Jiayun Wu

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.

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.

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.

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.