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Researcher profile

Fred Zhang

Fred Zhang contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Data Structures and AlgorithmsMachine Learningmath.STStatistics TheoryArtificial Intelligence

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Published work

4 published item(s)

preprint2026arXiv

Formal Conjectures: An Open and Evolving Benchmark for Verified Discovery in Mathematics

As automated reasoning systems advance rapidly, there is a growing need for research-level formal mathematical problems to accurately evaluate their capabilities. To address this, we present Formal Conjectures, an evolving benchmark of currently 2615 mathematical problem statements formalized in Lean 4. Sourced from areas of active mathematical research, the dataset features 1029 open research conjectures providing a zero-contamination benchmark for mathematical proof discovery, and 836 solved problems for proof autoformalization. Notably, the repository provides a structured interface connecting mathematicians who formalize and clarify problems with the AI systems and humans attempting to solve them. Demonstrating its immediate utility, the benchmark has already been leveraged to make new mathematical discoveries, including the resolution of open research conjectures. We describe our approach to ensuring the correctness of these formalizations in a collaborative open-source project where contributions stem from an active community. In this framework, AI-generated proofs and disproofs serve as a valuable auditing mechanism to iteratively improve the fidelity of the benchmark. Finally, we provide a standardized evaluation setup and report baseline results on frozen evaluation subsets, demonstrating a climbable signal that measures the current frontier of automated reasoning on research-level mathematics.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Faster Fundamental Graph Algorithms via Learned Predictions

We consider the question of speeding up classic graph algorithms with machine-learned predictions. In this model, algorithms are furnished with extra advice learned from past or similar instances. Given the additional information, we aim to improve upon the traditional worst-case run-time guarantees. Our contributions are the following: (i) We give a faster algorithm for minimum-weight bipartite matching via learned duals, improving the recent result by Dinitz, Im, Lavastida, Moseley and Vassilvitskii (NeurIPS, 2021); (ii) We extend the learned dual approach to the single-source shortest path problem (with negative edge lengths), achieving an almost linear runtime given sufficiently accurate predictions which improves upon the classic fastest algorithm due to Goldberg (SIAM J. Comput., 1995); (iii) We provide a general reduction-based framework for learning-based graph algorithms, leading to new algorithms for degree-constrained subgraph and minimum-cost $0$-$1$ flow, based on reductions to bipartite matching and the shortest path problem. Finally, we give a set of general learnability theorems, showing that the predictions required by our algorithms can be efficiently learned in a PAC fashion.

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preprint2021arXiv

Robust and Heavy-Tailed Mean Estimation Made Simple, via Regret Minimization

We study the problem of estimating the mean of a distribution in high dimensions when either the samples are adversarially corrupted or the distribution is heavy-tailed. Recent developments in robust statistics have established efficient and (near) optimal procedures for both settings. However, the algorithms developed on each side tend to be sophisticated and do not directly transfer to the other, with many of them having ad-hoc or complicated analyses. In this paper, we provide a meta-problem and a duality theorem that lead to a new unified view on robust and heavy-tailed mean estimation in high dimensions. We show that the meta-problem can be solved either by a variant of the Filter algorithm from the recent literature on robust estimation or by the quantum entropy scoring scheme (QUE), due to Dong, Hopkins and Li (NeurIPS '19). By leveraging our duality theorem, these results translate into simple and efficient algorithms for both robust and heavy-tailed settings. Furthermore, the QUE-based procedure has run-time that matches the fastest known algorithms on both fronts. Our analysis of Filter is through the classic regret bound of the multiplicative weights update method. This connection allows us to avoid the technical complications in previous works and improve upon the run-time analysis of a gradient-descent-based algorithm for robust mean estimation by Cheng, Diakonikolas, Ge and Soltanolkotabi (ICML '20).

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preprint2020arXiv

A Fast Spectral Algorithm for Mean Estimation with Sub-Gaussian Rates

We study the algorithmic problem of estimating the mean of heavy-tailed random vector in $\mathbb{R}^d$, given $n$ i.i.d. samples. The goal is to design an efficient estimator that attains the optimal sub-gaussian error bound, only assuming that the random vector has bounded mean and covariance. Polynomial-time solutions to this problem are known but have high runtime due to their use of semi-definite programming (SDP). Conceptually, it remains open whether convex relaxation is truly necessary for this problem. In this work, we show that it is possible to go beyond SDP and achieve better computational efficiency. In particular, we provide a spectral algorithm that achieves the optimal statistical performance and runs in time $\widetilde O\left(n^2 d \right)$, improving upon the previous fastest runtime $\widetilde O\left(n^{3.5}+ n^2d\right)$ by Cherapanamjeri el al. (COLT '19). Our algorithm is spectral in that it only requires (approximate) eigenvector computations, which can be implemented very efficiently by, for example, power iteration or the Lanczos method. At the core of our algorithm is a novel connection between the furthest hyperplane problem introduced by Karnin et al. (COLT '12) and a structural lemma on heavy-tailed distributions by Lugosi and Mendelson (Ann. Stat. '19). This allows us to iteratively reduce the estimation error at a geometric rate using only the information derived from the top singular vector of the data matrix, leading to a significantly faster running time.

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