tractable UCB-style algorithm applicable to both settings, which achieves the first optimal variance-dependent regret guarantees. Our regret bounds in both settings take the form $\tilde{O}( \sqrt{SA\,\text{Var}} + \text{lower-order terms})$, where $S,A$ are the state and action space sizes, and $\text{Var}$ captures cumulative transition variance. This implies minimax-optimal average-reward and $γ$-regret bounds in the worst case but also adapts to easier problem instances, for example yielding nearly constant regret in deterministic MDPs. Furthermore, our algorithm enjoys significantly improved lower-order terms for the average-reward setting. With prior knowledge of the optimal bias span $\Vert h^\star\Vert_\text{sp}$, our algorithm obtains lower-order terms scaling as $\Vert h^\star\Vert_\text{sp} S^2 A$, which we prove is optimal in both $\Vert h^\star\Vert_\text{sp}$ and $A$. Without prior knowledge, we prove that no algorithm can have lower-order terms smaller than $\Vert h^\star \Vert_\text{sp}^2 S A$, and we provide a prior-free algorithm whose lower-order terms scale as $\Vert h^\star\Vert_\text{sp}^2 S^3 A$, nearly matching this lower bound. Taken together, these results completely characterize the optimal dependence on $\Vert h^\star\Vert_\text{sp}$ in both leading and lower-order terms, and reveal a fundamental gap in what is achievable with and without prior knowledge.

Yudong Chen, Chaoyu Guan, Zhikun Wei et al.

Meta-learning aims at learning quickly on novel tasks with limited data by transferring generic experience learned from previous tasks. Naturally, few-shot learning has been one of the most popular applications for meta-learning. However, existing meta-learning algorithms rarely consider the time and resource efficiency or the generalization capacity for unknown datasets, which limits their applicability in real-world scenarios. In this paper, we propose MetaDelta, a novel practical meta-learning system for the few-shot image classification. MetaDelta consists of two core components: i) multiple meta-learners supervised by a central controller to ensure efficiency, and ii) a meta-ensemble module in charge of integrated inference and better generalization. In particular, each meta-learner in MetaDelta is composed of a unique pretrained encoder fine-tuned by batch training and parameter-free decoder used for prediction. MetaDelta ranks first in the final phase in the AAAI 2021 MetaDL Challenge\footnote{https://competitions.codalab.org/competitions/26638}, demonstrating the advantages of our proposed system. The codes are publicly available at https://github.com/Frozenmad/MetaDelta.

Xinyu Zhou, Delong Chen, Yudong Chen

This paper explores the potential of constructing an AI spoken dialogue system that "thinks how to respond" and "thinks how to speak" simultaneously, which more closely aligns with the human speech production process compared to the current cascade pipeline of independent chatbot and Text-to-Speech (TTS) modules. We hypothesize that Large Language Models (LLMs) with billions of parameters possess significant speech understanding capabilities and can jointly model dialogue responses and linguistic features. We conduct two sets of experiments: 1) Prosodic structure prediction, a typical front-end task in TTS, demonstrating the speech understanding ability of LLMs, and 2) Further integrating dialogue response and a wide array of linguistic features using a unified encoding format. Our results indicate that the LLM-based approach is a promising direction for building unified spoken dialogue systems.

Matthew Zurek, Yudong Chen

We study the sample complexity of learning an $\varepsilon$-optimal policy in an average-reward Markov decision process (MDP) under a generative model. For weakly communicating MDPs, we establish the complexity bound $\widetilde{O}(SA\frac{H}{\varepsilon^2} )$, where $H$ is the span of the bias function of the optimal policy and $SA$ is the cardinality of the state-action space. Our result is the first that is minimax optimal (up to log factors) in all parameters $S,A,H$, and $\varepsilon$, improving on existing work that either assumes uniformly bounded mixing times for all policies or has suboptimal dependence on the parameters. We also initiate the study of sample complexity in general (multichain) average-reward MDPs. We argue a new transient time parameter $B$ is necessary, establish an $\widetilde{O}(SA\frac{B + H}{\varepsilon^2})$ complexity bound, and prove a matching (up to log factors) minimax lower bound. Both results are based on reducing the average-reward MDP to a discounted MDP, which requires new ideas in the general setting. To optimally analyze this reduction, we develop improved bounds for $γ$-discounted MDPs, showing that $\widetilde{O}(SA\frac{H}{(1-γ)^2\varepsilon^2} )$ and $\widetilde{O}(SA\frac{B + H}{(1-γ)^2\varepsilon^2} )$ samples suffice to learn $\varepsilon$-optimal policies in weakly communicating and in general MDPs, respectively. Both these results circumvent the well-known minimax lower bound of $\widetildeΩ(SA\frac{1}{(1-γ)^3\varepsilon^2} )$ for $γ$-discounted MDPs, and establish a quadratic rather than cubic horizon dependence for a fixed MDP instance.

Yige Hong, Qiaomin Xie, Yudong Chen et al.

We consider the infinite-horizon, average-reward restless bandit problem in discrete time. We propose a new class of policies that are designed to drive a progressively larger subset of arms toward the optimal distribution. We show that our policies are asymptotically optimal with an $O(1/\sqrt{N})$ optimality gap for an $N$-armed problem, assuming only a unichain and aperiodicity assumption. Our approach departs from most existing work that focuses on index or priority policies, which rely on the Global Attractor Property (GAP) to guarantee convergence to the optimum, or a recently developed simulation-based policy, which requires a Synchronization Assumption (SA).

Dongyan Huo, Yudong Chen, Qiaomin Xie

In this paper, we study the effectiveness of using a constant stepsize in statistical inference via linear stochastic approximation (LSA) algorithms with Markovian data. After establishing a Central Limit Theorem (CLT), we outline an inference procedure that uses averaged LSA iterates to construct confidence intervals (CIs). Our procedure leverages the fast mixing property of constant-stepsize LSA for better covariance estimation and employs Richardson-Romberg (RR) extrapolation to reduce the bias induced by constant stepsize and Markovian data. We develop theoretical results for guiding stepsize selection in RR extrapolation, and identify several important settings where the bias provably vanishes even without extrapolation. We conduct extensive numerical experiments and compare against classical inference approaches. Our results show that using a constant stepsize enjoys easy hyperparameter tuning, fast convergence, and consistently better CI coverage, especially when data is limited.

Matthew Zurek, Yudong Chen

We study the sample complexity of finding an $\varepsilon$-optimal policy in average-reward Markov Decision Processes (MDPs) with a generative model. The minimax optimal span-based complexity of $\widetilde{O}(SAH/\varepsilon^2)$, where $H$ is the span of the optimal bias function, has only been achievable with prior knowledge of the value of $H$. Prior-knowledge-free algorithms have been the objective of intensive research, but several natural approaches provably fail to achieve this goal. We resolve this problem, developing the first algorithms matching the optimal span-based complexity without $H$ knowledge, both when the dataset size is fixed and when the suboptimality level $\varepsilon$ is fixed. Our main technique combines the discounted reduction approach with a method for automatically tuning the effective horizon based on empirical confidence intervals or lower bounds on performance, which we term horizon calibration. We also develop an empirical span penalization approach, inspired by sample variance penalization, which satisfies an oracle inequality performance guarantee. In particular this algorithm can outperform the minimax complexity in benign settings such as when there exist near-optimal policies with span much smaller than $H$.

Jeongyeol Kwon, Luke Dotson, Yudong Chen et al.

Previous studies on two-timescale stochastic approximation (SA) mainly focused on bounding mean-squared errors under diminishing stepsize schemes. In this work, we investigate {\it constant} stpesize schemes through the lens of Markov processes, proving that the iterates of both timescales converge to a unique joint stationary distribution in Wasserstein metric. We derive explicit geometric and non-asymptotic convergence rates, as well as the variance and bias introduced by constant stepsizes in the presence of Markovian noise. Specifically, with two constant stepsizes $α< β$, we show that the biases scale linearly with both stepsizes as $Θ(α)+Θ(β)$ up to higher-order terms, while the variance of the slower iterate (resp., faster iterate) scales only with its own stepsize as $O(α)$ (resp., $O(β)$). Unlike previous work, our results require no additional assumptions such as $β^2 \ll α$ nor extra dependence on dimensions. These fine-grained characterizations allow tail-averaging and extrapolation techniques to reduce variance and bias, improving mean-squared error bound to $O(β^4 + \frac{1}{t})$ for both iterates.

Yixuan Zhang, Dongyan Huo, Yudong Chen et al.

Motivated by Q-learning, we study nonsmooth contractive stochastic approximation (SA) with constant stepsize. We focus on two important classes of dynamics: 1) nonsmooth contractive SA with additive noise, and 2) synchronous and asynchronous Q-learning, which features both additive and multiplicative noise. For both dynamics, we establish weak convergence of the iterates to a stationary limit distribution in Wasserstein distance. Furthermore, we propose a prelimit coupling technique for establishing steady-state convergence and characterize the limit of the stationary distribution as the stepsize goes to zero. Using this result, we derive that the asymptotic bias of nonsmooth SA is proportional to the square root of the stepsize, which stands in sharp contrast to smooth SA. This bias characterization allows for the use of Richardson-Romberg extrapolation for bias reduction in nonsmooth SA.

Yixuan Zhang, Dongyan Huo, Yudong Chen et al.

Motivated by robust and quantile regression problems, we investigate the stochastic gradient descent (SGD) algorithm for minimizing an objective function $f$ that is locally strongly convex with a sub--quadratic tail. This setting covers many widely used online statistical methods. We introduce a novel piecewise Lyapunov function that enables us to handle functions $f$ with only first-order differentiability, which includes a wide range of popular loss functions such as Huber loss. Leveraging our proposed Lyapunov function, we derive finite-time moment bounds under general diminishing stepsizes, as well as constant stepsizes. We further establish the weak convergence, central limit theorem and bias characterization under constant stepsize, providing the first geometrical convergence result for sub--quadratic SGD. Our results have wide applications, especially in online statistical methods. In particular, we discuss two applications of our results. 1) Online robust regression: We consider a corrupted linear model with sub--exponential covariates and heavy--tailed noise. Our analysis provides convergence rates comparable to those for corrupted models with Gaussian covariates and noise. 2) Online quantile regression: Importantly, our results relax the common assumption in prior work that the conditional density is continuous and provide a more fine-grained analysis for the moment bounds.

Tyler Sam, Yudong Chen, Christina Lee Yu

Many reinforcement learning (RL) algorithms are too costly to use in practice due to the large sizes $S, A$ of the problem's state and action space. To resolve this issue, we study transfer RL with latent low rank structure. We consider the problem of transferring a latent low rank representation when the source and target MDPs have transition kernels with Tucker rank $(S , d, A )$, $(S , S , d), (d, S, A )$, or $(d , d , d )$. In each setting, we introduce the transfer-ability coefficient $α$ that measures the difficulty of representational transfer. Our algorithm learns latent representations in each source MDP and then exploits the linear structure to remove the dependence on $S, A $, or $S A$ in the target MDP regret bound. We complement our positive results with information theoretic lower bounds that show our algorithms (excluding the ($d, d, d$) setting) are minimax-optimal with respect to $α$.

Xumei Xi, Christina Lee Yu, Yudong Chen

Low-rank matrix completion concerns the problem of estimating unobserved entries in a matrix using a sparse set of observed entries. We consider the non-uniform setting where the observed entries are sampled with highly varying probabilities, potentially with different asymptotic scalings. We show that under structured sampling probabilities, it is often better and sometimes optimal to run estimation algorithms on a smaller submatrix rather than the entire matrix. In particular, we prove error upper bounds customized to each entry, which match the minimax lower bounds under certain conditions. Our bounds characterize the hardness of estimating each entry as a function of the localized sampling probabilities. We provide numerical experiments that confirm our theoretical findings.

Siya Qi, Yudong Chen, Runcong Zhao et al.

Hallucination detection is critical for ensuring the reliability of large language models (LLMs) in context-based generation. Prior work has explored intrinsic signals available during generation, among which attention offers a direct view of grounding behavior. However, existing approaches typically rely on coarse summaries that fail to capture fine-grained instabilities in attention. Inspired by signal processing, we introduce a frequency-aware perspective on attention by analyzing its variation during generation. We model attention distributions as discrete signals and extract high-frequency components that reflect rapid local changes in attention. Our analysis reveals that hallucinated tokens are associated with high-frequency attention energy, reflecting fragmented and unstable grounding behavior. Based on this insight, we develop a lightweight hallucination detector using high-frequency attention features. Experiments on the RAGTruth and HalluRAG benchmarks show that our approach achieves performance gains over verification-based, internal-representation-based, and attention-based methods across models and tasks.

Matthew Zurek, Yudong Chen

We study value-iteration (VI) algorithms for solving general (a.k.a. multichain) Markov decision processes (MDPs) under the average-reward criterion, a fundamental but theoretically challenging setting. Beyond the difficulties inherent to all average-reward problems posed by the lack of contractivity and non-uniqueness of solutions to the Bellman operator, in the multichain setting an optimal policy must solve the navigation subproblem of steering towards the best connected component, in addition to optimizing long-run performance within each component. We develop algorithms which better solve this navigational subproblem in order to achieve faster convergence for multichain MDPs, obtaining improved rates of convergence and sharper measures of complexity relative to prior work. Many key components of our results are of potential independent interest, including novel connections between average-reward and discounted problems, optimal fixed-point methods for discounted VI which extend to general Banach spaces, new sublinear convergence rates for the discounted value error, and refined suboptimality decompositions for multichain MDPs. Overall our results yield faster convergence rates for discounted and average-reward problems and expand the theoretical foundations of VI approaches.

Matthew Zurek, Guy Zamir, Yudong Chen

We study offline reinforcement learning in average-reward MDPs, which presents increased challenges from the perspectives of distribution shift and non-uniform coverage, and has been relatively underexamined from a theoretical perspective. While previous work obtains performance guarantees under single-policy data coverage assumptions, such guarantees utilize additional complexity measures which are uniform over all policies, such as the uniform mixing time. We develop sharp guarantees depending only on the target policy, specifically the bias span and a novel policy hitting radius, yielding the first fully single-policy sample complexity bound for average-reward offline RL. We are also the first to handle general weakly communicating MDPs, contrasting restrictive structural assumptions made in prior work. To achieve this, we introduce an algorithm based on pessimistic discounted value iteration enhanced by a novel quantile clipping technique, which enables the use of a sharper empirical-span-based penalty function. Our algorithm also does not require any prior parameter knowledge for its implementation. Remarkably, we show via hard examples that learning under our conditions requires coverage assumptions beyond the stationary distribution of the target policy, distinguishing single-policy complexity measures from previously examined cases. We also develop lower bounds nearly matching our main result.

Jeremy McMahan, Young Wu, Yudong Chen et al.

In this work, we develop a reward design framework for installing a desired behavior as a strict equilibrium across standard solution concepts: dominant strategy equilibrium, Nash equilibrium, correlated equilibrium, and coarse correlated equilibrium. We also extend our framework to capture the Markov-perfect equivalents of each solution concept. Central to our framework is a comprehensive mathematical characterization of strictly installable, based on the desired solution concept and the behavior's structure. These characterizations lead to efficient iterative algorithms, which we generalize to handle optimization objectives through linear programming. Finally, we explore how our results generalize to bounded rational agents.

Yichen Wang, Yudong Chen, Lorenzo Rosasco et al.

Understanding how the test risk scales with model complexity is a central question in machine learning. Classical theory is challenged by the learning curves observed for large over-parametrized deep networks. Capacity measures based on parameter count typically fail to account for these empirical observations. To tackle this challenge, we consider norm-based capacity measures and develop our study for random features based estimators, widely used as simplified theoretical models for more complex networks. In this context, we provide a precise characterization of how the estimator's norm concentrates and how it governs the associated test error. Our results show that the predicted learning curve admits a phase transition from under- to over-parameterization, but no double descent behavior. This confirms that more classical U-shaped behavior is recovered considering appropriate capacity measures based on models norms rather than size. From a technical point of view, we leverage deterministic equivalence as the key tool and further develop new deterministic quantities which are of independent interest.

Jeremy McMahan, Young Wu, Yudong Chen et al.

We study security threats to Markov games due to information asymmetry and misinformation. We consider an attacker player who can spread misinformation about its reward function to influence the robust victim player's behavior. Given a fixed fake reward function, we derive the victim's policy under worst-case rationality and present polynomial-time algorithms to compute the attacker's optimal worst-case policy based on linear programming and backward induction. Then, we provide an efficient inception ("planting an idea in someone's mind") attack algorithm to find the optimal fake reward function within a restricted set of reward functions with dominant strategies. Importantly, our methods exploit the universal assumption of rationality to compute attacks efficiently. Thus, our work exposes a security vulnerability arising from standard game assumptions under misinformation.

Li Ge, Lin Chen, Yudong Chen et al.

In this work, a tensor completion problem is studied, which aims to perfectly recover the tensor from partial observations. The existing theoretical guarantee requires the involved transform to be orthogonal, which hinders its applications. In this paper, jumping out of the constraints of isotropy and self-adjointness, the theoretical guarantee of exact tensor completion with arbitrary linear transforms is established by directly operating the tensors in the transform domain. With the enriched choices of transforms, a new analysis obtained by the proof discloses why slim transforms outperform their square counterparts from a theoretical level. Our model and proof greatly enhance the flexibility of tensor completion and extensive experiments validate the superiority of the proposed method.

Yige Hong, Qiaomin Xie, Yudong Chen et al.

We study the infinite-horizon restless bandit problem with the average reward criterion, in both discrete-time and continuous-time settings. A fundamental goal is to efficiently compute policies that achieve a diminishing optimality gap as the number of arms, $N$, grows large. Existing results on asymptotic optimality all rely on the uniform global attractor property (UGAP), a complex and challenging-to-verify assumption. In this paper, we propose a general, simulation-based framework, Follow-the-Virtual-Advice, that converts any single-armed policy into a policy for the original $N$-armed problem. This is done by simulating the single-armed policy on each arm and carefully steering the real state towards the simulated state. Our framework can be instantiated to produce a policy with an $O(1/\sqrt{N})$ optimality gap. In the discrete-time setting, our result holds under a simpler synchronization assumption, which covers some problem instances that violate UGAP. More notably, in the continuous-time setting, we do not require \emph{any} additional assumptions beyond the standard unichain condition. In both settings, our work is the first asymptotic optimality result that does not require UGAP.

Xumei Xi, Christina Lee Yu, Yudong Chen

We consider offline Reinforcement Learning (RL), where the agent does not interact with the environment and must rely on offline data collected using a behavior policy. Previous works provide policy evaluation guarantees when the target policy to be evaluated is covered by the behavior policy, that is, state-action pairs visited by the target policy must also be visited by the behavior policy. We show that when the MDP has a latent low-rank structure, this coverage condition can be relaxed. Building on the connection to weighted matrix completion with non-uniform observations, we propose an offline policy evaluation algorithm that leverages the low-rank structure to estimate the values of uncovered state-action pairs. Our algorithm does not require a known feature representation, and our finite-sample error bound involves a novel discrepancy measure quantifying the discrepancy between the behavior and target policies in the spectral space. We provide concrete examples where our algorithm achieves accurate estimation while existing coverage conditions are not satisfied. Building on the above evaluation algorithm, we further design an offline policy optimization algorithm and provide non-asymptotic performance guarantees.

Jiazhen Hong, Wei Qian, Yudong Chen et al.

\kmeans clustering is a fundamental problem in many scientific and engineering domains. The optimization problem associated with \kmeans clustering is nonconvex, for which standard algorithms are only guaranteed to find a local optimum. Leveraging the hidden structure of local solutions, we propose a general algorithmic framework for escaping undesirable local solutions and recovering the global solution or the ground truth clustering. This framework consists of iteratively alternating between two steps: (i) detect mis-specified clusters in a local solution, and (ii) improve the local solution by non-local operations. We discuss specific implementation of these steps, and elucidate how the proposed framework unifies many existing variants of \kmeans algorithms through a geometric perspective. We also present two natural variants of the proposed framework, where the initial number of clusters may be over- or under-specified. We provide theoretical justifications and extensive experiments to demonstrate the efficacy of the proposed approach.

Yingjie Fei, Zhuoran Yang, Yudong Chen et al.

We study risk-sensitive reinforcement learning (RL) based on the entropic risk measure. Although existing works have established non-asymptotic regret guarantees for this problem, they leave open an exponential gap between the upper and lower bounds. We identify the deficiencies in existing algorithms and their analysis that result in such a gap. To remedy these deficiencies, we investigate a simple transformation of the risk-sensitive Bellman equations, which we call the exponential Bellman equation. The exponential Bellman equation inspires us to develop a novel analysis of Bellman backup procedures in risk-sensitive RL algorithms, and further motivates the design of a novel exploration mechanism. We show that these analytic and algorithmic innovations together lead to improved regret upper bounds over existing ones.

Lijun Ding, Liwei Jiang, Yudong Chen et al.

We study the robust recovery of a low-rank matrix from sparsely and grossly corrupted Gaussian measurements, with no prior knowledge on the intrinsic rank. We consider the robust matrix factorization approach. We employ a robust $\ell_1$ loss function and deal with the challenge of the unknown rank by using an overspecified factored representation of the matrix variable. We then solve the associated nonconvex nonsmooth problem using a subgradient method with diminishing stepsizes. We show that under a regularity condition on the sensing matrices and corruption, which we call restricted direction preserving property (RDPP), even with rank overspecified, the subgradient method converges to the exact low-rank solution at a sublinear rate. Moreover, our result is more general in the sense that it automatically speeds up to a linear rate once the factor rank matches the unknown rank. On the other hand, we show that the RDPP condition holds under generic settings, such as Gaussian measurements under independent or adversarial sparse corruptions, where the result could be of independent interest. Both the exact recovery and the convergence rate of the proposed subgradient method are numerically verified in the overspecified regime. Moreover, our experiment further shows that our particular design of diminishing stepsize effectively prevents overfitting for robust recovery under overparameterized models, such as robust matrix sensing and learning robust deep image prior. This regularization effect is worth further investigation.

Qinghong Lin, Xiaojun Chen, Qin Zhang et al.

Hashing technology has been widely used in image retrieval due to its computational and storage efficiency. Recently, deep unsupervised hashing methods have attracted increasing attention due to the high cost of human annotations in the real world and the superiority of deep learning technology. However, most deep unsupervised hashing methods usually pre-compute a similarity matrix to model the pairwise relationship in the pre-trained feature space. Then this similarity matrix would be used to guide hash learning, in which most of the data pairs are treated equivalently. The above process is confronted with the following defects: 1) The pre-computed similarity matrix is inalterable and disconnected from the hash learning process, which cannot explore the underlying semantic information. 2) The informative data pairs may be buried by the large number of less-informative data pairs. To solve the aforementioned problems, we propose a Deep Self-Adaptive Hashing (DSAH) model to adaptively capture the semantic information with two special designs: Adaptive Neighbor Discovery (AND) and Pairwise Information Content (PIC). Firstly, we adopt the AND to initially construct a neighborhood-based similarity matrix, and then refine this initial similarity matrix with a novel update strategy to further investigate the semantic structure behind the learned representation. Secondly, we measure the priorities of data pairs with PIC and assign adaptive weights to them, which is relies on the assumption that more dissimilar data pairs contain more discriminative information for hash learning. Extensive experiments on several datasets demonstrate that the above two technologies facilitate the deep hashing model to achieve superior performance.

Fanghui Liu, Xiaolin Huang, Yudong Chen et al.

In this paper, we develop a quadrature framework for large-scale kernel machines via a numerical integration representation. Considering that the integration domain and measure of typical kernels, e.g., Gaussian kernels, arc-cosine kernels, are fully symmetric, we leverage deterministic fully symmetric interpolatory rules to efficiently compute quadrature nodes and associated weights for kernel approximation. The developed interpolatory rules are able to reduce the number of needed nodes while retaining a high approximation accuracy. Further, we randomize the above deterministic rules by the classical Monte-Carlo sampling and control variates techniques with two merits: 1) The proposed stochastic rules make the dimension of the feature mapping flexibly varying, such that we can control the discrepancy between the original and approximate kernels by tuning the dimnension. 2) Our stochastic rules have nice statistical properties of unbiasedness and variance reduction with fast convergence rate. In addition, we elucidate the relationship between our deterministic/stochastic interpolatory rules and current quadrature rules for kernel approximation, including the sparse grids quadrature and stochastic spherical-radial rules, thereby unifying these methods under our framework. Experimental results on several benchmark datasets show that our methods compare favorably with other representative kernel approximation based methods.

Xin Wang, Yudong Chen, Wenwu Zhu

Curriculum learning (CL) is a training strategy that trains a machine learning model from easier data to harder data, which imitates the meaningful learning order in human curricula. As an easy-to-use plug-in, the CL strategy has demonstrated its power in improving the generalization capacity and convergence rate of various models in a wide range of scenarios such as computer vision and natural language processing etc. In this survey article, we comprehensively review CL from various aspects including motivations, definitions, theories, and applications. We discuss works on curriculum learning within a general CL framework, elaborating on how to design a manually predefined curriculum or an automatic curriculum. In particular, we summarize existing CL designs based on the general framework of Difficulty Measurer+Training Scheduler and further categorize the methodologies for automatic CL into four groups, i.e., Self-paced Learning, Transfer Teacher, RL Teacher, and Other Automatic CL. We also analyze principles to select different CL designs that may benefit practical applications. Finally, we present our insights on the relationships connecting CL and other machine learning concepts including transfer learning, meta-learning, continual learning and active learning, etc., then point out challenges in CL as well as potential future research directions deserving further investigations.

Yudong Chen, Dogyoon Song, Xumei Xi et al.

We investigate the landscape of the negative log-likelihood function of Gaussian Mixture Models (GMMs) with a general number of components in the population limit. As the objective function is non-convex, there can be multiple local minima that are not globally optimal, even for well-separated mixture models. Our study reveals that all local minima share a common structure that partially identifies the cluster centers (i.e., means of the Gaussian components) of the true location mixture. Specifically, each local minimum can be represented as a non-overlapping combination of two types of sub-configurations: fitting a single mean estimate to multiple Gaussian components or fitting multiple estimates to a single true component. These results apply to settings where the true mixture components satisfy a certain separation condition, and are valid even when the number of components is over- or under-specified. We also present a more fine-grained analysis for the setting of one-dimensional GMMs with three components, which provide sharper approximation error bounds with improved dependence on the separation.

Lijun Ding, Yuqian Zhang, Yudong Chen

Existing results for low-rank matrix recovery largely focus on quadratic loss, which enjoys favorable properties such as restricted strong convexity/smoothness (RSC/RSM) and well conditioning over all low rank matrices. However, many interesting problems involve more general, non-quadratic losses, which do not satisfy such properties. For these problems, standard nonconvex approaches such as rank-constrained projected gradient descent (a.k.a. iterative hard thresholding) and Burer-Monteiro factorization could have poor empirical performance, and there is no satisfactory theory guaranteeing global and fast convergence for these algorithms. In this paper, we show that a critical component in provable low-rank recovery with non-quadratic loss is a regularity projection oracle. This oracle restricts iterates to low-rank matrices within an appropriate bounded set, over which the loss function is well behaved and satisfies a set of approximate RSC/RSM conditions. Accordingly, we analyze an (averaged) projected gradient method equipped with such an oracle, and prove that it converges globally and linearly. Our results apply to a wide range of non-quadratic low-rank estimation problems including one bit matrix sensing/completion, individualized rank aggregation, and more broadly generalized linear models with rank constraints.