Daily Papers

Test-time adaptation (TTA) addresses distribution shifts for streaming test data in unsupervised settings. Currently, most TTA methods can only deal with minor shifts and rely heavily on heuristic and empirical studies. To advance TTA under domain shifts, we propose the novel problem setting of active test-time adaptation (ATTA) that integrates active learning within the fully TTA setting. We provide a learning theory analysis, demonstrating that incorporating limited labeled test instances enhances overall performances across test domains with a theoretical guarantee. We also present a sample entropy balancing for implementing ATTA while avoiding catastrophic forgetting (CF). We introduce a simple yet effective ATTA algorithm, known as SimATTA, using real-time sample selection techniques. Extensive experimental results confirm consistency with our theoretical analyses and show that the proposed ATTA method yields substantial performance improvements over TTA methods while maintaining efficiency and shares similar effectiveness to the more demanding active domain adaptation (ADA) methods. Our code is available at https://github.com/divelab/ATTA

A variety of Auto-Regressive Video Diffusion Models (ARVDM) have achieved remarkable successes in generating realistic long-form videos. However, theoretical analyses of these models remain scant. In this work, we develop theoretical underpinnings for these models and use our insights to improve the performance of existing models. We first develop Meta-ARVDM, a unified framework of ARVDMs that subsumes most existing methods. Using Meta-ARVDM, we analyze the KL-divergence between the videos generated by Meta-ARVDM and the true videos. Our analysis uncovers two important phenomena inherent to ARVDM -- error accumulation and memory bottleneck. By deriving an information-theoretic impossibility result, we show that the memory bottleneck phenomenon cannot be avoided. To mitigate the memory bottleneck, we design various network structures to explicitly use more past frames. We also achieve a significantly improved trade-off between the mitigation of the memory bottleneck and the inference efficiency by compressing the frames. Experimental results on DMLab and Minecraft validate the efficacy of our methods. Our experiments also demonstrate a Pareto-frontier between the error accumulation and memory bottleneck across different methods.

Recent reinforcement learning (RL) methods have substantially enhanced the planning capabilities of Large Language Models (LLMs), yet the theoretical basis for their effectiveness remains elusive. In this work, we investigate RL's benefits and limitations through a tractable graph-based abstraction, focusing on policy gradient (PG) and Q-learning methods. Our theoretical analyses reveal that supervised fine-tuning (SFT) may introduce co-occurrence-based spurious solutions, whereas RL achieves correct planning primarily through exploration, underscoring exploration's role in enabling better generalization. However, we also show that PG suffers from diversity collapse, where output diversity decreases during training and persists even after perfect accuracy is attained. By contrast, Q-learning provides two key advantages: off-policy learning and diversity preservation at convergence. We further demonstrate that careful reward design is necessary to prevent reward hacking in Q-learning. Finally, applying our framework to the real-world planning benchmark Blocksworld, we confirm that these behaviors manifest in practice.

Hallucination in Large Language Models (LLMs) refers to the generation of content that is not faithful to the input or the real-world facts. This paper provides a rigorous treatment of hallucination in LLMs, including formal definitions and theoretical analyses. We distinguish between intrinsic and extrinsic hallucinations, and define a hallucination risk for models. We derive bounds on this risk using learning-theoretic frameworks (PAC-Bayes and Rademacher complexity). We then survey detection strategies for hallucinations, such as token-level uncertainty estimation, confidence calibration, and attention alignment checks. On the mitigation side, we discuss approaches including retrieval-augmented generation, hallucination-aware fine-tuning, logit calibration, and the incorporation of fact-verification modules. We propose a unified detection and mitigation workflow, illustrated with a diagram, to integrate these strategies. Finally, we outline evaluation protocols for hallucination, recommending datasets, metrics, and experimental setups to quantify and reduce hallucinations. Our work lays a theoretical foundation and practical guidelines for addressing the crucial challenge of hallucination in LLMs.

Federated Learning (FL) aims to protect data privacy by enabling clients to collectively train machine learning models without sharing their raw data. However, recent studies demonstrate that information exchanged during FL is subject to Gradient Inversion Attacks (GIA) and, consequently, a variety of privacy-preserving methods have been integrated into FL to thwart such attacks, such as Secure Multi-party Computing (SMC), Homomorphic Encryption (HE), and Differential Privacy (DP). Despite their ability to protect data privacy, these approaches inherently involve substantial privacy-utility trade-offs. By revisiting the key to privacy exposure in FL under GIA, which lies in the frequent sharing of model gradients that contain private data, we take a new perspective by designing a novel privacy preserve FL framework that effectively ``breaks the direct connection'' between the shared parameters and the local private data to defend against GIA. Specifically, we propose a Hypernetwork Federated Learning (HyperFL) framework that utilizes hypernetworks to generate the parameters of the local model and only the hypernetwork parameters are uploaded to the server for aggregation. Theoretical analyses demonstrate the convergence rate of the proposed HyperFL, while extensive experimental results show the privacy-preserving capability and comparable performance of HyperFL. Code is available at https://github.com/Pengxin-Guo/HyperFL.

Machine unlearning aims to remove knowledge of the specific training data in a well-trained model. Currently, machine unlearning methods typically handle all forgetting data in a single batch, removing the corresponding knowledge all at once upon request. However, in practical scenarios, requests for data removal often arise in a streaming manner rather than in a single batch, leading to reduced efficiency and effectiveness in existing methods. Such challenges of streaming forgetting have not been the focus of much research. In this paper, to address the challenges of performance maintenance, efficiency, and data access brought about by streaming unlearning requests, we introduce a streaming unlearning paradigm, formalizing the unlearning as a distribution shift problem. We then estimate the altered distribution and propose a novel streaming unlearning algorithm to achieve efficient streaming forgetting without requiring access to the original training data. Theoretical analyses confirm an O(T + V_T) error bound on the streaming unlearning regret, where V_T represents the cumulative total variation in the optimal solution over T learning rounds. This theoretical guarantee is achieved under mild conditions without the strong restriction of convex loss function. Experiments across various models and datasets validate the performance of our proposed method.

Chain-of-thought (CoT) reasoning enables large language models (LLMs) to break down complex problems into interpretable intermediate steps, significantly enhancing model transparency and performance in reasoning tasks. However, conventional CoT methods rely on heuristic sampling without structured modeling of reasoning transitions, constraining their ability to systematically explore and discover diverse and effective reasoning trajectories. In this work, we introduce CTRLS, a framework that formulates CoT reasoning as a Markov decision process (MDP) with latent state transitions, enabling principled and state-aware exploration via distributional reinforcement learning. By modelling reasoning actions as explicit probability distributions in latent space, our approach explicitly models epistemic uncertainty, facilitating robust exploration of the reasoning space. As part of our framework, we introduce an on-policy reinforcement learning strategy incorporating epsilon-greedy exploration and entropy-based regularization to iteratively refine latent state transitions without requiring additional fine-tuning of the underlying LLM. Theoretical analyses provide evidence lower bounds (ELBO), theoretically grounding our transition-aware modeling of latent reasoning dynamics. Further experiments demonstrate improvements in reasoning accuracy, diversity, and exploration efficiency across benchmark reasoning tasks.

Unsupervised Multiplex Graph Learning (UMGL) aims to learn node representations on various edge types without manual labeling. However, existing research overlooks a key factor: the reliability of the graph structure. Real-world data often exhibit a complex nature and contain abundant task-irrelevant noise, severely compromising UMGL's performance. Moreover, existing methods primarily rely on contrastive learning to maximize mutual information across different graphs, limiting them to multiplex graph redundant scenarios and failing to capture view-unique task-relevant information. In this paper, we focus on a more realistic and challenging task: to unsupervisedly learn a fused graph from multiple graphs that preserve sufficient task-relevant information while removing task-irrelevant noise. Specifically, our proposed Information-aware Unsupervised Multiplex Graph Fusion framework (InfoMGF) uses graph structure refinement to eliminate irrelevant noise and simultaneously maximizes view-shared and view-unique task-relevant information, thereby tackling the frontier of non-redundant multiplex graph. Theoretical analyses further guarantee the effectiveness of InfoMGF. Comprehensive experiments against various baselines on different downstream tasks demonstrate its superior performance and robustness. Surprisingly, our unsupervised method even beats the sophisticated supervised approaches. The source code and datasets are available at https://github.com/zxlearningdeep/InfoMGF.

Recently, large language models (LLMs) have demonstrated excellent performance in understanding human instructions and generating code, which has inspired researchers to explore the feasibility of generating RTL code with LLMs. However, the existing approaches to fine-tune LLMs on RTL codes typically are conducted on fixed datasets, which do not fully stimulate the capability of LLMs and require large amounts of reference data. To mitigate these issues , we introduce a simple yet effective iterative training paradigm named ITERTL. During each iteration, samples are drawn from the model trained in the previous cycle. Then these new samples are employed for training in this loop. Through this iterative approach, the distribution mismatch between the model and the training samples is reduced. Additionally, the model is thus enabled to explore a broader generative space and receive more comprehensive feedback. Theoretical analyses are conducted to investigate the mechanism of the effectiveness. Experimental results show the model trained through our proposed approach can compete with and even outperform the state-of-the-art (SOTA) open-source model with nearly 37\% reference samples, achieving remarkable 42.9\% and 62.2\% pass@1 rate on two VerilogEval evaluation datasets respectively. While using the same amount of reference samples, our method can achieved a relative improvement of 16.9\% and 12.5\% in pass@1 compared to the non-iterative method. This study facilitates the application of LLMs for generating RTL code in practical scenarios with limited data.

Aligning foundation models is essential for their safe and trustworthy deployment. However, traditional fine-tuning methods are computationally intensive and require updating billions of model parameters. A promising alternative, alignment via decoding, adjusts the response distribution directly without model updates to maximize a target reward r, thus providing a lightweight and adaptable framework for alignment. However, principled decoding methods rely on oracle access to an optimal Q-function (Q^*), which is often unavailable in practice. Hence, prior SoTA methods either approximate this Q^* using Q^{pi_{sft}} (derived from the reference SFT model) or rely on short-term rewards, resulting in sub-optimal decoding performance. In this work, we propose Transfer Q^*, which implicitly estimates the optimal value function for a target reward r through a baseline model rho_{BL} aligned with a baseline reward rho_{BL} (which can be different from the target reward r). Theoretical analyses of Transfer Q^* provide a rigorous characterization of its optimality, deriving an upper bound on the sub-optimality gap and identifying a hyperparameter to control the deviation from the pre-trained reference SFT model based on user needs. Our approach significantly reduces the sub-optimality gap observed in prior SoTA methods and demonstrates superior empirical performance across key metrics such as coherence, diversity, and quality in extensive tests on several synthetic and real datasets.

To obtain a near-optimal policy with fewer interactions in Reinforcement Learning (RL), a promising approach involves the combination of offline RL, which enhances sample efficiency by leveraging offline datasets, and online RL, which explores informative transitions by interacting with the environment. Offline-to-Online (O2O) RL provides a paradigm for improving an offline trained agent within limited online interactions. However, due to the significant distribution shift between online experiences and offline data, most offline RL algorithms suffer from performance drops and fail to achieve stable policy improvement in O2O adaptation. To address this problem, we propose the Robust Offline-to-Online (RO2O) algorithm, designed to enhance offline policies through uncertainty and smoothness, and to mitigate the performance drop in online adaptation. Specifically, RO2O incorporates Q-ensemble for uncertainty penalty and adversarial samples for policy and value smoothness, which enable RO2O to maintain a consistent learning procedure in online adaptation without requiring special changes to the learning objective. Theoretical analyses in linear MDPs demonstrate that the uncertainty and smoothness lead to a tighter optimality bound in O2O against distribution shift. Experimental results illustrate the superiority of RO2O in facilitating stable offline-to-online learning and achieving significant improvement with limited online interactions.

Existing theoretical studies on offline reinforcement learning (RL) mostly consider a dataset sampled directly from the target task. In practice, however, data often come from several heterogeneous but related sources. Motivated by this gap, this work aims at rigorously understanding offline RL with multiple datasets that are collected from randomly perturbed versions of the target task instead of from itself. An information-theoretic lower bound is derived, which reveals a necessary requirement on the number of involved sources in addition to that on the number of data samples. Then, a novel HetPEVI algorithm is proposed, which simultaneously considers the sample uncertainties from a finite number of data samples per data source and the source uncertainties due to a finite number of available data sources. Theoretical analyses demonstrate that HetPEVI can solve the target task as long as the data sources collectively provide a good data coverage. Moreover, HetPEVI is demonstrated to be optimal up to a polynomial factor of the horizon length. Finally, the study is extended to offline Markov games and offline robust RL, which demonstrates the generality of the proposed designs and theoretical analyses.

Recent advances in Audio-Language Models (ALMs) have significantly improved multimodal understanding capabilities. However, the introduction of the audio modality also brings new and unique vulnerability vectors. Previous studies have proposed jailbreak attacks that specifically target ALMs, revealing that defenses directly transferred from traditional audio adversarial attacks or text-based Large Language Model (LLM) jailbreaks are largely ineffective against these ALM-specific threats. To address this issue, we propose ALMGuard, the first defense framework tailored to ALMs. Based on the assumption that safety-aligned shortcuts naturally exist in ALMs, we design a method to identify universal Shortcut Activation Perturbations (SAPs) that serve as triggers that activate the safety shortcuts to safeguard ALMs at inference time. To better sift out effective triggers while preserving the model's utility on benign tasks, we further propose Mel-Gradient Sparse Mask (M-GSM), which restricts perturbations to Mel-frequency bins that are sensitive to jailbreaks but insensitive to speech understanding. Both theoretical analyses and empirical results demonstrate the robustness of our method against both seen and unseen attacks. Overall, \MethodName reduces the average success rate of advanced ALM-specific jailbreak attacks to 4.6% across four models, while maintaining comparable utility on benign benchmarks, establishing it as the new state of the art. Our code and data are available at https://github.com/WeifeiJin/ALMGuard.

Loss spikes often occur during pre-training of large language models. The spikes degrade the performance of large language models and sometimes ruin the pre-training. Since the pre-training needs a vast computational budget, we should avoid such spikes. To investigate the cause of loss spikes, we focus on gradients of internal layers. Through theoretical analyses, we reveal two causes of the exploding gradients, and provide requirements to prevent the explosion. In addition, we propose a method to satisfy the requirements by combining the initialization method and a simple modification to embeddings. We conduct various experiments to verify our theoretical analyses empirically. Experimental results indicate that the combination is effective in preventing spikes during pre-training.

Inspired by human conscious planning, we propose Skipper, a model-based reinforcement learning framework utilizing spatio-temporal abstractions to generalize better in novel situations. It automatically decomposes the given task into smaller, more manageable subtasks, and thus enables sparse decision-making and focused computation on the relevant parts of the environment. The decomposition relies on the extraction of an abstracted proxy problem represented as a directed graph, in which vertices and edges are learned end-to-end from hindsight. Our theoretical analyses provide performance guarantees under appropriate assumptions and establish where our approach is expected to be helpful. Generalization-focused experiments validate Skipper's significant advantage in zero-shot generalization, compared to some existing state-of-the-art hierarchical planning methods.

Evaluating fairness can be challenging in practice because the sensitive attributes of data are often inaccessible due to privacy constraints. The go-to approach that the industry frequently adopts is using off-the-shelf proxy models to predict the missing sensitive attributes, e.g. Meta [Alao et al., 2021] and Twitter [Belli et al., 2022]. Despite its popularity, there are three important questions unanswered: (1) Is directly using proxies efficacious in measuring fairness? (2) If not, is it possible to accurately evaluate fairness using proxies only? (3) Given the ethical controversy over inferring user private information, is it possible to only use weak (i.e. inaccurate) proxies in order to protect privacy? Our theoretical analyses show that directly using proxy models can give a false sense of (un)fairness. Second, we develop an algorithm that is able to measure fairness (provably) accurately with only three properly identified proxies. Third, we show that our algorithm allows the use of only weak proxies (e.g. with only 68.85%accuracy on COMPAS), adding an extra layer of protection on user privacy. Experiments validate our theoretical analyses and show our algorithm can effectively measure and mitigate bias. Our results imply a set of practical guidelines for practitioners on how to use proxies properly. Code is available at github.com/UCSC-REAL/fair-eval.

Multi-agent systems (MAS) extend large language models (LLMs) from independent single-model reasoning to coordinative system-level intelligence. While existing LLM agents depend on text-based mediation for reasoning and communication, we take a step forward by enabling models to collaborate directly within the continuous latent space. We introduce LatentMAS, an end-to-end training-free framework that enables pure latent collaboration among LLM agents. In LatentMAS, each agent first performs auto-regressive latent thoughts generation through last-layer hidden embeddings. A shared latent working memory then preserves and transfers each agent's internal representations, ensuring lossless information exchange. We provide theoretical analyses establishing that LatentMAS attains higher expressiveness and lossless information preservation with substantially lower complexity than vanilla text-based MAS. In addition, empirical evaluations across 9 comprehensive benchmarks spanning math and science reasoning, commonsense understanding, and code generation show that LatentMAS consistently outperforms strong single-model and text-based MAS baselines, achieving up to 14.6% higher accuracy, reducing output token usage by 70.8%-83.7%, and providing 4x-4.3x faster end-to-end inference. These results demonstrate that our new latent collaboration framework enhances system-level reasoning quality while offering substantial efficiency gains without any additional training. Code and data are fully open-sourced at https://github.com/Gen-Verse/LatentMAS.

Large Language Models (LLMs) have become indispensable in numerous real-world applications. Unfortunately, fine-tuning these models at scale, especially in federated settings where data privacy and communication efficiency are critical, presents significant challenges. Existing methods often resort to parameter-efficient fine-tuning (PEFT) to mitigate communication overhead, but this typically comes at the cost of model accuracy. To address these limitations, we propose federated full-parameter tuning at scale for LLMs (Ferret), the first first-order method with shared randomness to enable scalable full-parameter tuning of LLMs across decentralized data sources while maintaining competitive model accuracy. Ferret accomplishes this through three aspects: (1) it employs widely applied first-order methods for efficient local updates; (2) it projects these updates into a low-dimensional space to considerably reduce communication overhead; and (3) it reconstructs local updates from this low-dimensional space with shared randomness to facilitate effective full-parameter global aggregation, ensuring fast convergence and competitive final performance. Our rigorous theoretical analyses and insights along with extensive experiments, show that Ferret significantly enhances the scalability of existing federated full-parameter tuning approaches by achieving high computational efficiency, reduced communication overhead, and fast convergence, all while maintaining competitive model accuracy. Our implementation is available at https://github.com/allen4747/Ferret.

Batch Normalization is a key component in almost all state-of-the-art image classifiers, but it also introduces practical challenges: it breaks the independence between training examples within a batch, can incur compute and memory overhead, and often results in unexpected bugs. Building on recent theoretical analyses of deep ResNets at initialization, we propose a simple set of analysis tools to characterize signal propagation on the forward pass, and leverage these tools to design highly performant ResNets without activation normalization layers. Crucial to our success is an adapted version of the recently proposed Weight Standardization. Our analysis tools show how this technique preserves the signal in networks with ReLU or Swish activation functions by ensuring that the per-channel activation means do not grow with depth. Across a range of FLOP budgets, our networks attain performance competitive with the state-of-the-art EfficientNets on ImageNet.

In theoretical neuroscience, recent work leverages deep learning tools to explore how some network attributes critically influence its learning dynamics. Notably, initial weight distributions with small (resp. large) variance may yield a rich (resp. lazy) regime, where significant (resp. minor) changes to network states and representation are observed over the course of learning. However, in biology, neural circuit connectivity could exhibit a low-rank structure and therefore differs markedly from the random initializations generally used for these studies. As such, here we investigate how the structure of the initial weights -- in particular their effective rank -- influences the network learning regime. Through both empirical and theoretical analyses, we discover that high-rank initializations typically yield smaller network changes indicative of lazier learning, a finding we also confirm with experimentally-driven initial connectivity in recurrent neural networks. Conversely, low-rank initialization biases learning towards richer learning. Importantly, however, as an exception to this rule, we find lazier learning can still occur with a low-rank initialization that aligns with task and data statistics. Our research highlights the pivotal role of initial weight structures in shaping learning regimes, with implications for metabolic costs of plasticity and risks of catastrophic forgetting.

A Membership Inference Attack (MIA) assesses how much a trained machine learning model reveals about its training data by determining whether specific query instances were included in the dataset. We classify existing MIAs into adaptive or non-adaptive, depending on whether the adversary is allowed to train shadow models on membership queries. In the adaptive setting, where the adversary can train shadow models after accessing query instances, we highlight the importance of exploiting membership dependencies between instances and propose an attack-agnostic framework called Cascading Membership Inference Attack (CMIA), which incorporates membership dependencies via conditional shadow training to boost membership inference performance. In the non-adaptive setting, where the adversary is restricted to training shadow models before obtaining membership queries, we introduce Proxy Membership Inference Attack (PMIA). PMIA employs a proxy selection strategy that identifies samples with similar behaviors to the query instance and uses their behaviors in shadow models to perform a membership posterior odds test for membership inference. We provide theoretical analyses for both attacks, and extensive experimental results demonstrate that CMIA and PMIA substantially outperform existing MIAs in both settings, particularly in the low false-positive regime, which is crucial for evaluating privacy risks.

We present Prover Agent, a novel AI agent for automated theorem proving that integrates large language models (LLMs) with a formal proof assistant, Lean. Prover Agent coordinates an informal reasoning LLM, a formal prover model, and feedback from Lean while also generating auxiliary lemmas. These auxiliary lemmas are not limited to subgoals in the formal proof but can also include special cases or potentially useful facts derived from the assumptions, which help in discovering a viable proof strategy. It achieves an 88.1% success rate on the MiniF2F benchmark, establishing a new state-of-the-art among methods using small language models (SLMs) with a much lower sample budget than previous approaches. We also present theoretical analyses and case studies that illustrate how these generated lemmas contribute to solving challenging problems. Our code is publicly available at: https://github.com/kAIto47802/Prover-Agent.

In this work, we explore the mechanism of in-context learning (ICL) on out-of-distribution (OOD) tasks that were not encountered during training. To achieve this, we conduct synthetic experiments where the objective is to learn OOD mathematical functions through ICL using a GPT-2 model. We reveal that Transformers may struggle to learn OOD task functions through ICL. Specifically, ICL performance resembles implementing a function within the pretraining hypothesis space and optimizing it with gradient descent based on the in-context examples. Additionally, we investigate ICL's well-documented ability to learn unseen abstract labels in context. We demonstrate that such ability only manifests in the scenarios without distributional shifts and, therefore, may not serve as evidence of new-task-learning ability. Furthermore, we assess ICL's performance on OOD tasks when the model is pretrained on multiple tasks. Both empirical and theoretical analyses demonstrate the existence of the low-test-error preference of ICL, where it tends to implement the pretraining function that yields low test error in the testing context. We validate this through numerical experiments. This new theoretical result, combined with our empirical findings, elucidates the mechanism of ICL in addressing OOD tasks.

In this study, we investigate the DIstribution Correction Estimation (DICE) methods, an important line of work in offline reinforcement learning (RL) and imitation learning (IL). DICE-based methods impose state-action-level behavior constraint, which is an ideal choice for offline learning. However, they typically perform much worse than current state-of-the-art (SOTA) methods that solely use action-level behavior constraint. After revisiting DICE-based methods, we find there exist two gradient terms when learning the value function using true-gradient update: forward gradient (taken on the current state) and backward gradient (taken on the next state). Using forward gradient bears a large similarity to many offline RL methods, and thus can be regarded as applying action-level constraint. However, directly adding the backward gradient may degenerate or cancel out its effect if these two gradients have conflicting directions. To resolve this issue, we propose a simple yet effective modification that projects the backward gradient onto the normal plane of the forward gradient, resulting in an orthogonal-gradient update, a new learning rule for DICE-based methods. We conduct thorough theoretical analyses and find that the projected backward gradient brings state-level behavior regularization, which reveals the mystery of DICE-based methods: the value learning objective does try to impose state-action-level constraint, but needs to be used in a corrected way. Through toy examples and extensive experiments on complex offline RL and IL tasks, we demonstrate that DICE-based methods using orthogonal-gradient updates (O-DICE) achieve SOTA performance and great robustness.

Domain generalization (DG) seeks to learn robust models that generalize well under unknown distribution shifts. As a critical aspect of DG, optimizer selection has not been explored in depth. Currently, most DG methods follow the widely used benchmark, DomainBed, and utilize Adam as the default optimizer for all datasets. However, we reveal that Adam is not necessarily the optimal choice for the majority of current DG methods and datasets. Based on the perspective of loss landscape flatness, we propose a novel approach, Flatness-Aware Minimization for Domain Generalization (FAD), which can efficiently optimize both zeroth-order and first-order flatness simultaneously for DG. We provide theoretical analyses of the FAD's out-of-distribution (OOD) generalization error and convergence. Our experimental results demonstrate the superiority of FAD on various DG datasets. Additionally, we confirm that FAD is capable of discovering flatter optima in comparison to other zeroth-order and first-order flatness-aware optimization methods.

Recent advancements in diffusion models have revolutionized generative modeling. However, the impressive and vivid outputs they produce often come at the cost of significant model scaling and increased computational demands. Consequently, building personalized diffusion models based on off-the-shelf models has emerged as an appealing alternative. In this paper, we introduce a novel perspective on conditional generation for transferring a pre-trained model. From this viewpoint, we propose *Domain Guidance*, a straightforward transfer approach that leverages pre-trained knowledge to guide the sampling process toward the target domain. Domain Guidance shares a formulation similar to advanced classifier-free guidance, facilitating better domain alignment and higher-quality generations. We provide both empirical and theoretical analyses of the mechanisms behind Domain Guidance. Our experimental results demonstrate its substantial effectiveness across various transfer benchmarks, achieving over a 19.6% improvement in FID and a 23.4% improvement in FD_DINOv2 compared to standard fine-tuning. Notably, existing fine-tuned models can seamlessly integrate Domain Guidance to leverage these benefits, without additional training.

Expert parallelism has been introduced as a strategy to distribute the computational workload of sparsely-gated mixture-of-experts (MoE) models across multiple computing devices, facilitating the execution of these increasingly large-scale models. However, the All-to-All communication intrinsic to expert parallelism constitutes a significant overhead, diminishing the MoE models' efficiency. Current optimization approaches offer some relief, yet they are constrained by the sequential interdependence of communication and computation operations. To address this limitation, we present a novel shortcut-connected MoE architecture with overlapping parallel strategy, designated as ScMoE, which effectively decouples communication from its conventional sequence, allowing for a substantial overlap of 70% to 100% with computation. When compared with the prevalent top-2 MoE architecture, ScMoE demonstrates training speed improvements of 30% and 11%, and inference improvements of 40% and 15%, in our PCIe and NVLink hardware environments, respectively, where communication constitutes 60% and 15% of the total MoE time consumption. On the other hand, extensive experiments and theoretical analyses indicate that ScMoE not only achieves comparable but in some instances surpasses the model quality of existing approaches in vision and language tasks.

Understanding the performance of machine learning (ML) models across diverse data distributions is critically important for reliable applications. Despite recent empirical studies positing a near-perfect linear correlation between in-distribution (ID) and out-of-distribution (OOD) accuracies, we empirically demonstrate that this correlation is more nuanced under subpopulation shifts. Through rigorous experimentation and analysis across a variety of datasets, models, and training epochs, we demonstrate that OOD performance often has a nonlinear correlation with ID performance in subpopulation shifts. Our findings, which contrast previous studies that have posited a linear correlation in model performance during distribution shifts, reveal a "moon shape" correlation (parabolic uptrend curve) between the test performance on the majority subpopulation and the minority subpopulation. This non-trivial nonlinear correlation holds across model architectures, hyperparameters, training durations, and the imbalance between subpopulations. Furthermore, we found that the nonlinearity of this "moon shape" is causally influenced by the degree of spurious correlations in the training data. Our controlled experiments show that stronger spurious correlation in the training data creates more nonlinear performance correlation. We provide complementary experimental and theoretical analyses for this phenomenon, and discuss its implications for ML reliability and fairness. Our work highlights the importance of understanding the nonlinear effects of model improvement on performance in different subpopulations, and has the potential to inform the development of more equitable and responsible machine learning models.

DeepSeek-R1 has successfully enhanced Large Language Model (LLM) reasoning capabilities through its rule-based reward system. While it's a ''perfect'' reward system that effectively mitigates reward hacking, such reward functions are often discrete. Our experimental observations suggest that discrete rewards can lead to gradient anomaly, unstable optimization, and slow convergence. To address this issue, we propose ReDit (Reward Dithering), a method that dithers the discrete reward signal by adding simple random noise. With this perturbed reward, exploratory gradients are continuously provided throughout the learning process, enabling smoother gradient updates and accelerating convergence. The injected noise also introduces stochasticity into flat reward regions, encouraging the model to explore novel policies and escape local optima. Experiments across diverse tasks demonstrate the effectiveness and efficiency of ReDit. On average, ReDit achieves performance comparable to vanilla GRPO with only approximately 10% the training steps, and furthermore, still exhibits a 4% performance improvement over vanilla GRPO when trained for a similar duration. Visualizations confirm significant mitigation of gradient issues with ReDit. Moreover, theoretical analyses are provided to further validate these advantages.

Interpretability is essential for machine learning models to be trusted and deployed in critical domains. However, existing methods for interpreting text models are often complex, lack mathematical foundations, and their performance is not guaranteed. In this paper, we propose FRED (Faithful and Robust Explainer for textual Documents), a novel method for interpreting predictions over text. FRED offers three key insights to explain a model prediction: (1) it identifies the minimal set of words in a document whose removal has the strongest influence on the prediction, (2) it assigns an importance score to each token, reflecting its influence on the model's output, and (3) it provides counterfactual explanations by generating examples similar to the original document, but leading to a different prediction. We establish the reliability of FRED through formal definitions and theoretical analyses on interpretable classifiers. Additionally, our empirical evaluation against state-of-the-art methods demonstrates the effectiveness of FRED in providing insights into text models.

Recent research has revealed that high compression of Deep Neural Networks (DNNs), e.g., massive pruning of the weight matrix of a DNN, leads to a severe drop in accuracy and susceptibility to adversarial attacks. Integration of network pruning into an adversarial training framework has been proposed to promote adversarial robustness. It has been observed that a highly pruned weight matrix tends to be ill-conditioned, i.e., increasing the condition number of the weight matrix. This phenomenon aggravates the vulnerability of a DNN to input noise. Although a highly pruned weight matrix is considered to be able to lower the upper bound of the local Lipschitz constant to tolerate large distortion, the ill-conditionedness of such a weight matrix results in a non-robust DNN model. To overcome this challenge, this work develops novel joint constraints to adjust the weight distribution of networks, namely, the Transformed Sparse Constraint joint with Condition Number Constraint (TSCNC), which copes with smoothing distribution and differentiable constraint functions to reduce condition number and thus avoid the ill-conditionedness of weight matrices. Furthermore, our theoretical analyses unveil the relevance between the condition number and the local Lipschitz constant of the weight matrix, namely, the sharply increasing condition number becomes the dominant factor that restricts the robustness of over-sparsified models. Extensive experiments are conducted on several public datasets, and the results show that the proposed constraints significantly improve the robustness of a DNN with high pruning rates.

In the presence of noisy labels, designing robust loss functions is critical for securing the generalization performance of deep neural networks. Cross Entropy (CE) loss has been shown to be not robust to noisy labels due to its unboundedness. To alleviate this issue, existing works typically design specialized robust losses with the symmetric condition, which usually lead to the underfitting issue. In this paper, our key idea is to induce a loss bound at the logit level, thus universally enhancing the noise robustness of existing losses. Specifically, we propose logit clipping (LogitClip), which clamps the norm of the logit vector to ensure that it is upper bounded by a constant. In this manner, CE loss equipped with our LogitClip method is effectively bounded, mitigating the overfitting to examples with noisy labels. Moreover, we present theoretical analyses to certify the noise-tolerant ability of LogitClip. Extensive experiments show that LogitClip not only significantly improves the noise robustness of CE loss, but also broadly enhances the generalization performance of popular robust losses.

A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most prior theoretical analyses have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as nonlinear elliptic variational PDEs, whose solutions minimize an Euler-Lagrange energy functional E(u) = int_Omega L(x, u(x), nabla u(x)) - f(x) u(x)dx. We show that if composing a function with Barron norm b with partial derivatives of L produces a function of Barron norm at most B_L b^p, the solution to the PDE can be epsilon-approximated in the L^2 sense by a function with Barron norm Oleft(left(dB_Lright)^{max{p log(1/ epsilon), p^{log(1/epsilon)}}}right). By a classical result due to Barron [1993], this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating p, epsilon, B_L as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs over a unit hypercube.

Multi-Task Learning (MTL) is a learning paradigm in machine learning and its aim is to leverage useful information contained in multiple related tasks to help improve the generalization performance of all the tasks. In this paper, we give a survey for MTL from the perspective of algorithmic modeling, applications and theoretical analyses. For algorithmic modeling, we give a definition of MTL and then classify different MTL algorithms into five categories, including feature learning approach, low-rank approach, task clustering approach, task relation learning approach and decomposition approach as well as discussing the characteristics of each approach. In order to improve the performance of learning tasks further, MTL can be combined with other learning paradigms including semi-supervised learning, active learning, unsupervised learning, reinforcement learning, multi-view learning and graphical models. When the number of tasks is large or the data dimensionality is high, we review online, parallel and distributed MTL models as well as dimensionality reduction and feature hashing to reveal their computational and storage advantages. Many real-world applications use MTL to boost their performance and we review representative works in this paper. Finally, we present theoretical analyses and discuss several future directions for MTL.

The Muon optimizer is consistently faster than Adam in training Large Language Models (LLMs), yet the mechanism underlying its success remains unclear. This paper demystifies this mechanism through the lens of associative memory. By ablating the transformer components optimized by Muon, we reveal that the associative memory parameters of LLMs, namely the Value and Output (VO) attention weights and Feed-Forward Networks (FFNs), are the primary contributors to Muon's superiority. Motivated by this associative memory view, we then explain Muon's superiority on real-world corpora, which are intrinsically heavy-tailed: a few classes (tail classes) appear far less frequently than others. The superiority is explained through two key properties: (i) its update rule consistently yields a more isotropic singular spectrum than Adam; and as a result, (ii) on heavy-tailed data, it optimizes tail classes more effectively than Adam. Beyond empirical evidence, we theoretically confirm these findings by analyzing a one-layer associative memory model under class-imbalanced data. We prove that Muon consistently achieves balanced learning across classes regardless of feature embeddings, whereas Adam can induce large disparities in learning errors depending on embedding properties. In summary, our empirical observations and theoretical analyses reveal Muon's core advantage: its update rule aligns with the outer-product structure of linear associative memories, enabling more balanced and effective learning of tail classes in heavy-tailed distributions than Adam.

Large Language Models (LLMs) have revolutionized natural language processing and hold immense potential for advancing Artificial Intelligence. However, the core architecture of most mainstream LLMs -- the Transformer -- has inherent limitations in computational depth, rendering them theoretically incapable of solving many reasoning tasks that demand increasingly deep computations. Chain of Thought (CoT) prompting has emerged as a technique to address these architectural limitations, as evidenced by several theoretical studies. It offers a promising approach to solving complex reasoning tasks that were previously beyond the capabilities of these models. Despite its successes, CoT and its variants (such as Tree of Thought, Graph of Thought, etc.) rely on a "one-prompt-for-all" approach, using a single prompt structure (e.g., "think step by step") for a wide range of tasks -- from counting and sorting to solving mathematical and algorithmic problems. This approach poses significant challenges for models to generate the correct reasoning steps, as the model must navigate through a vast prompt template space to find the appropriate template for each task. In this work, we build upon previous theoretical analyses of CoT to demonstrate how the one-prompt-for-all approach can negatively affect the computability of LLMs. We partition the solution search space into two: the prompt space and the answer space. Our findings show that task-specific supervision is essential for navigating the prompt space accurately and achieving optimal performance. Through experiments with state-of-the-art LLMs, we reveal a gap in reasoning performance when supervision is applied versus when it is not.

Offline reinforcement learning (RL) presents a promising approach for learning reinforced policies from offline datasets without the need for costly or unsafe interactions with the environment. However, datasets collected by humans in real-world environments are often noisy and may even be maliciously corrupted, which can significantly degrade the performance of offline RL. In this work, we first investigate the performance of current offline RL algorithms under comprehensive data corruption, including states, actions, rewards, and dynamics. Our extensive experiments reveal that implicit Q-learning (IQL) demonstrates remarkable resilience to data corruption among various offline RL algorithms. Furthermore, we conduct both empirical and theoretical analyses to understand IQL's robust performance, identifying its supervised policy learning scheme as the key factor. Despite its relative robustness, IQL still suffers from heavy-tail targets of Q functions under dynamics corruption. To tackle this challenge, we draw inspiration from robust statistics to employ the Huber loss to handle the heavy-tailedness and utilize quantile estimators to balance penalization for corrupted data and learning stability. By incorporating these simple yet effective modifications into IQL, we propose a more robust offline RL approach named Robust IQL (RIQL). Extensive experiments demonstrate that RIQL exhibits highly robust performance when subjected to diverse data corruption scenarios.

Modern deep neural networks have achieved impressive performance on tasks from image classification to natural language processing. Surprisingly, these complex systems with massive amounts of parameters exhibit the same structural properties in their last-layer features and classifiers across canonical datasets when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means, and those class-means are the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is known as Neural Collapse (NC). Recent papers have theoretically shown that NC emerges in the global minimizers of training problems with the simplified "unconstrained feature model". In this context, we take a step further and prove the NC occurrences in deep linear networks for the popular mean squared error (MSE) and cross entropy (CE) losses, showing that global solutions exhibit NC properties across the linear layers. Furthermore, we extend our study to imbalanced data for MSE loss and present the first geometric analysis of NC under bias-free setting. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of orthogonal vectors, whose lengths depend on the amount of data in their corresponding classes. Finally, we empirically validate our theoretical analyses on synthetic and practical network architectures with both balanced and imbalanced scenarios.

State-of-the-art results on image recognition tasks are achieved using over-parameterized learning algorithms that (nearly) perfectly fit the training set and are known to fit well even random labels. This tendency to memorize the labels of the training data is not explained by existing theoretical analyses. Memorization of the training data also presents significant privacy risks when the training data contains sensitive personal information and thus it is important to understand whether such memorization is necessary for accurate learning. We provide the first conceptual explanation and a theoretical model for this phenomenon. Specifically, we demonstrate that for natural data distributions memorization of labels is necessary for achieving close-to-optimal generalization error. Crucially, even labels of outliers and noisy labels need to be memorized. The model is motivated and supported by the results of several recent empirical works. In our model, data is sampled from a mixture of subpopulations and our results show that memorization is necessary whenever the distribution of subpopulation frequencies is long-tailed. Image and text data is known to be long-tailed and therefore our results establish a formal link between these empirical phenomena. Our results allow to quantify the cost of limiting memorization in learning and explain the disparate effects that privacy and model compression have on different subgroups.

Language model deployments in consumer-facing applications introduce numerous risks. While existing research on harms and hazards of such applications follows top-down approaches derived from regulatory frameworks and theoretical analyses, empirical evidence of real-world failure modes remains underexplored. In this work, we introduce RealHarm, a dataset of annotated problematic interactions with AI agents built from a systematic review of publicly reported incidents. Analyzing harms, causes, and hazards specifically from the deployer's perspective, we find that reputational damage constitutes the predominant organizational harm, while misinformation emerges as the most common hazard category. We empirically evaluate state-of-the-art guardrails and content moderation systems to probe whether such systems would have prevented the incidents, revealing a significant gap in the protection of AI applications.

Dense retrieval requires high-quality text sequence embeddings to support effective search in the representation space. Autoencoder-based language models are appealing in dense retrieval as they train the encoder to output high-quality embedding that can reconstruct the input texts. However, in this paper, we provide theoretical analyses and show empirically that an autoencoder language model with a low reconstruction loss may not provide good sequence representations because the decoder may take shortcuts by exploiting language patterns. To address this, we propose a new self-learning method that pre-trains the autoencoder using a weak decoder, with restricted capacity and attention flexibility to push the encoder to provide better text representations. Our experiments on web search, news recommendation, and open domain question answering show that our pre-trained model significantly boosts the effectiveness and few-shot ability of dense retrieval models. Our code is available at https://github.com/microsoft/SEED-Encoder/.

Despite the empirical success of prompt tuning in adapting pretrained language models to new tasks, theoretical analyses of its capabilities remain limited. Existing theoretical work primarily addresses universal approximation properties, demonstrating results comparable to standard weight tuning. In this paper, we explore a different aspect of the theory of transformers: the memorization capability of prompt tuning. We provide two principal theoretical contributions. First, we prove that the amount of information memorized by a transformer cannot scale faster than linearly with the prompt length. Second, and more importantly, we present the first formal proof of a phenomenon empirically observed in large language models: performance degradation in transformers with extended contexts. We rigorously demonstrate that transformers inherently have limited memory, constraining the amount of information they can retain, regardless of the context size. This finding offers a fundamental understanding of the intrinsic limitations of transformer architectures, particularly their ability to handle long sequences.

Information is often stored in a distributed and proprietary form, and agents who own information are often self-interested and require incentives to reveal their information. Suitable mechanisms are required to elicit and aggregate such distributed information for decision making. In this paper, we use simulations to investigate the use of decision markets as mechanisms in a multi-agent learning system to aggregate distributed information for decision-making in a contextual bandit problem. The system utilises strictly proper decision scoring rules to assess the accuracy of probabilistic reports from agents, which allows agents to learn to solve the contextual bandit problem jointly. Our simulations show that our multi-agent system with distributed information can be trained as efficiently as a centralised counterpart with a single agent that receives all information. Moreover, we use our system to investigate scenarios with deterministic decision scoring rules which are not incentive compatible. We observe the emergence of more complex dynamics with manipulative behaviour, which agrees with existing theoretical analyses.

As large language models continue to develop in the field of AI, text generation systems are susceptible to a worrisome phenomenon known as hallucination. In this study, we summarize recent compelling insights into hallucinations in LLMs. We present a novel taxonomy of hallucinations from various text generation tasks, thus provide theoretical insights, detection methods and improvement approaches. Based on this, future research directions are proposed. Our contribution are threefold: (1) We provide a detailed and complete taxonomy for hallucinations appearing in text generation tasks; (2) We provide theoretical analyses of hallucinations in LLMs and provide existing detection and improvement methods; (3) We propose several research directions that can be developed in the future. As hallucinations garner significant attention from the community, we will maintain updates on relevant research progress.

Widely adopted in modern Vision Transformer designs, Softmax attention can effectively capture long-range visual information; however, it incurs excessive computational cost when dealing with high-resolution inputs. In contrast, linear attention naturally enjoys linear complexity and has great potential to scale up to higher-resolution images. Nonetheless, the unsatisfactory performance of linear attention greatly limits its practical application in various scenarios. In this paper, we take a step forward to close the gap between the linear and Softmax attention with novel theoretical analyses, which demystify the core factors behind the performance deviations. Specifically, we present two key perspectives to understand and alleviate the limitations of linear attention: the injective property and the local modeling ability. Firstly, we prove that linear attention is not injective, which is prone to assign identical attention weights to different query vectors, thus adding to severe semantic confusion since different queries correspond to the same outputs. Secondly, we confirm that effective local modeling is essential for the success of Softmax attention, in which linear attention falls short. The aforementioned two fundamental differences significantly contribute to the disparities between these two attention paradigms, which is demonstrated by our substantial empirical validation in the paper. In addition, more experiment results indicate that linear attention, as long as endowed with these two properties, can outperform Softmax attention across various tasks while maintaining lower computation complexity. Code is available at https://github.com/LeapLabTHU/InLine.

With the rapid advancement of large language models (LLMs), aligning policy models with human preferences has become increasingly critical. Direct Preference Optimization (DPO) has emerged as a promising approach for alignment, acting as an RL-free alternative to Reinforcement Learning from Human Feedback (RLHF). Despite DPO's various advancements and inherent limitations, an in-depth review of these aspects is currently lacking in the literature. In this work, we present a comprehensive review of the challenges and opportunities in DPO, covering theoretical analyses, variants, relevant preference datasets, and applications. Specifically, we categorize recent studies on DPO based on key research questions to provide a thorough understanding of DPO's current landscape. Additionally, we propose several future research directions to offer insights on model alignment for the research community.

Large language models (LLMs) have attracted significant attention for their remarkable abilities in various natural language processing tasks, but they suffer from hallucinations that will cause performance degradation. One promising solution to improve the LLMs' performance is to ask LLMs to revise their answer after generation, a technique known as self-correction. Among the two types of self-correction, intrinsic self-correction is considered a promising direction because it does not utilize external knowledge. However, recent works doubt the validity of LLM's ability to conduct intrinsic self-correction. In this paper, we present a novel perspective on the intrinsic self-correction capabilities of LLMs through theoretical analyses and empirical experiments. In addition, we identify two critical factors for successful self-correction: zero temperature and fair prompts. Leveraging these factors, we demonstrate that intrinsic self-correction ability is exhibited across multiple existing LLMs. Our findings offer insights into the fundamental theories underlying the self-correction behavior of LLMs and remark on the importance of unbiased prompts and zero temperature settings in harnessing their full potential.

Momentum is known to accelerate the convergence of gradient descent in strongly convex settings without stochastic gradient noise. In stochastic optimization, such as training neural networks, folklore suggests that momentum may help deep learning optimization by reducing the variance of the stochastic gradient update, but previous theoretical analyses do not find momentum to offer any provable acceleration. Theoretical results in this paper clarify the role of momentum in stochastic settings where the learning rate is small and gradient noise is the dominant source of instability, suggesting that SGD with and without momentum behave similarly in the short and long time horizons. Experiments show that momentum indeed has limited benefits for both optimization and generalization in practical training regimes where the optimal learning rate is not very large, including small- to medium-batch training from scratch on ImageNet and fine-tuning language models on downstream tasks.

Real-world multichannel time series prediction faces growing demands for efficiency across edge and cloud environments, making channel compression a timely and essential problem. Motivated by success of Multiple-Input Multiple-Output (MIMO) methods, we propose a predictability-aware compression-decompression framework to reduce runtime, lower communication cost, and maintain prediction accuracy across diverse predictors. The core idea involves using a circular periodicity key matrix with orthogonality to capture underlying time series predictability during compression and to mitigate reconstruction errors during decompression by relaxing oversimplified data assumptions. Theoretical and empirical analyses show that the proposed framework is both time-efficient and scalable under a large number of channels. Extensive experiments on six datasets across various predictors demonstrate that the proposed method achieves superior overall performance by jointly considering prediction accuracy and runtime, while maintaining strong compatibility with diverse predictors.

Implicit Neural Representations (INRs), as a versatile representation paradigm, have achieved success in various computer vision tasks. Due to the spectral bias of the vanilla multi-layer perceptrons (MLPs), existing methods focus on designing MLPs with sophisticated architectures or repurposing training techniques for highly accurate INRs. In this paper, we delve into the linear dynamics model of MLPs and theoretically identify the empirical Neural Tangent Kernel (eNTK) matrix as a reliable link between spectral bias and training dynamics. Based on this insight, we propose a practical Inductive Gradient Adjustment (IGA) method, which could purposefully improve the spectral bias via inductive generalization of eNTK-based gradient transformation matrix. Theoretical and empirical analyses validate impacts of IGA on spectral bias. Further, we evaluate our method on different INRs tasks with various INR architectures and compare to existing training techniques. The superior and consistent improvements clearly validate the advantage of our IGA. Armed with our gradient adjustment method, better INRs with more enhanced texture details and sharpened edges can be learned from data by tailored impacts on spectral bias.

We introduce Sigma, an efficient large language model specialized for the system domain, empowered by a novel architecture including DiffQKV attention, and pre-trained on our meticulously collected system domain data. DiffQKV attention significantly enhances the inference efficiency of Sigma by optimizing the Query (Q), Key (K), and Value (V) components in the attention mechanism differentially, based on their varying impacts on the model performance and efficiency indicators. Specifically, we (1) conduct extensive experiments that demonstrate the model's varying sensitivity to the compression of K and V components, leading to the development of differentially compressed KV, and (2) propose augmented Q to expand the Q head dimension, which enhances the model's representation capacity with minimal impacts on the inference speed. Rigorous theoretical and empirical analyses reveal that DiffQKV attention significantly enhances efficiency, achieving up to a 33.36% improvement in inference speed over the conventional grouped-query attention (GQA) in long-context scenarios. We pre-train Sigma on 6T tokens from various sources, including 19.5B system domain data that we carefully collect and 1T tokens of synthesized and rewritten data. In general domains, Sigma achieves comparable performance to other state-of-arts models. In the system domain, we introduce the first comprehensive benchmark AIMicius, where Sigma demonstrates remarkable performance across all tasks, significantly outperforming GPT-4 with an absolute improvement up to 52.5%.

In this paper, we present the findings of our Project ALPINE which stands for ``Autoregressive Learning for Planning In NEtworks." Project ALPINE initiates a theoretical investigation into the development of planning capabilities in Transformer-based language models through their autoregressive learning mechanisms, aiming to identify any potential limitations in their planning abilities. We abstract planning as a network path-finding task where the objective is to generate a valid path from a specified source node to a designated target node. In terms of expressiveness, we show that the Transformer is capable of executing path-finding by embedding the adjacency and reachability matrices within its weights. Our theoretical analysis of the gradient-based learning dynamic of the Transformer reveals that the Transformer is capable of learning both the adjacency matrix and a limited form of the reachability matrix. These theoretical insights are then validated through experiments, which demonstrate that the Transformer indeed learns the adjacency matrix and an incomplete reachability matrix, which aligns with the predictions made in our theoretical analysis. Additionally, when applying our methodology to a real-world planning benchmark, called Blocksworld, our observations remain consistent. Our theoretical and empirical analyses further unveil a potential limitation of Transformer in path-finding: it cannot identify reachability relationships through transitivity, and thus would fail when path concatenation is needed to generate a path. In summary, our findings shed new light on how the internal mechanisms of autoregressive learning enable planning in networks. This study may contribute to our understanding of the general planning capabilities in other related domains.

The dominant approach to generating from language models subject to some constraint is locally constrained decoding (LCD), incrementally sampling tokens at each time step such that the constraint is never violated. Typically, this is achieved through token masking: looping over the vocabulary and excluding non-conforming tokens. There are two important problems with this approach. (i) Evaluating the constraint on every token can be prohibitively expensive -- LM vocabularies often exceed 100,000 tokens. (ii) LCD can distort the global distribution over strings, sampling tokens based only on local information, even if they lead down dead-end paths. This work introduces a new algorithm that addresses both these problems. First, to avoid evaluating a constraint on the full vocabulary at each step of generation, we propose an adaptive rejection sampling algorithm that typically requires orders of magnitude fewer constraint evaluations. Second, we show how this algorithm can be extended to produce low-variance, unbiased estimates of importance weights at a very small additional cost -- estimates that can be soundly used within previously proposed sequential Monte Carlo algorithms to correct for the myopic behavior of local constraint enforcement. Through extensive empirical evaluation in text-to-SQL, molecular synthesis, goal inference, pattern matching, and JSON domains, we show that our approach is superior to state-of-the-art baselines, supporting a broader class of constraints and improving both runtime and performance. Additional theoretical and empirical analyses show that our method's runtime efficiency is driven by its dynamic use of computation, scaling with the divergence between the unconstrained and constrained LM, and as a consequence, runtime improvements are greater for better models.

Graph neural networks (GNNs) learn the representation of graph-structured data, and their expressiveness can be further enhanced by inferring node relations for propagation. Attention-based GNNs infer neighbor importance to manipulate the weight of its propagation. Despite their popularity, the discussion on deep graph attention and its unique challenges has been limited. In this work, we investigate some problematic phenomena related to deep graph attention, including vulnerability to over-smoothed features and smooth cumulative attention. Through theoretical and empirical analyses, we show that various attention-based GNNs suffer from these problems. Motivated by our findings, we propose AEROGNN, a novel GNN architecture designed for deep graph attention. AERO-GNN provably mitigates the proposed problems of deep graph attention, which is further empirically demonstrated with (a) its adaptive and less smooth attention functions and (b) higher performance at deep layers (up to 64). On 9 out of 12 node classification benchmarks, AERO-GNN outperforms the baseline GNNs, highlighting the advantages of deep graph attention. Our code is available at https://github.com/syleeheal/AERO-GNN.

Feature selection is the problem of selecting a subset of features for a machine learning model that maximizes model quality subject to a budget constraint. For neural networks, prior methods, including those based on ell_1 regularization, attention, and other techniques, typically select the entire feature subset in one evaluation round, ignoring the residual value of features during selection, i.e., the marginal contribution of a feature given that other features have already been selected. We propose a feature selection algorithm called Sequential Attention that achieves state-of-the-art empirical results for neural networks. This algorithm is based on an efficient one-pass implementation of greedy forward selection and uses attention weights at each step as a proxy for feature importance. We give theoretical insights into our algorithm for linear regression by showing that an adaptation to this setting is equivalent to the classical Orthogonal Matching Pursuit (OMP) algorithm, and thus inherits all of its provable guarantees. Our theoretical and empirical analyses offer new explanations towards the effectiveness of attention and its connections to overparameterization, which may be of independent interest.

Transformers, the backbone of modern large language models (LLMs), face inherent architectural limitations that impede their reasoning capabilities. Unlike recurrent networks, Transformers lack recurrent connections, confining them to constant-depth computation. This restriction places them in the complexity class TC^0, making them theoretically incapable of solving tasks that demand increasingly deep reasoning as input length grows. Counting, a fundamental component of many reasoning tasks, also requires reasoning depth to grow linearly to be performed inductively. While previous studies have established the upper limits of counting ability in Transformer-based expert models (i.e., models specifically trained for counting tasks), these findings do not directly extend to general-purpose LLMs due to differences in reasoning mechanisms. Recent work has highlighted how Chain of Thought (CoT) reasoning can help alleviate some of the architectural limitations of Transformers in counting tasks. However, little attention has been paid to the role of tokenization in these models. Unlike expert models that often use character-level tokenization, LLMs typically rely on byte-level (BPE) tokenizers, which fundamentally alters the way reasoning is processed. Our work investigates the impact of tokenization on the counting abilities of LLMs, uncovering substantial performance variations based on input tokenization differences. We provide both theoretical and experimental analyses, offering insights into how tokenization choices can undermine models' theoretical computability, thereby inspiring the design of new tokenization methods to enhance reasoning in LLMs.

Large language models are typically adapted to downstream tasks through supervised fine-tuning on domain-specific data. While standard fine-tuning focuses on minimizing generation loss to optimize model parameters, we take a deeper step by retaining and leveraging the model's own learning signals, analogous to how human learners reflect on past mistakes to improve future performance. We first introduce the concept of Mistake Log to systematically track the model's learning behavior and recurring errors throughout fine-tuning. Treating the original transformer-based model as the Pilot, we correspondingly design a Copilot model to refine the Pilot's inference performance via logits rectification. We name the overall Pilot-Copilot framework the Transformer Copilot, which introduces (i) a novel Copilot model design, (ii) a joint training paradigm where the Copilot continuously learns from the evolving Mistake Log alongside the Pilot, and (iii) a fused inference paradigm where the Copilot rectifies the Pilot's logits for enhanced generation. We provide both theoretical and empirical analyses on our new learning framework. Experiments on 12 benchmarks spanning commonsense, arithmetic, and recommendation tasks demonstrate that Transformer Copilot consistently improves performance by up to 34.5%, while introducing marginal computational overhead to Pilot models and exhibiting strong scalability and transferability.

Optimizing inference for long-context Large Language Models (LLMs) is increasingly important due to the quadratic compute and linear memory complexity of Transformers. Existing approximation methods, such as key-value (KV) cache dropping, sparse attention, and prompt compression, typically rely on rough predictions of token or KV pair importance. We propose a novel framework for approximate LLM inference that leverages small draft models to more accurately predict the importance of tokens and KV pairs. Specifically, we introduce two instantiations of our proposed framework: (i) SpecKV, which leverages a draft output to accurately assess the importance of each KV pair for more effective KV cache dropping, and (ii) SpecPC, which uses the draft model's attention activations to identify and discard unimportant prompt tokens. To the best of our knowledge, this is the first work to use draft models for approximate LLM inference acceleration, extending their utility beyond traditional lossless speculative decoding. We motivate our methods with theoretical and empirical analyses, and show a strong correlation between the attention patterns of draft and target models. Extensive experiments on long-context benchmarks show that our methods consistently achieve higher accuracy than existing baselines, while preserving the same improvements in memory usage, latency, and throughput. Our code is available at https://github.com/furiosa-ai/draft-based-approx-llm.

The alignment of language models with human preferences is vital for their application in real-world tasks. The problem is formulated as optimizing the model's policy to maximize the expected reward that reflects human preferences with minimal deviation from the initial policy. While considered as a straightforward solution, reinforcement learning (RL) suffers from high variance in policy updates, which impedes efficient policy improvement. Recently, direct preference optimization (DPO) was proposed to directly optimize the policy from preference data. Though simple to implement, DPO is derived based on the optimal policy that is not assured to be achieved in practice, which undermines its convergence to the intended solution. In this paper, we propose efficient exact optimization (EXO) of the alignment objective. We prove that EXO is guaranteed to optimize in the same direction as the RL algorithms asymptotically for arbitary parametrization of the policy, while enables efficient optimization by circumventing the complexities associated with RL algorithms. We compare our method to DPO with both theoretical and empirical analyses, and further demonstrate the advantages of our method over existing approaches on realistic human preference data.

Text-guided diffusion models have revolutionized image generation and editing, offering exceptional realism and diversity. Specifically, in the context of diffusion-based editing, where a source image is edited according to a target prompt, the process commences by acquiring a noisy latent vector corresponding to the source image via the diffusion model. This vector is subsequently fed into separate source and target diffusion branches for editing. The accuracy of this inversion process significantly impacts the final editing outcome, influencing both essential content preservation of the source image and edit fidelity according to the target prompt. Prior inversion techniques aimed at finding a unified solution in both the source and target diffusion branches. However, our theoretical and empirical analyses reveal that disentangling these branches leads to a distinct separation of responsibilities for preserving essential content and ensuring edit fidelity. Building on this insight, we introduce "Direct Inversion," a novel technique achieving optimal performance of both branches with just three lines of code. To assess image editing performance, we present PIE-Bench, an editing benchmark with 700 images showcasing diverse scenes and editing types, accompanied by versatile annotations and comprehensive evaluation metrics. Compared to state-of-the-art optimization-based inversion techniques, our solution not only yields superior performance across 8 editing methods but also achieves nearly an order of speed-up.

Task arithmetic has recently emerged as a cost-effective and scalable approach to edit pre-trained models directly in weight space: By adding the fine-tuned weights of different tasks, the model's performance can be improved on these tasks, while negating them leads to task forgetting. Yet, our understanding of the effectiveness of task arithmetic and its underlying principles remains limited. We present a comprehensive study of task arithmetic in vision-language models and show that weight disentanglement is the crucial factor that makes it effective. This property arises during pre-training and manifests when distinct directions in weight space govern separate, localized regions in function space associated with the tasks. Notably, we show that fine-tuning models in their tangent space by linearizing them amplifies weight disentanglement. This leads to substantial performance improvements across multiple task arithmetic benchmarks and diverse models. Building on these findings, we provide theoretical and empirical analyses of the neural tangent kernel (NTK) of these models and establish a compelling link between task arithmetic and the spatial localization of the NTK eigenfunctions. Overall, our work uncovers novel insights into the fundamental mechanisms of task arithmetic and offers a more reliable and effective approach to edit pre-trained models through the NTK linearization.

Conditional Average Treatment Effects (CATE) estimation is one of the main challenges in causal inference with observational data. In addition to Machine Learning based-models, nonparametric estimators called meta-learners have been developed to estimate the CATE with the main advantage of not restraining the estimation to a specific supervised learning method. This task becomes, however, more complicated when the treatment is not binary as some limitations of the naive extensions emerge. This paper looks into meta-learners for estimating the heterogeneous effects of multi-valued treatments. We consider different meta-learners, and we carry out a theoretical analysis of their error upper bounds as functions of important parameters such as the number of treatment levels, showing that the naive extensions do not always provide satisfactory results. We introduce and discuss meta-learners that perform well as the number of treatments increases. We empirically confirm the strengths and weaknesses of those methods with synthetic and semi-synthetic datasets.

The existing active learning methods select the samples by evaluating the sample's uncertainty or its effect on the diversity of labeled datasets based on different task-specific or model-specific criteria. In this paper, we propose the Influence Selection for Active Learning(ISAL) which selects the unlabeled samples that can provide the most positive Influence on model performance. To obtain the Influence of the unlabeled sample in the active learning scenario, we design the Untrained Unlabeled sample Influence Calculation(UUIC) to estimate the unlabeled sample's expected gradient with which we calculate its Influence. To prove the effectiveness of UUIC, we provide both theoretical and experimental analyses. Since the UUIC just depends on the model gradients, which can be obtained easily from any neural network, our active learning algorithm is task-agnostic and model-agnostic. ISAL achieves state-of-the-art performance in different active learning settings for different tasks with different datasets. Compared with previous methods, our method decreases the annotation cost at least by 12%, 13% and 16% on CIFAR10, VOC2012 and COCO, respectively.

Existing approaches to language model alignment often treat safety as a tradeoff against helpfulness, which can lead to unacceptable responses in sensitive domains. To ensure reliable performance in such settings, we propose High-Confidence Safe Reinforcement Learning from Human Feedback (HC-RLHF), a method that provides high-confidence safety guarantees while maximizing helpfulness. Similar to previous methods, HC-RLHF explicitly decouples human preferences into helpfulness and harmlessness (safety), which are learned by training a reward model and a cost model, respectively. It then employs a two-step process to find safe solutions. In the first step, it optimizes the reward function under an intentionally pessimistic version of the cost constraint. In the second step, the trained model undergoes a safety test to verify whether its performance stays within an upper-confidence bound of the actual cost constraint. We provide a theoretical analysis of HC-RLHF, including proof that it will not return an unsafe solution with a probability greater than a user-specified threshold. For our empirical analysis, we apply HC-RLHF to align three different language models (Qwen2-1.5B, Qwen2.5-3B, and LLaMa3.2-3B) with human preferences. Our results demonstrate that HC-RLHF produces safe models with high probability and can improve harmlessness and helpfulness compared to previous methods.

Extending the context length of Language Models (LMs) by improving Rotary Position Embedding (RoPE) has become a trend. While existing works mainly address RoPE's limitations within attention mechanism, this paper provides an analysis across nearly all parts of LMs, uncovering their adverse effects on length generalization for RoPE-based attention. Using Discrete Signal Processing theory, we show that RoPE enables periodic attention by implicitly achieving Non-Uniform Discrete Fourier Transform. However, this periodicity is undermined by the spectral damage caused by: 1) linear layers and activation functions outside of attention; 2) insufficiently trained frequency components brought by time-domain truncation. Building on our observations, we propose Fourier Position Embedding (FoPE), which enhances attention's frequency-domain properties to improve both its periodic extension and length generalization. FoPE constructs Fourier Series and zero-outs the destructive frequency components, increasing model robustness against the spectrum damage. Experiments across various model scales show that, within varying context windows, FoPE can maintain a more stable perplexity and a more consistent accuracy in a needle-in-haystack task compared to RoPE and ALiBi. Several analyses and ablations bring further support to our method and theoretical modeling.

We revisit the theoretical properties of Hamiltonian stochastic differential equations (SDES) for Bayesian posterior sampling, and we study the two types of errors that arise from numerical SDE simulation: the discretization error and the error due to noisy gradient estimates in the context of data subsampling. Our main result is a novel analysis for the effect of mini-batches through the lens of differential operator splitting, revising previous literature results. The stochastic component of a Hamiltonian SDE is decoupled from the gradient noise, for which we make no normality assumptions. This leads to the identification of a convergence bottleneck: when considering mini-batches, the best achievable error rate is O(eta^2), with eta being the integrator step size. Our theoretical results are supported by an empirical study on a variety of regression and classification tasks for Bayesian neural networks.

In real world, large language models (LLMs) can serve as the assistant to help users accomplish their jobs, and also support the development of advanced applications. For the wide application of LLMs, the inference efficiency is an essential concern, which has been widely studied in existing work, and numerous optimization algorithms and code libraries have been proposed to improve it. Nonetheless, users still find it challenging to compare the effectiveness of all the above methods and understand the underlying mechanisms. In this work, we perform a detailed coarse-to-fine analysis of the inference performance of various code libraries. To evaluate the overall effectiveness, we examine four usage scenarios within two practical applications. We further provide both theoretical and empirical fine-grained analyses of each module in the Transformer architecture. Our experiments yield comprehensive results that are invaluable for researchers to evaluate code libraries and improve inference strategies.

We present the analytic evaluation of the gravitational energy and of the angular momentum flux with tidal effects for inspiraling compact binaries, at next-to-next-to-leading post-Newtoian (2PN) order, within the effective field theory diagrammatic approach. We first compute the stress-energy tensor for a binary system, that requires the evaluation of two-point Feynman integrals, up to two loops. Then, we extract the multipole moments of the system, which we present for generic orbits in center-of-mass coordinates, and which are needed for the evaluation of the total gravitational energy and the angular momentum flux, for generic orbits. Finally, we provide the expression of gauge invariant quantities such as the fluxes, and the mode amplitudes and phase of the emitted gravitational wave, for circular orbits. Our findings are useful to update earlier theoretical studies as well as related phenomenological analyses, and waveform models

Fine-tuning language models (LMs) has yielded success on diverse downstream tasks, but as LMs grow in size, backpropagation requires a prohibitively large amount of memory. Zeroth-order (ZO) methods can in principle estimate gradients using only two forward passes but are theorized to be catastrophically slow for optimizing large models. In this work, we propose a memory-efficient zerothorder optimizer (MeZO), adapting the classical ZO-SGD method to operate in-place, thereby fine-tuning LMs with the same memory footprint as inference. For example, with a single A100 80GB GPU, MeZO can train a 30-billion parameter model, whereas fine-tuning with backpropagation can train only a 2.7B LM with the same budget. We conduct comprehensive experiments across model types (masked and autoregressive LMs), model scales (up to 66B), and downstream tasks (classification, multiple-choice, and generation). Our results demonstrate that (1) MeZO significantly outperforms in-context learning and linear probing; (2) MeZO achieves comparable performance to fine-tuning with backpropagation across multiple tasks, with up to 12x memory reduction; (3) MeZO is compatible with both full-parameter and parameter-efficient tuning techniques such as LoRA and prefix tuning; (4) MeZO can effectively optimize non-differentiable objectives (e.g., maximizing accuracy or F1). We support our empirical findings with theoretical insights, highlighting how adequate pre-training and task prompts enable MeZO to fine-tune huge models, despite classical ZO analyses suggesting otherwise.

A great deal of effort has been devoted to reducing the risk of spurious scientific discoveries, from the use of sophisticated validation techniques, to deep statistical methods for controlling the false discovery rate in multiple hypothesis testing. However, there is a fundamental disconnect between the theoretical results and the practice of data analysis: the theory of statistical inference assumes a fixed collection of hypotheses to be tested, or learning algorithms to be applied, selected non-adaptively before the data are gathered, whereas in practice data is shared and reused with hypotheses and new analyses being generated on the basis of data exploration and the outcomes of previous analyses. In this work we initiate a principled study of how to guarantee the validity of statistical inference in adaptive data analysis. As an instance of this problem, we propose and investigate the question of estimating the expectations of m adaptively chosen functions on an unknown distribution given n random samples. We show that, surprisingly, there is a way to estimate an exponential in n number of expectations accurately even if the functions are chosen adaptively. This gives an exponential improvement over standard empirical estimators that are limited to a linear number of estimates. Our result follows from a general technique that counter-intuitively involves actively perturbing and coordinating the estimates, using techniques developed for privacy preservation. We give additional applications of this technique to our question.

Precise segmentation of brain tumors, particularly contrast-enhancing regions visible in post-contrast MRI (areas highlighted by contrast agent injection), is crucial for accurate clinical diagnosis and treatment planning but remains challenging. However, current methods exhibit notable performance degradation in segmenting these enhancing brain tumor areas, largely due to insufficient consideration of MRI-specific tumor features such as complex textures and directional variations. To address this, we propose the Harmonized Frequency Fusion Network (HFF-Net), which rethinks brain tumor segmentation from a frequency-domain perspective. To comprehensively characterize tumor regions, we develop a Frequency Domain Decomposition (FDD) module that separates MRI images into low-frequency components, capturing smooth tumor contours and high-frequency components, highlighting detailed textures and directional edges. To further enhance sensitivity to tumor boundaries, we introduce an Adaptive Laplacian Convolution (ALC) module that adaptively emphasizes critical high-frequency details using dynamically updated convolution kernels. To effectively fuse tumor features across multiple scales, we design a Frequency Domain Cross-Attention (FDCA) integrating semantic, positional, and slice-specific information. We further validate and interpret frequency-domain improvements through visualization, theoretical reasoning, and experimental analyses. Extensive experiments on four public datasets demonstrate that HFF-Net achieves an average relative improvement of 4.48\% (ranging from 2.39\% to 7.72\%) in the mean Dice scores across the three major subregions, and an average relative improvement of 7.33% (ranging from 5.96% to 8.64%) in the segmentation of contrast-enhancing tumor regions, while maintaining favorable computational efficiency and clinical applicability. Code: https://github.com/VinyehShaw/HFF.

Cross-entropy is a widely used loss function in applications. It coincides with the logistic loss applied to the outputs of a neural network, when the softmax is used. But, what guarantees can we rely on when using cross-entropy as a surrogate loss? We present a theoretical analysis of a broad family of loss functions, comp-sum losses, that includes cross-entropy (or logistic loss), generalized cross-entropy, the mean absolute error and other cross-entropy-like loss functions. We give the first H-consistency bounds for these loss functions. These are non-asymptotic guarantees that upper bound the zero-one loss estimation error in terms of the estimation error of a surrogate loss, for the specific hypothesis set H used. We further show that our bounds are tight. These bounds depend on quantities called minimizability gaps. To make them more explicit, we give a specific analysis of these gaps for comp-sum losses. We also introduce a new family of loss functions, smooth adversarial comp-sum losses, that are derived from their comp-sum counterparts by adding in a related smooth term. We show that these loss functions are beneficial in the adversarial setting by proving that they admit H-consistency bounds. This leads to new adversarial robustness algorithms that consist of minimizing a regularized smooth adversarial comp-sum loss. While our main purpose is a theoretical analysis, we also present an extensive empirical analysis comparing comp-sum losses. We further report the results of a series of experiments demonstrating that our adversarial robustness algorithms outperform the current state-of-the-art, while also achieving a superior non-adversarial accuracy.

Adversarial training (AT) is a canonical method for enhancing the robustness of deep neural networks (DNNs). However, recent studies empirically demonstrated that it suffers from robust overfitting, i.e., a long time AT can be detrimental to the robustness of DNNs. This paper presents a theoretical explanation of robust overfitting for DNNs. Specifically, we non-trivially extend the neural tangent kernel (NTK) theory to AT and prove that an adversarially trained wide DNN can be well approximated by a linearized DNN. Moreover, for squared loss, closed-form AT dynamics for the linearized DNN can be derived, which reveals a new AT degeneration phenomenon: a long-term AT will result in a wide DNN degenerates to that obtained without AT and thus cause robust overfitting. Based on our theoretical results, we further design a method namely Adv-NTK, the first AT algorithm for infinite-width DNNs. Experiments on real-world datasets show that Adv-NTK can help infinite-width DNNs enhance comparable robustness to that of their finite-width counterparts, which in turn justifies our theoretical findings. The code is available at https://github.com/fshp971/adv-ntk.

To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Frechet mean. In this work, we equip a set of graphs with the pseudometric defined by the norm between the eigenvalues of their respective adjacency matrix. Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems for graph-valued data. We describe an algorithm to compute an approximation to the sample Frechet mean of a set of undirected unweighted graphs with a fixed size using this pseudometric.

Continual learning (CL) is a setting in which an agent has to learn from an incoming stream of data during its entire lifetime. Although major advances have been made in the field, one recurring problem which remains unsolved is that of Catastrophic Forgetting (CF). While the issue has been extensively studied empirically, little attention has been paid from a theoretical angle. In this paper, we show that the impact of CF increases as two tasks increasingly align. We introduce a measure of task similarity called the NTK overlap matrix which is at the core of CF. We analyze common projected gradient algorithms and demonstrate how they mitigate forgetting. Then, we propose a variant of Orthogonal Gradient Descent (OGD) which leverages structure of the data through Principal Component Analysis (PCA). Experiments support our theoretical findings and show how our method can help reduce CF on classical CL datasets.

Despite the great empirical success of deep reinforcement learning, its theoretical foundation is less well understood. In this work, we make the first attempt to theoretically understand the deep Q-network (DQN) algorithm (Mnih et al., 2015) from both algorithmic and statistical perspectives. In specific, we focus on a slight simplification of DQN that fully captures its key features. Under mild assumptions, we establish the algorithmic and statistical rates of convergence for the action-value functions of the iterative policy sequence obtained by DQN. In particular, the statistical error characterizes the bias and variance that arise from approximating the action-value function using deep neural network, while the algorithmic error converges to zero at a geometric rate. As a byproduct, our analysis provides justifications for the techniques of experience replay and target network, which are crucial to the empirical success of DQN. Furthermore, as a simple extension of DQN, we propose the Minimax-DQN algorithm for zero-sum Markov game with two players. Borrowing the analysis of DQN, we also quantify the difference between the policies obtained by Minimax-DQN and the Nash equilibrium of the Markov game in terms of both the algorithmic and statistical rates of convergence.

Data imbalance is a common problem in machine learning that can have a critical effect on the performance of a model. Various solutions exist but their impact on the convergence of the learning dynamics is not understood. Here, we elucidate the significant negative impact of data imbalance on learning, showing that the learning curves for minority and majority classes follow sub-optimal trajectories when training with a gradient-based optimizer. This slowdown is related to the imbalance ratio and can be traced back to a competition between the optimization of different classes. Our main contribution is the analysis of the convergence of full-batch (GD) and stochastic gradient descent (SGD), and of variants that renormalize the contribution of each per-class gradient. We find that GD is not guaranteed to decrease the loss for each class but that this problem can be addressed by performing a per-class normalization of the gradient. With SGD, class imbalance has an additional effect on the direction of the gradients: the minority class suffers from a higher directional noise, which reduces the effectiveness of the per-class gradient normalization. Our findings not only allow us to understand the potential and limitations of strategies involving the per-class gradients, but also the reason for the effectiveness of previously used solutions for class imbalance such as oversampling.

Recent empirical works have successfully used unlabeled data to learn feature representations that are broadly useful in downstream classification tasks. Several of these methods are reminiscent of the well-known word2vec embedding algorithm: leveraging availability of pairs of semantically "similar" data points and "negative samples," the learner forces the inner product of representations of similar pairs with each other to be higher on average than with negative samples. The current paper uses the term contrastive learning for such algorithms and presents a theoretical framework for analyzing them by introducing latent classes and hypothesizing that semantically similar points are sampled from the same latent class. This framework allows us to show provable guarantees on the performance of the learned representations on the average classification task that is comprised of a subset of the same set of latent classes. Our generalization bound also shows that learned representations can reduce (labeled) sample complexity on downstream tasks. We conduct controlled experiments in both the text and image domains to support the theory.

Reinforcement learning (RL) plays a crucial role in shaping the behavior of large language and reasoning models (LLMs/LRMs). However, it often produces brittle and unstable policies, leading to critical failures such as spurious reasoning, deceptive alignment, and instruction disobedience that undermine the trustworthiness and safety of LLMs/LRMs. Currently, these issues lack a unified theoretical explanation and are typically addressed using ad-hoc heuristics. This paper presents a rigorous mathematical framework for analyzing the stability of the mapping from a reward function to the optimal policy. We show that policy brittleness often stems from non-unique optimal actions, a common occurrence when multiple valid traces exist in a reasoning task. This theoretical lens provides a unified explanation for a range of seemingly disparate failures, reframing them as rational outcomes of optimizing rewards that may be incomplete or noisy, especially in the presence of action degeneracy. We extend this analysis from the fundamental single-reward setting to the more realistic multi-reward RL across diverse domains, showing how stability is governed by an "effective reward" aggregation mechanism. We also prove that entropy regularization restores policy stability at the cost of increased stochasticity. Our framework provides a unified explanation for recent empirical findings on deceptive reasoning, instruction-following trade-offs, and RLHF-induced sophistry, and is further validated through perturbation experiments in multi-reward RL. This work advances policy-stability analysis from empirical heuristics towards a principled theory, offering essential insights for designing safer and more trustworthy AI systems.

The reasoning abilities of large language models (LLMs) have improved with chain-of-thought (CoT) prompting, allowing models to solve complex tasks in a stepwise manner. However, training CoT capabilities requires detailed reasoning data, which is often scarce. The self-taught reasoner (STaR) framework addresses this by using reinforcement learning to automatically generate reasoning steps, reducing reliance on human-labeled data. Although STaR and its variants have demonstrated empirical success, a theoretical foundation explaining these improvements is lacking. This work provides a theoretical framework for understanding the effectiveness of reinforcement learning on CoT reasoning and STaR. Our contributions are: (1) an analysis of policy improvement, showing why LLM reasoning improves iteratively with STaR; (2) conditions for convergence to an optimal reasoning policy; (3) an examination of STaR's robustness, explaining how it can improve reasoning even when incorporating occasional incorrect steps; and (4) criteria for the quality of pre-trained models necessary to initiate effective reasoning improvement. This framework aims to bridge empirical findings with theoretical insights, advancing reinforcement learning approaches for reasoning in LLMs.

Owing to the capability of in-context learning, large language models (LLMs) have shown impressive performance across diverse mathematical reasoning benchmarks. However, we find that few-shot demonstrations can sometimes bring negative performance and their effectiveness on LLMs' reasoning abilities remains unreliable. To this end, in this paper, we aim to theoretically analyze the impact of in-context demonstrations on LLMs' reasoning performance. We prove that the reasoning efficacy (measured by empirical prediction loss) can be bounded by a LLM-oriented semantic similarity and an inference stability of demonstrations, which is general for both one-shot and few-shot scenarios. Based on this finding, we propose a straightforward, generalizable, and low-complexity demonstration selection method named LMS3. It can adaptively facilitate to select the most pertinent samples for different LLMs and includes a novel demonstration rejection mechanism to automatically filter out samples that are unsuitable for few-shot learning. Through experiments on three representative benchmarks, two LLM backbones, and multiple few-shot settings, we verify that our LMS3 has superiority and achieves consistent improvements on all datasets, which existing methods have been unable to accomplish.

The Mamba model has gained significant attention for its computational advantages over Transformer-based models, while achieving comparable performance across a wide range of language tasks. Like Transformers, Mamba exhibits in-context learning (ICL) capabilities, i.e., making predictions for new tasks based on a prompt containing input-label pairs and a query, without requiring fine-tuning. Despite its empirical success, the theoretical understanding of Mamba remains limited, largely due to the nonlinearity introduced by its gating mechanism. To the best of our knowledge, this paper presents the first theoretical analysis of the training dynamics of a one-layer Mamba model, which consists of a linear attention component followed by a nonlinear gating layer, and its ICL generalization on unseen binary classification tasks, even when the prompt includes additive outliers. Our analysis shows that Mamba leverages the linear attention layer to select informative context examples and uses the nonlinear gating layer to suppress the influence of outliers. By establishing and comparing to the analysis of linear Transformers under the same setting, we show that although Mamba may require more training iterations to converge, it maintains accurate predictions even when the proportion of outliers exceeds the threshold that a linear Transformer can tolerate. These theoretical findings are supported by empirical experiments.

Pretext-based self-supervised learning learns the semantic representation via a handcrafted pretext task over unlabeled data and then uses the learned representation for downstream tasks, which effectively reduces the sample complexity of downstream tasks under Conditional Independence (CI) condition. However, the downstream sample complexity gets much worse if the CI condition does not hold. One interesting question is whether we can make the CI condition hold by using downstream data to refine the unlabeled data to boost self-supervised learning. At first glance, one might think that seeing downstream data in advance would always boost the downstream performance. However, we show that it is not intuitively true and point out that in some cases, it hurts the final performance instead. In particular, we prove both model-free and model-dependent lower bounds of the number of downstream samples used for data refinement. Moreover, we conduct various experiments on both synthetic and real-world datasets to verify our theoretical results.

Anchors (Ribeiro et al., 2018) is a post-hoc, rule-based interpretability method. For text data, it proposes to explain a decision by highlighting a small set of words (an anchor) such that the model to explain has similar outputs when they are present in a document. In this paper, we present the first theoretical analysis of Anchors, considering that the search for the best anchor is exhaustive. After formalizing the algorithm for text classification, we present explicit results on different classes of models when the vectorization step is TF-IDF, and words are replaced by a fixed out-of-dictionary token when removed. Our inquiry covers models such as elementary if-then rules and linear classifiers. We then leverage this analysis to gain insights on the behavior of Anchors for any differentiable classifiers. For neural networks, we empirically show that the words corresponding to the highest partial derivatives of the model with respect to the input, reweighted by the inverse document frequencies, are selected by Anchors.

State Space Models (SSMs) have recently emerged as efficient alternatives to Transformer-Based Models (TBMs) for long-sequence processing, offering linear scaling and lower memory use. Yet, how contextual information flows across layers and tokens in these architectures remains understudied. We present the first unified, token- and layer-level analysis of representation propagation in SSMs and TBMs. Using centered kernel alignment, stability metrics, and probing, we characterize how representations evolve within and across layers. We find a key divergence: TBMs rapidly homogenize token representations, with diversity reemerging only in later layers, while SSMs preserve token uniqueness early but converge to homogenization deeper. Theoretical analysis and parameter randomization further reveal that oversmoothing in TBMs stems from architectural design, whereas in SSMs it arises mainly from training dynamics. These insights clarify the inductive biases of both architectures and inform future model and training designs for long-context reasoning.

Zero-Input AI (ZIA) introduces a novel framework for human-computer interaction by enabling proactive intent prediction without explicit user commands. It integrates gaze tracking, bio-signals (EEG, heart rate), and contextual data (time, location, usage history) into a multi-modal model for real-time inference, targeting <100 ms latency. The proposed architecture employs a transformer-based model with cross-modal attention, variational Bayesian inference for uncertainty estimation, and reinforcement learning for adaptive optimization. To support deployment on edge devices (CPUs, TPUs, NPUs), ZIA utilizes quantization, weight pruning, and linear attention to reduce complexity from quadratic to linear with sequence length. Theoretical analysis establishes an information-theoretic bound on prediction error and demonstrates how multi-modal fusion improves accuracy over single-modal approaches. Expected performance suggests 85-90% accuracy with EEG integration and 60-100 ms inference latency. ZIA provides a scalable, privacy-preserving framework for accessibility, healthcare, and consumer applications, advancing AI toward anticipatory intelligence.

Direct Preference Optimization (DPO), which derives reward signals directly from pairwise preference data, has shown its effectiveness on aligning Large Language Models (LLMs) with human preferences. Despite its widespread use across various tasks, DPO has been criticized for its sensitivity to the SFT's effectiveness and its hindrance to the learning capacity towards human-preferred responses, leading to less satisfactory performance. To overcome those limitations, the theoretical understanding of DPO are indispensable but still lacking. To this end, we take a step towards theoretically analyzing and understanding the limitations of DPO. Specifically, we provide an analytical framework using the field theory to analyze the optimization process of DPO. By analyzing the gradient vector field of the DPO loss function, we find that the DPO loss function decreases the probability of producing human dispreferred data at a faster rate than it increases the probability of producing preferred data. This provides theoretical insights for understanding the limitations of DPO discovered in the related research experiments, thereby setting the foundation for its improvement.

It is not fully understood why adversarial examples can deceive neural networks and transfer between different networks. To elucidate this, several studies have hypothesized that adversarial perturbations, while appearing as noises, contain class features. This is supported by empirical evidence showing that networks trained on mislabeled adversarial examples can still generalize well to correctly labeled test samples. However, a theoretical understanding of how perturbations include class features and contribute to generalization is limited. In this study, we provide a theoretical framework for understanding learning from perturbations using a one-hidden-layer network trained on mutually orthogonal samples. Our results highlight that various adversarial perturbations, even perturbations of a few pixels, contain sufficient class features for generalization. Moreover, we reveal that the decision boundary when learning from perturbations matches that from standard samples except for specific regions under mild conditions. The code is available at https://github.com/s-kumano/learning-from-adversarial-perturbations.

Recently, Ainsworth et al. showed that using weight matching (WM) to minimize the L_2 distance in a permutation search of model parameters effectively identifies permutations that satisfy linear mode connectivity (LMC), in which the loss along a linear path between two independently trained models with different seeds remains nearly constant. This paper provides a theoretical analysis of LMC using WM, which is crucial for understanding stochastic gradient descent's effectiveness and its application in areas like model merging. We first experimentally and theoretically show that permutations found by WM do not significantly reduce the L_2 distance between two models and the occurrence of LMC is not merely due to distance reduction by WM in itself. We then provide theoretical insights showing that permutations can change the directions of the singular vectors, but not the singular values, of the weight matrices in each layer. This finding shows that permutations found by WM mainly align the directions of singular vectors associated with large singular values across models. This alignment brings the singular vectors with large singular values, which determine the model functionality, closer between pre-merged and post-merged models, so that the post-merged model retains functionality similar to the pre-merged models, making it easy to satisfy LMC. Finally, we analyze the difference between WM and straight-through estimator (STE), a dataset-dependent permutation search method, and show that WM outperforms STE, especially when merging three or more models.

Score-based generative modeling with probability flow ordinary differential equations (ODEs) has achieved remarkable success in a variety of applications. While various fast ODE-based samplers have been proposed in the literature and employed in practice, the theoretical understandings about convergence properties of the probability flow ODE are still quite limited. In this paper, we provide the first non-asymptotic convergence analysis for a general class of probability flow ODE samplers in 2-Wasserstein distance, assuming accurate score estimates. We then consider various examples and establish results on the iteration complexity of the corresponding ODE-based samplers.

Ensembling is among the most popular tools in machine learning (ML) due to its effectiveness in minimizing variance and thus improving generalization. Most ensembling methods for black-box base learners fall under the umbrella of "stacked generalization," namely training an ML algorithm that takes the inferences from the base learners as input. While stacking has been widely applied in practice, its theoretical properties are poorly understood. In this paper, we prove a novel result, showing that choosing the best stacked generalization from a (finite or finite-dimensional) family of stacked generalizations based on cross-validated performance does not perform "much worse" than the oracle best. Our result strengthens and significantly extends the results in Van der Laan et al. (2007). Inspired by the theoretical analysis, we further propose a particular family of stacked generalizations in the context of probabilistic forecasting, each one with a different sensitivity for how much the ensemble weights are allowed to vary across items, timestamps in the forecast horizon, and quantiles. Experimental results demonstrate the performance gain of the proposed method.

Backpropagation-based visualizations have been proposed to interpret convolutional neural networks (CNNs), however a theory is missing to justify their behaviors: Guided backpropagation (GBP) and deconvolutional network (DeconvNet) generate more human-interpretable but less class-sensitive visualizations than saliency map. Motivated by this, we develop a theoretical explanation revealing that GBP and DeconvNet are essentially doing (partial) image recovery which is unrelated to the network decisions. Specifically, our analysis shows that the backward ReLU introduced by GBP and DeconvNet, and the local connections in CNNs are the two main causes of compelling visualizations. Extensive experiments are provided that support the theoretical analysis.

Test-time scaling seeks to improve the reasoning performance of large language models (LLMs) by adding computational resources. A prevalent approach within the field is sampling-based test-time scaling methods, which enhance reasoning by generating multiple reasoning paths for a given input during inference. However, despite its practical success, the theoretical foundations remain underexplored. In this paper, we provide the first theoretical framework for analyzing sampling-based test-time scaling methods, grounded in the perspective of confidence estimation. Based on the framework, we analyze two dominant paradigms: self-consistency and perplexity, and reveal key limitations: self-consistency suffers from high estimation error while perplexity exhibits substantial modeling error and possible degradation of the estimation error convergence. To address these limitations, we introduce RPC, a hybrid method that leverages our theoretical insights through two key components: Perplexity Consistency and Reasoning Pruning. Perplexity Consistency combines the strengths of self-consistency and perplexity, boosting the convergence rate of estimation error from linear to exponential while preserving model error. Reasoning Pruning prevents degradation by eliminating low-probability reasoning paths. Both theoretical analysis and empirical results across seven benchmark datasets demonstrate that RPC has a strong potential for reducing reasoning error. Notably, RPC achieves reasoning performance comparable to self-consistency while not only enhancing confidence reliability but also reducing sampling costs by 50%. The code and resources are available at https://wnjxyk.github.io/RPC.

We consider multi-draft speculative sampling, where the proposal sequences are sampled independently from different draft models. At each step, a token-level draft selection scheme takes a list of valid tokens as input and produces an output token whose distribution matches that of the target model. Previous works have demonstrated that the optimal scheme (which maximizes the probability of accepting one of the input tokens) can be cast as a solution to a linear program. In this work we show that the optimal scheme can be decomposed into a two-step solution: in the first step an importance sampling (IS) type scheme is used to select one intermediate token; in the second step (single-draft) speculative sampling is applied to generate the output token. For the case of two identical draft models we further 1) establish a necessary and sufficient condition on the distributions of the target and draft models for the acceptance probability to equal one and 2) provide an explicit expression for the optimal acceptance probability. Our theoretical analysis also motives a new class of token-level selection scheme based on weighted importance sampling. Our experimental results demonstrate consistent improvements in the achievable block efficiency and token rates over baseline schemes in a number of scenarios.

Curriculum learning plays a crucial role in enhancing the training efficiency of large language models (LLMs) on reasoning tasks. However, existing methods often fail to adequately account for variations in prompt difficulty or rely on simplistic filtering mechanisms to select prompt datasets within a narrow criterion range, resulting in significant computational waste. In this work, we approach the problem from the perspective of reinforcement learning gradient optimization, offering a systematic and theoretical investigation into how to improve the training efficiency of LLMs. We identify two key factors influencing training efficiency: the selection of training prompts and the allocation of rollout quantities across different prompts. Our theoretical analysis reveals that the sampling distribution of prompts dictates the convergence rate of gradient descent, while the allocation of the rollout quantity influences the consistency and stability of overall gradient updates. Based on these insights, we propose CurES, an efficient training method that accelerates convergence and employs Bayesian posterior estimation to minimize computational overhead. Experiments demonstrate that our CurES outperforms Group Relative Policy Optimization (GRPO) by +3.30 points and +4.82 points with 1.5B and 7B models, respectively. Additionally, CurES exhibits faster convergence compared to baselines, including GRPO.

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L^2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in epsilon-accuracy can be done in tilde Oleft(d log (1/delta){epsilon}right) steps: 1) the variance-delta Gaussian perturbation of any data distribution; 2) data distributions with 1/delta-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

State-space models (SSMs) are a new class of foundation models that have emerged as a compelling alternative to Transformers and their attention mechanisms for sequence processing tasks. This paper provides a detailed theoretical analysis of selective SSMs, the core components of the Mamba and Mamba-2 architectures. We leverage the connection between selective SSMs and the self-attention mechanism to highlight the fundamental similarities between these models. Building on this connection, we establish a length independent covering number-based generalization bound for selective SSMs, providing a deeper understanding of their theoretical performance guarantees. We analyze the effects of state matrix stability and input-dependent discretization, shedding light on the critical role played by these factors in the generalization capabilities of selective SSMs. Finally, we empirically demonstrate the sequence length independence of the derived bounds on two tasks.

Diffusion language models have emerged as a promising approach for text generation. One would naturally expect this method to be an efficient replacement for autoregressive models since multiple tokens can be sampled in parallel during each diffusion step. However, its efficiency-accuracy trade-off is not yet well understood. In this paper, we present a rigorous theoretical analysis of a widely used type of diffusion language model, the Masked Diffusion Model (MDM), and find that its effectiveness heavily depends on the target evaluation metric. Under mild conditions, we prove that when using perplexity as the metric, MDMs can achieve near-optimal perplexity in sampling steps regardless of sequence length, demonstrating that efficiency can be achieved without sacrificing performance. However, when using the sequence error rate--which is important for understanding the "correctness" of a sequence, such as a reasoning chain--we show that the required sampling steps must scale linearly with sequence length to obtain "correct" sequences, thereby eliminating MDM's efficiency advantage over autoregressive models. Our analysis establishes the first theoretical foundation for understanding the benefits and limitations of MDMs. All theoretical findings are supported by empirical studies.

Recent Reinforcement Learning (RL) algorithms making use of Kullback-Leibler (KL) regularization as a core component have shown outstanding performance. Yet, only little is understood theoretically about why KL regularization helps, so far. We study KL regularization within an approximate value iteration scheme and show that it implicitly averages q-values. Leveraging this insight, we provide a very strong performance bound, the very first to combine two desirable aspects: a linear dependency to the horizon (instead of quadratic) and an error propagation term involving an averaging effect of the estimation errors (instead of an accumulation effect). We also study the more general case of an additional entropy regularizer. The resulting abstract scheme encompasses many existing RL algorithms. Some of our assumptions do not hold with neural networks, so we complement this theoretical analysis with an extensive empirical study.

Pretrained language models have achieved state-of-the-art performance when adapted to a downstream NLP task. However, theoretical analysis of these models is scarce and challenging since the pretraining and downstream tasks can be very different. We propose an analysis framework that links the pretraining and downstream tasks with an underlying latent variable generative model of text -- the downstream classifier must recover a function of the posterior distribution over the latent variables. We analyze head tuning (learning a classifier on top of the frozen pretrained model) and prompt tuning in this setting. The generative model in our analysis is either a Hidden Markov Model (HMM) or an HMM augmented with a latent memory component, motivated by long-term dependencies in natural language. We show that 1) under certain non-degeneracy conditions on the HMM, simple classification heads can solve the downstream task, 2) prompt tuning obtains downstream guarantees with weaker non-degeneracy conditions, and 3) our recovery guarantees for the memory-augmented HMM are stronger than for the vanilla HMM because task-relevant information is easier to recover from the long-term memory. Experiments on synthetically generated data from HMMs back our theoretical findings.

The prevalent deployment of learning from human preferences through reinforcement learning (RLHF) relies on two important approximations: the first assumes that pairwise preferences can be substituted with pointwise rewards. The second assumes that a reward model trained on these pointwise rewards can generalize from collected data to out-of-distribution data sampled by the policy. Recently, Direct Preference Optimisation (DPO) has been proposed as an approach that bypasses the second approximation and learn directly a policy from collected data without the reward modelling stage. However, this method still heavily relies on the first approximation. In this paper we try to gain a deeper theoretical understanding of these practical algorithms. In particular we derive a new general objective called PsiPO for learning from human preferences that is expressed in terms of pairwise preferences and therefore bypasses both approximations. This new general objective allows us to perform an in-depth analysis of the behavior of RLHF and DPO (as special cases of PsiPO) and to identify their potential pitfalls. We then consider another special case for PsiPO by setting Psi simply to Identity, for which we can derive an efficient optimisation procedure, prove performance guarantees and demonstrate its empirical superiority to DPO on some illustrative examples.

Designing a high-efficiency and high-quality expressive network architecture has always been the most important research topic in the field of deep learning. Most of today's network design strategies focus on how to integrate features extracted from different layers, and how to design computing units to effectively extract these features, thereby enhancing the expressiveness of the network. This paper proposes a new network design strategy, i.e., to design the network architecture based on gradient path analysis. On the whole, most of today's mainstream network design strategies are based on feed forward path, that is, the network architecture is designed based on the data path. In this paper, we hope to enhance the expressive ability of the trained model by improving the network learning ability. Due to the mechanism driving the network parameter learning is the backward propagation algorithm, we design network design strategies based on back propagation path. We propose the gradient path design strategies for the layer-level, the stage-level, and the network-level, and the design strategies are proved to be superior and feasible from theoretical analysis and experiments.

Recent empirical evidence indicates that transformer based in-context learning performs better when using a prefix language model (prefixLM), in which in-context samples can all attend to each other, compared to causal language models (causalLM), which use auto-regressive attention that prohibits in-context samples to attend to future samples. While this result is intuitive, it is not understood from a theoretical perspective. In this paper we take a theoretical approach and analyze the convergence behavior of prefixLM and causalLM under a certain parameter construction. Our analysis shows that both LM types converge to their stationary points at a linear rate, but that while prefixLM converges to the optimal solution of linear regression, causalLM convergence dynamics follows that of an online gradient descent algorithm, which is not guaranteed to be optimal even as the number of samples grows infinitely. We supplement our theoretical claims with empirical experiments over synthetic and real tasks and using various types of transformers. Our experiments verify that causalLM consistently underperforms prefixLM in all settings.

We provide a theoretical framework for Reinforcement Learning with Human Feedback (RLHF). Our analysis shows that when the true reward function is linear, the widely used maximum likelihood estimator (MLE) converges under both the Bradley-Terry-Luce (BTL) model and the Plackett-Luce (PL) model. However, we show that when training a policy based on the learned reward model, MLE fails while a pessimistic MLE provides policies with improved performance under certain coverage assumptions. Additionally, we demonstrate that under the PL model, the true MLE and an alternative MLE that splits the K-wise comparison into pairwise comparisons both converge. Moreover, the true MLE is asymptotically more efficient. Our results validate the empirical success of existing RLHF algorithms in InstructGPT and provide new insights for algorithm design. Furthermore, our results unify the problem of RLHF and max-entropy Inverse Reinforcement Learning (IRL), and provide the first sample complexity bound for max-entropy IRL.

Graph neural networks (GNNs) are the most widely adopted model in graph-structured data oriented learning and representation. Despite their extraordinary success in real-world applications, understanding their working mechanism by theory is still on primary stage. In this paper, we move towards this goal from the perspective of generalization. To be specific, we first establish high probability bounds of generalization gap and gradients in transductive learning with consideration of stochastic optimization. After that, we provide high probability bounds of generalization gap for popular GNNs. The theoretical results reveal the architecture specific factors affecting the generalization gap. Experimental results on benchmark datasets show the consistency between theoretical results and empirical evidence. Our results provide new insights in understanding the generalization of GNNs.

Federated Averaging (FedAvg) and its variants are the most popular optimization algorithms in federated learning (FL). Previous convergence analyses of FedAvg either assume full client participation or partial client participation where the clients can be uniformly sampled. However, in practical cross-device FL systems, only a subset of clients that satisfy local criteria such as battery status, network connectivity, and maximum participation frequency requirements (to ensure privacy) are available for training at a given time. As a result, client availability follows a natural cyclic pattern. We provide (to our knowledge) the first theoretical framework to analyze the convergence of FedAvg with cyclic client participation with several different client optimizers such as GD, SGD, and shuffled SGD. Our analysis discovers that cyclic client participation can achieve a faster asymptotic convergence rate than vanilla FedAvg with uniform client participation under suitable conditions, providing valuable insights into the design of client sampling protocols.

This paper investigates conservative exploration in reinforcement learning where the performance of the learning agent is guaranteed to be above a certain threshold throughout the learning process. It focuses on the tabular episodic Markov Decision Process (MDP) setting that has finite states and actions. With the knowledge of an existing safe baseline policy, an algorithm termed as StepMix is proposed to balance the exploitation and exploration while ensuring that the conservative constraint is never violated in each episode with high probability. StepMix features a unique design of a mixture policy that adaptively and smoothly interpolates between the baseline policy and the optimistic policy. Theoretical analysis shows that StepMix achieves near-optimal regret order as in the constraint-free setting, indicating that obeying the stringent episode-wise conservative constraint does not compromise the learning performance. Besides, a randomization-based EpsMix algorithm is also proposed and shown to achieve the same performance as StepMix. The algorithm design and theoretical analysis are further extended to the setting where the baseline policy is not given a priori but must be learned from an offline dataset, and it is proved that similar conservative guarantee and regret can be achieved if the offline dataset is sufficiently large. Experiment results corroborate the theoretical analysis and demonstrate the effectiveness of the proposed conservative exploration strategies.

We analyze continual learning on a sequence of separable linear classification tasks with binary labels. We show theoretically that learning with weak regularization reduces to solving a sequential max-margin problem, corresponding to a special case of the Projection Onto Convex Sets (POCS) framework. We then develop upper bounds on the forgetting and other quantities of interest under various settings with recurring tasks, including cyclic and random orderings of tasks. We discuss several practical implications to popular training practices like regularization scheduling and weighting. We point out several theoretical differences between our continual classification setting and a recently studied continual regression setting.

We consider the block coordinate descent methods of Gauss-Seidel type with proximal regularization (BCD-PR), which is a classical method of minimizing general nonconvex objectives under constraints that has a wide range of practical applications. We theoretically establish the worst-case complexity bound for this algorithm. Namely, we show that for general nonconvex smooth objectives with block-wise constraints, the classical BCD-PR algorithm converges to an epsilon-stationary point within O(1/epsilon) iterations. Under a mild condition, this result still holds even if the algorithm is executed inexactly in each step. As an application, we propose a provable and efficient algorithm for `Wasserstein CP-dictionary learning', which seeks a set of elementary probability distributions that can well-approximate a given set of d-dimensional joint probability distributions. Our algorithm is a version of BCD-PR that operates in the dual space, where the primal problem is regularized both entropically and proximally.

Despite the great success of Transformer networks in various applications such as natural language processing and computer vision, their theoretical aspects are not well understood. In this paper, we study the approximation and estimation ability of Transformers as sequence-to-sequence functions with infinite dimensional inputs. Although inputs and outputs are both infinite dimensional, we show that when the target function has anisotropic smoothness, Transformers can avoid the curse of dimensionality due to their feature extraction ability and parameter sharing property. In addition, we show that even if the smoothness changes depending on each input, Transformers can estimate the importance of features for each input and extract important features dynamically. Then, we proved that Transformers achieve similar convergence rate as in the case of the fixed smoothness. Our theoretical results support the practical success of Transformers for high dimensional data.

Macro-AUC is the arithmetic mean of the class-wise AUCs in multi-label learning and is commonly used in practice. However, its theoretical understanding is far lacking. Toward solving it, we characterize the generalization properties of various learning algorithms based on the corresponding surrogate losses w.r.t. Macro-AUC. We theoretically identify a critical factor of the dataset affecting the generalization bounds: the label-wise class imbalance. Our results on the imbalance-aware error bounds show that the widely-used univariate loss-based algorithm is more sensitive to the label-wise class imbalance than the proposed pairwise and reweighted loss-based ones, which probably implies its worse performance. Moreover, empirical results on various datasets corroborate our theory findings. To establish it, technically, we propose a new (and more general) McDiarmid-type concentration inequality, which may be of independent interest.

Reliable application of machine learning is of primary importance to the practical deployment of deep learning methods. A fundamental challenge is that models are often unreliable due to overconfidence. In this paper, we estimate a model's reliability by measuring the agreement between its latent space, and the latent space of a foundation model. However, it is challenging to measure the agreement between two different latent spaces due to their incoherence, \eg, arbitrary rotations and different dimensionality. To overcome this incoherence issue, we design a neighborhood agreement measure between latent spaces and find that this agreement is surprisingly well-correlated with the reliability of a model's predictions. Further, we show that fusing neighborhood agreement into a model's predictive confidence in a post-hoc way significantly improves its reliability. Theoretical analysis and extensive experiments on failure detection across various datasets verify the effectiveness of our method on both in-distribution and out-of-distribution settings.

Continual learning (CL), which aims to learn a sequence of tasks, has attracted significant recent attention. However, most work has focused on the experimental performance of CL, and theoretical studies of CL are still limited. In particular, there is a lack of understanding on what factors are important and how they affect "catastrophic forgetting" and generalization performance. To fill this gap, our theoretical analysis, under overparameterized linear models, provides the first-known explicit form of the expected forgetting and generalization error. Further analysis of such a key result yields a number of theoretical explanations about how overparameterization, task similarity, and task ordering affect both forgetting and generalization error of CL. More interestingly, by conducting experiments on real datasets using deep neural networks (DNNs), we show that some of these insights even go beyond the linear models and can be carried over to practical setups. In particular, we use concrete examples to show that our results not only explain some interesting empirical observations in recent studies, but also motivate better practical algorithm designs of CL.

Message Passing Neural Networks (MPNNs) are instances of Graph Neural Networks that leverage the graph to send messages over the edges. This inductive bias leads to a phenomenon known as over-squashing, where a node feature is insensitive to information contained at distant nodes. Despite recent methods introduced to mitigate this issue, an understanding of the causes for over-squashing and of possible solutions are lacking. In this theoretical work, we prove that: (i) Neural network width can mitigate over-squashing, but at the cost of making the whole network more sensitive; (ii) Conversely, depth cannot help mitigate over-squashing: increasing the number of layers leads to over-squashing being dominated by vanishing gradients; (iii) The graph topology plays the greatest role, since over-squashing occurs between nodes at high commute (access) time. Our analysis provides a unified framework to study different recent methods introduced to cope with over-squashing and serves as a justification for a class of methods that fall under graph rewiring.

With infinitely many high-quality data points, infinite computational power, an infinitely large foundation model with a perfect training algorithm and guaranteed zero generalization error on the pretext task, can the model be used for everything? This question cannot be answered by the existing theory of representation, optimization or generalization, because the issues they mainly investigate are assumed to be nonexistent here. In this paper, we show that category theory provides powerful machinery to answer this question. We have proved three results. The first one limits the power of prompt-based learning, saying that the model can solve a downstream task with prompts if and only if the task is representable. The second one says fine tuning does not have this limit, as a foundation model with the minimum required power (up to symmetry) can theoretically solve downstream tasks for the category defined by pretext task, with fine tuning and enough resources. Our final result can be seen as a new type of generalization theorem, showing that the foundation model can generate unseen objects from the target category (e.g., images) using the structural information from the source category (e.g., texts). Along the way, we provide a categorical framework for supervised and self-supervised learning, which might be of independent interest.

We theoretically study the impact of differential privacy on fairness in classification. We prove that, given a class of models, popular group fairness measures are pointwise Lipschitz-continuous with respect to the parameters of the model. This result is a consequence of a more general statement on accuracy conditioned on an arbitrary event (such as membership to a sensitive group), which may be of independent interest. We use the aforementioned Lipschitz property to prove a high probability bound showing that, given enough examples, the fairness level of private models is close to the one of their non-private counterparts.

Differential privacy (DP) is by far the most widely accepted framework for mitigating privacy risks in machine learning. However, exactly how small the privacy parameter epsilon needs to be to protect against certain privacy risks in practice is still not well-understood. In this work, we study data reconstruction attacks for discrete data and analyze it under the framework of multiple hypothesis testing. We utilize different variants of the celebrated Fano's inequality to derive upper bounds on the inferential power of a data reconstruction adversary when the model is trained differentially privately. Importantly, we show that if the underlying private data takes values from a set of size M, then the target privacy parameter epsilon can be O(log M) before the adversary gains significant inferential power. Our analysis offers theoretical evidence for the empirical effectiveness of DP against data reconstruction attacks even at relatively large values of epsilon.

Diffusion models are a class of generative models that learn to synthesize samples by inverting a diffusion process that gradually maps data into noise. While these models have enjoyed great success recently, a full theoretical understanding of their observed properties is still lacking, in particular, their weak sensitivity to the choice of noise family and the role of adequate scheduling of noise levels for good synthesis. By identifying a correspondence between diffusion models and a well-known paradigm in cognitive science known as serial reproduction, whereby human agents iteratively observe and reproduce stimuli from memory, we show how the aforementioned properties of diffusion models can be explained as a natural consequence of this correspondence. We then complement our theoretical analysis with simulations that exhibit these key features. Our work highlights how classic paradigms in cognitive science can shed light on state-of-the-art machine learning problems.

We identify and formalize a fundamental gradient descent phenomenon resulting in a learning proclivity in over-parameterized neural networks. Gradient Starvation arises when cross-entropy loss is minimized by capturing only a subset of features relevant for the task, despite the presence of other predictive features that fail to be discovered. This work provides a theoretical explanation for the emergence of such feature imbalance in neural networks. Using tools from Dynamical Systems theory, we identify simple properties of learning dynamics during gradient descent that lead to this imbalance, and prove that such a situation can be expected given certain statistical structure in training data. Based on our proposed formalism, we develop guarantees for a novel regularization method aimed at decoupling feature learning dynamics, improving accuracy and robustness in cases hindered by gradient starvation. We illustrate our findings with simple and real-world out-of-distribution (OOD) generalization experiments.

We present new intuitions and theoretical assessments of the emergence of disentangled representation in variational autoencoders. Taking a rate-distortion theory perspective, we show the circumstances under which representations aligned with the underlying generative factors of variation of data emerge when optimising the modified ELBO bound in beta-VAE, as training progresses. From these insights, we propose a modification to the training regime of beta-VAE, that progressively increases the information capacity of the latent code during training. This modification facilitates the robust learning of disentangled representations in beta-VAE, without the previous trade-off in reconstruction accuracy.