Jiayun Wu

Jiashuo Liu, Jiayun Wu, Jie Peng et al.

Enhancing the stability of machine learning algorithms under distributional shifts is at the heart of the Out-of-Distribution (OOD) Generalization problem. Derived from causal learning, recent works of invariant learning pursue strict invariance with multiple training environments. Although intuitively reasonable, strong assumptions on the availability and quality of environments are made to learn the strict invariance property. In this work, we come up with the ``distributional stability" notion to mitigate such limitations. It quantifies the stability of prediction mechanisms among sub-populations down to a prescribed scale. Based on this, we propose the learnability assumption and derive the generalization error bound under distribution shifts. Inspired by theoretical analyses, we propose our novel stable risk minimization (SRM) algorithm to enhance the model's stability w.r.t. shifts in prediction mechanisms ($Y|X$-shifts). Experimental results are consistent with our intuition and validate the effectiveness of our algorithm. The code can be found at https://github.com/LJSthu/SRM.

Jiayun Wu, Tao Jia, Yansong Wang et al. · pku

Temporal networks are suitable for modeling complex evolving systems. It has a wide range of applications, such as social network analysis, recommender systems, and epidemiology. Recently, modeling such dynamic systems has drawn great attention in many domains. However, most existing approaches resort to taking discrete snapshots of the temporal networks and modeling all events with equal importance. This paper proposes Significant Ties Graph Neural Networks (STGNN), a novel framework that captures and describes significant ties. To better model the diversity of interactions, STGNN introduces a novel aggregation mechanism to organize the most significant historical neighbors' information and adaptively obtain the significance of node pairs. Experimental results on four real networks demonstrate the effectiveness of the proposed framework.

Shuo Han, Jiacheng Liu, Jiayun Wu et al. · pku

Graph Representation Learning (GRL) is an influential methodology, enabling a more profound understanding of graph-structured data and aiding graph clustering, a critical task across various domains. The recent incursion of attention mechanisms, originally an artifact of Natural Language Processing (NLP), into the realm of graph learning has spearheaded a notable shift in research trends. Consequently, Graph Attention Networks (GATs) and Graph Attention Auto-Encoders have emerged as preferred tools for graph clustering tasks. Yet, these methods primarily employ a local attention mechanism, thereby curbing their capacity to apprehend the intricate global dependencies between nodes within graphs. Addressing these impediments, this study introduces an innovative method known as the Graph Transformer Auto-Encoder for Graph Clustering (GTAGC). By melding the Graph Auto-Encoder with the Graph Transformer, GTAGC is adept at capturing global dependencies between nodes. This integration amplifies the graph representation and surmounts the constraints posed by the local attention mechanism. The architecture of GTAGC encompasses graph embedding, integration of the Graph Transformer within the autoencoder structure, and a clustering component. It strategically alternates between graph embedding and clustering, thereby tailoring the Graph Transformer for clustering tasks, whilst preserving the graph's global structural information. Through extensive experimentation on diverse benchmark datasets, GTAGC has exhibited superior performance against existing state-of-the-art graph clustering methodologies.

Jiashuo Liu, Jiayun Wu, Bo Li et al.

As an intrinsic and fundamental property of big data, data heterogeneity exists in a variety of real-world applications, such as precision medicine, autonomous driving, financial applications, etc. For machine learning algorithms, the ignorance of data heterogeneity will greatly hurt the generalization performance and the algorithmic fairness, since the prediction mechanisms among different sub-populations are likely to differ from each other. In this work, we focus on the data heterogeneity that affects the prediction of machine learning models, and firstly propose the \emph{usable predictive heterogeneity}, which takes into account the model capacity and computational constraints. We prove that it can be reliably estimated from finite data with probably approximately correct (PAC) bounds. Additionally, we design a bi-level optimization algorithm to explore the usable predictive heterogeneity from data. Empirically, the explored heterogeneity provides insights for sub-population divisions in income prediction, crop yield prediction and image classification tasks, and leveraging such heterogeneity benefits the out-of-distribution generalization performance.

Yong Lin, Shange Tang, Bohan Lyu et al.

We introduce Goedel-Prover, an open-source language model that achieves state-of-the-art (as of April 5 2025) performance in automated formal proof generation for mathematical problems. A key challenge in this field is the scarcity of formalized mathematical statements and proofs, which we address through the following approaches. First, we train LLMs to convert natural language math problems from the Numina dataset to equivalent formal statements in Lean 4. This process creates the dataset Goedel-Pset-v1, which includes 1.64 million formal statements. Next, we develop a large dataset of formal proofs by training a series of provers. Each new prover can prove many statements that previous ones could not, and these new proofs are added to the training set for the next prover. Finally, we obtain the dataset Goedel-Pset-v1-solved, which contains proofs for over 800K statements from Goedel-Pset-v1. Supervised fine-tuning (SFT) of DeepSeek-Prover-V1.5-Base on Goedel-Pset-v1-solved (i.e., no RL) yields a Goedel-Prover-SFT that achieves a success rate of 57.6% (Pass@32) on miniF2F, surpassing the previous leader DeepSeek-Prover-V1.5-RL (trained using SFT + RL on a proprietary dataset) by 7.6%. On PutnamBench, Goedel-Prover-SFT successfully solves 7 problems (Pass@512), ranking first on the leaderboard. We provide extensive discussion of our training methodology, highlighting the key design choices that contribute to Goedel-Prover's strong performance. Further RL training (including DPO) improves Goedel-Prover-SFT's success rate to over 60% (Pass@32) on miniF2F. To aid future research, we provide extensive discussion of our training methodology and design choices. We also fully open-source our codes, models, and datasets. Additionally, we open-source formal proofs for 29.7K problems in Lean Workbook, nearly doubling the 15.7K solved by prior provers.

Jingchu Gai, Nai-Chieh Huang, Jiayun Wu

The residual update of a pre-norm Transformer layer admits an interpretation as one step of a first-order optimizer acting on a surrogate token energy, wherein the attention and MLP sublayers function as gradient oracles. Based on this observation, we build a family of optimizer-inspired Transformers (triple-momentum, Adam/AdamW, Muon, SOAP) and compare them under matched compute. In our main pretraining experiment, the triple-momentum TMMFormer achieves the lowest validation loss, outperforming the vanilla Transformer and prior architectural variants. A controlled ablation and supporting theory show that momentum, not preconditioning, is the main source of the gain. We further show that TMMFormer and other momentum-based designs reach flatter minima than the vanilla Transformer, which leads to less forgetting and better generalization.

Jiashuo Liu, Jiayun Wu, Tianyu Wang et al.

Machine learning algorithms minimizing average risk are susceptible to distributional shifts. Distributionally Robust Optimization (DRO) addresses this issue by optimizing the worst-case risk within an uncertainty set. However, DRO suffers from over-pessimism, leading to low-confidence predictions, poor parameter estimations as well as poor generalization. In this work, we conduct a theoretical analysis of a probable root cause of over-pessimism: excessive focus on noisy samples. To alleviate the impact of noise, we incorporate data geometry into calibration terms in DRO, resulting in our novel Geometry-Calibrated DRO (GCDRO) for regression. We establish the connection between our risk objective and the Helmholtz free energy in statistical physics, and this free-energy-based risk can extend to standard DRO methods. Leveraging gradient flow in Wasserstein space, we develop an approximate minimax optimization algorithm with a bounded error ratio and elucidate how our approach mitigates noisy sample effects. Comprehensive experiments confirm GCDRO's superiority over conventional DRO methods.

Yong Lin, Shange Tang, Bohan Lyu et al. · uw

We introduce Goedel-Prover-V2, a series of open-source language models that set a new state-of-the-art in automated theorem proving. Built on the standard expert iteration and reinforcement learning pipeline, our approach incorporates three key innovations: (1) Scaffolded data synthesis: We generate synthetic tasks of increasing difficulty to train the model to master increasingly complex theorems; (2) Verifier-guided self-correction: We enable the model to iteratively revise its proofs by leveraging feedback from the Lean compiler; (3) Model averaging: We merge model checkpoints to mitigate the decrease in model output diversity in later stages of training. Our small model, Goedel-Prover-V2-8B, reaches 84.6% pass@32 on MiniF2F and outperforms DeepSeek-Prover-V2-671B under the same metric, despite being 80X smaller. Our flagship model, Goedel-Prover-V2-32B, achieves 88.1% on MiniF2F at pass@32 in standard mode and 90.4% in self-correction mode, outperforming prior SOTA by a large margin. Additionally, our flagship model solves 86 problems on PutnamBench at pass@184, securing the first place among open-source models on the leaderboard, surpassing DeepSeek-Prover-V2-671B's record of solving 47 problems by pass@1024 with a significantly smaller model size and compute budget. At the time of its release (July-August 2025), Goedel-Prover-V2 achieves the strongest overall performance among all open-source theorem provers. It also ranks among the top-performing models--including closed-source systems with publicly reported performance--under a constrained test-time compute budget. Our models, code, and data are released at https://github.com/Goedel-LM/Goedel-Prover-V2.

Jiashuo Liu, Jiayun Wu, Chunjie Wu et al.

The rapid proliferation of Large Language Models (LLMs) and diverse specialized benchmarks necessitates a shift from fragmented, task-specific metrics to a holistic, competitive ranking system that effectively aggregates performance across multiple ability dimensions. Primarily using static scoring, current evaluation methods are fundamentally limited. They struggle to determine the proper mix ratio across diverse benchmarks, and critically, they fail to capture a model's dynamic competitive fitness or its vulnerability when confronted with sequential, high-stakes tasks. To address this, we introduce the novel Competitive Swiss-System Dynamics (CSD) framework. CSD simulates a multi-round, sequential contest where models are dynamically paired across a curated sequence of benchmarks based on their accumulated win-loss record. And Monte Carlo Simulation ($N=100,000$ iterations) is used to approximate the statistically robust Expected Win Score ($E[S_m]$), which eliminates the noise of random pairing and early-round luck. Furthermore, we implement a Failure Sensitivity Analysis by parameterizing the per-round elimination quantity ($T_k$), which allows us to profile models based on their risk appetite--distinguishing between robust generalists and aggressive specialists. We demonstrate that CSD provides a more nuanced and context-aware ranking than traditional aggregate scoring and static pairwise models, representing a vital step towards risk-informed, next-generation LLM evaluation.

Jiayun Wu, Peixu Hou, Shan Qu et al.

Reward models (RMs) are critical for aligning Large Language Models via Reinforcement Learning from Human Feedback (RLHF). While Generative Reward Models (GRMs) achieve superior accuracy through chain-of-thought (CoT) reasoning, they incur substantial computational costs. Conversely, Scalar Reward Models (SRMs) offer efficiency but suffer from limited performance and adaptability in complex scenarios. We introduce Fast-Slow Thinking Reward Models (F/S-RM), a hybrid RM architecture inspired by Dual Process Theory. It trains a single model to integrate two distinct reward paradigms: first-token prediction as a scalar score (fast thinking) and CoT-based judgment (slow thinking), regulated by a dual-confidence activation mechanism that determines when to activate slow thinking. F/S-RM achieves a 1.2% relative performance improvement over state-of-the-art models while reducing token consumption by 20.8%. Code and data will be publicly available.

Jiayun Wu, Jiashuo Liu, Zhiyuan Zeng et al.

LLM deployment in critical domains is currently impeded by persistent hallucinations--generating plausible but factually incorrect assertions. While scaling laws drove significant improvements in general capabilities, theoretical frameworks suggest hallucination is not merely stochastic error but a predictable statistical consequence of training objectives prioritizing mimicking data distribution over epistemic honesty. Standard RLVR paradigms, utilizing binary reward signals, inadvertently incentivize models as good test-takers rather than honest communicators, encouraging guessing whenever correctness probability exceeds zero. This paper presents an exhaustive investigation into behavioral calibration, which incentivizes models to stochastically admit uncertainty by abstaining when not confident, aligning model behavior with accuracy. Synthesizing recent advances, we propose and evaluate training interventions optimizing strictly proper scoring rules for models to output a calibrated probability of correctness. Our methods enable models to either abstain from producing a complete response or flag individual claims where uncertainty remains. Utilizing Qwen3-4B-Instruct, empirical analysis reveals behavior-calibrated reinforcement learning allows smaller models to surpass frontier models in uncertainty quantification--a transferable meta-skill decouplable from raw predictive accuracy. Trained on math reasoning tasks, our model's log-scale Accuracy-to-Hallucination Ratio gain (0.806) exceeds GPT-5's (0.207) in a challenging in-domain evaluation (BeyondAIME). Moreover, in cross-domain factual QA (SimpleQA), our 4B LLM achieves zero-shot calibration error on par with frontier models including Grok-4 and Gemini-2.5-Pro, even though its factual accuracy is much lower.

Han Yu, Jiashuo Liu, Xingxuan Zhang et al.

Machine learning models, while progressively advanced, rely heavily on the IID assumption, which is often unfulfilled in practice due to inevitable distribution shifts. This renders them susceptible and untrustworthy for deployment in risk-sensitive applications. Such a significant problem has consequently spawned various branches of works dedicated to developing algorithms capable of Out-of-Distribution (OOD) generalization. Despite these efforts, much less attention has been paid to the evaluation of OOD generalization, which is also a complex and fundamental problem. Its goal is not only to assess whether a model's OOD generalization capability is strong or not, but also to evaluate where a model generalizes well or poorly. This entails characterizing the types of distribution shifts that a model can effectively address, and identifying the safe and risky input regions given a model. This paper serves as the first effort to conduct a comprehensive review of OOD evaluation. We categorize existing research into three paradigms: OOD performance testing, OOD performance prediction, and OOD intrinsic property characterization, according to the availability of test data. Additionally, we briefly discuss OOD evaluation in the context of pretrained models. In closing, we propose several promising directions for future research in OOD evaluation.

Bing Yuan, Zhang Jiang, Aobo Lyu et al.

Emergence and causality are two fundamental concepts for understanding complex systems. They are interconnected. On one hand, emergence refers to the phenomenon where macroscopic properties cannot be solely attributed to the cause of individual properties. On the other hand, causality can exhibit emergence, meaning that new causal laws may arise as we increase the level of abstraction. Causal emergence theory aims to bridge these two concepts and even employs measures of causality to quantify emergence. This paper provides a comprehensive review of recent advancements in quantitative theories and applications of causal emergence. Two key problems are addressed: quantifying causal emergence and identifying it in data. Addressing the latter requires the use of machine learning techniques, thus establishing a connection between causal emergence and artificial intelligence. We highlighted that the architectures used for identifying causal emergence are shared by causal representation learning, causal model abstraction, and world model-based reinforcement learning. Consequently, progress in any of these areas can benefit the others. Potential applications and future perspectives are also discussed in the final section of the review.

Xingxuan Zhang, Gang Ren, Han Yu et al.

We argue that progress toward general intelligence requires complementary foundation models grounded in language, the physical world, and structured data. This report presents LimiX-16M and LimiX-2M, two instantiations of our large structured-data models (LDMs). Both models treat structured data as a joint distribution over variables and missingness, thus capable of addressing a wide range of tabular tasks through query-based conditional prediction via a single model. They are pretrained using masked joint-distribution modeling with an episodic, context-conditional objective, supporting rapid, training-free adaptation at inference. We evaluate LimiX models across 11 large structured-data benchmarks with broad regimes of sample size, feature dimensionality, class number, categorical-to-numerical feature ratio, missingness, and sample-to-feature ratios. LimiX-16M consistently surpasses strong baselines, as shown in Figure 1 and Figure 2. The superiority holds across a wide range of tasks, such as classification, regression, missing value imputation, and data generation, often by substantial margins, while avoiding task-specific architectures or bespoke training per task. Notably, LimiX-2M delivers strong results under tight compute and memory budgets. We also present the first scaling law study for LDMs, revealing how data and model scaling jointly influence downstream performance and offering quantitative guidance for tabular foundation modeling. All LimiX models are publicly accessible under Apache 2.0.

Jingwu Tang, Jiayun Wu, Zhiwei Steven Wu et al.

When model predictions inform downstream decision making, a natural question is under what conditions can the decision-makers simply respond to the predictions as if they were the true outcomes. Calibration suffices to guarantee that simple best-response to predictions is optimal. However, calibration for high-dimensional prediction outcome spaces requires exponential computational and statistical complexity. The recent relaxation known as decision calibration ensures the optimality of the simple best-response rule while requiring only polynomial sample complexity in the dimension of outcomes. However, known results on calibration and decision calibration crucially rely on linear loss functions for establishing best-response optimality. A natural approach to handle nonlinear losses is to map outcomes $y$ into a feature space $φ(y)$ of dimension $m$, then approximate losses with linear functions of $φ(y)$. Unfortunately, even simple classes of nonlinear functions can demand exponentially large or infinite feature dimensions $m$. A key open problem is whether it is possible to achieve decision calibration with sample complexity independent of~$m$. We begin with a negative result: even verifying decision calibration under standard deterministic best response inherently requires sample complexity polynomial in~$m$. Motivated by this lower bound, we investigate a smooth version of decision calibration in which decision-makers follow a smooth best-response. This smooth relaxation enables dimension-free decision calibration algorithms. We introduce algorithms that, given $\mathrm{poly}(|A|,1/ε)$ samples and any initial predictor~$p$, can efficiently post-process it to satisfy decision calibration without worsening accuracy. Our algorithms apply broadly to function classes that can be well-approximated by bounded-norm functions in (possibly infinite-dimensional) separable RKHS.

Konstantina Bairaktari, Jiayun Wu, Zhiwei Steven Wu

Conformal prediction is a powerful distribution-free framework for constructing prediction sets with coverage guarantees. Classical methods, such as split conformal prediction, provide marginal coverage, ensuring that the prediction set contains the label of a random test point with a target probability. However, these guarantees may not hold uniformly across different subpopulations, leading to disparities in coverage. Prior work has explored coverage guarantees conditioned on events related to the covariates and label of the test point. We present Kandinsky conformal prediction, a framework that significantly expands the scope of conditional coverage guarantees. In contrast to Mondrian conformal prediction, which restricts its coverage guarantees to disjoint groups -- reminiscent of the rigid, structured grids of Piet Mondrian's art -- our framework flexibly handles overlapping and fractional group memberships defined jointly on covariates and labels, reflecting the layered, intersecting forms in Wassily Kandinsky's compositions. Our algorithm unifies and extends existing methods, encompassing covariate-based group conditional, class conditional, and Mondrian conformal prediction as special cases, while achieving a minimax-optimal high-probability conditional coverage bound. Finally, we demonstrate the practicality of our approach through empirical evaluation on real-world datasets.

Chenxi Zhang, Qing Zhang, Jiayun Wu et al.

Camouflaged Object Detection (COD) aims to identify objects that blend seamlessly into their surroundings. The inherent visual complexity of camouflaged objects, including their low contrast with the background, diverse textures, and subtle appearance variations, often obscures semantic cues, making accurate segmentation highly challenging. Existing methods primarily rely on visual features, which are insufficient to handle the variability and intricacy of camouflaged objects, leading to unstable object perception and ambiguous segmentation results. To tackle these limitations, we introduce a novel task, class-guided camouflaged object detection (CGCOD), which extends traditional COD task by incorporating object-specific class knowledge to enhance detection robustness and accuracy. To facilitate this task, we present a new dataset, CamoClass, comprising real-world camouflaged objects with class annotations. Furthermore, we propose a multi-stage framework, CGNet, which incorporates a plug-and-play class prompt generator and a simple yet effective class-guided detector. This establishes a new paradigm for COD, bridging the gap between contextual understanding and class-guided detection. Extensive experimental results demonstrate the effectiveness of our flexible framework in improving the performance of proposed and existing detectors by leveraging class-level textual information.

Shange Tang, Jiayun Wu, Jianqing Fan et al.

Benign overfitting refers to the phenomenon where an over-parameterized model fits the training data perfectly, including noise in the data, but still generalizes well to the unseen test data. While prior work provides some theoretical understanding of this phenomenon under the in-distribution setup, modern machine learning often operates in a more challenging Out-of-Distribution (OOD) regime, where the target (test) distribution can be rather different from the source (training) distribution. In this work, we take an initial step towards understanding benign overfitting in the OOD regime by focusing on the basic setup of over-parameterized linear models under covariate shift. We provide non-asymptotic guarantees proving that benign overfitting occurs in standard ridge regression, even under the OOD regime when the target covariance satisfies certain structural conditions. We identify several vital quantities relating to source and target covariance, which govern the performance of OOD generalization. Our result is sharp, which provably recovers prior in-distribution benign overfitting guarantee [Tsigler and Bartlett, 2023], as well as under-parameterized OOD guarantee [Ge et al., 2024] when specializing to each setup. Moreover, we also present theoretical results for a more general family of target covariance matrix, where standard ridge regression only achieves a slow statistical rate of $O(1/\sqrt{n})$ for the excess risk, while Principal Component Regression (PCR) is guaranteed to achieve the fast rate $O(1/n)$, where $n$ is the number of samples.

Chenxi Zhang, Jiayun Wu, Qing Zhang et al.

Camouflaged object detection (COD) primarily focuses on learning subtle yet discriminative representations from complex scenes. Existing methods predominantly follow the parametric feedforward architecture based on static visual representation modeling. However, they lack explicit mechanisms for acquiring historical context, limiting their adaptation and effectiveness in handling challenging camouflage scenes. In this paper, we propose a recall-augmented COD architecture, namely RetroMem, which dynamically modulates camouflage pattern perception and inference by integrating relevant historical knowledge into the process. Specifically, RetroMem employs a two-stage training paradigm consisting of a learning stage and a recall stage to construct, update, and utilize memory representations effectively. During the learning stage, we design a dense multi-scale adapter (DMA) to improve the pretrained encoder's capability to capture rich multi-scale visual information with very few trainable parameters, thereby providing foundational inferences. In the recall stage, we propose a dynamic memory mechanism (DMM) and an inference pattern reconstruction (IPR). These components fully leverage the latent relationships between learned knowledge and current sample context to reconstruct the inference of camouflage patterns, thereby significantly improving the model's understanding of camouflage scenes. Extensive experiments on several widely used datasets demonstrate that our RetroMem significantly outperforms existing state-of-the-art methods.

Weihuang Zheng, Jiashuo Liu, Jiaxing Li et al.

Graph Neural Networks (GNNs) are widely used for node classification tasks but often fail to generalize when training and test nodes come from different distributions, limiting their practicality. To overcome this, recent approaches adopt invariant learning techniques from the out-of-distribution (OOD) generalization field, which seek to establish stable prediction methods across environments. However, the applicability of these invariant assumptions to graph data remains unverified, and such methods often lack solid theoretical support. In this work, we introduce the Topology-Aware Dynamic Reweighting (TAR) framework, which dynamically adjusts sample weights through gradient flow in the geometric Wasserstein space during training. Instead of relying on strict invariance assumptions, we prove that our method is able to provide distributional robustness, thereby enhancing the out-of-distribution generalization performance on graph data. By leveraging the inherent graph structure, TAR effectively addresses distribution shifts. Our framework's superiority is demonstrated through standard testing on four graph OOD datasets and three class-imbalanced node classification datasets, exhibiting marked improvements over existing methods.

Jiayun Wu, Jiashuo Liu, Peng Cui et al.

We establish a new model-agnostic optimization framework for out-of-distribution generalization via multicalibration, a criterion that ensures a predictor is calibrated across a family of overlapping groups. Multicalibration is shown to be associated with robustness of statistical inference under covariate shift. We further establish a link between multicalibration and robustness for prediction tasks both under and beyond covariate shift. We accomplish this by extending multicalibration to incorporate grouping functions that consider covariates and labels jointly. This leads to an equivalence of the extended multicalibration and invariance, an objective for robust learning in existence of concept shift. We show a linear structure of the grouping function class spanned by density ratios, resulting in a unifying framework for robust learning by designing specific grouping functions. We propose MC-Pseudolabel, a post-processing algorithm to achieve both extended multicalibration and out-of-distribution generalization. The algorithm, with lightweight hyperparameters and optimization through a series of supervised regression steps, achieves superior performance on real-world datasets with distribution shift.