Chao Gao

Hongjin Qian, Zheng Liu, Chao Gao et al.

In real-world information-seeking scenarios, users have dynamic and diverse needs, requiring RAG systems to demonstrate adaptable resilience. To comprehensively evaluate the resilience of current RAG methods, we introduce HawkBench, a human-labeled, multi-domain benchmark designed to rigorously assess RAG performance across categorized task types. By stratifying tasks based on information-seeking behaviors, HawkBench provides a systematic evaluation of how well RAG systems adapt to diverse user needs. Unlike existing benchmarks, which focus primarily on specific task types (mostly factoid queries) and rely on varying knowledge bases, HawkBench offers: (1) systematic task stratification to cover a broad range of query types, including both factoid and rationale queries, (2) integration of multi-domain corpora across all task types to mitigate corpus bias, and (3) rigorous annotation for high-quality evaluation. HawkBench includes 1,600 high-quality test samples, evenly distributed across domains and task types. Using this benchmark, we evaluate representative RAG methods, analyzing their performance in terms of answer quality and response latency. Our findings highlight the need for dynamic task strategies that integrate decision-making, query interpretation, and global knowledge understanding to improve RAG generalizability. We believe HawkBench serves as a pivotal benchmark for advancing the resilience of RAG methods and their ability to achieve general-purpose information seeking.

Chao Gao, Anderson Y. Zhang

We study the statistical estimation problem of orthogonal group synchronization and rotation group synchronization. The model is $Y_{ij} = Z_i^* Z_j^{*T} + σW_{ij}\in\mathbb{R}^{d\times d}$ where $W_{ij}$ is a Gaussian random matrix and $Z_i^*$ is either an orthogonal matrix or a rotation matrix, and each $Y_{ij}$ is observed independently with probability $p$. We analyze an iterative polar decomposition algorithm for the estimation of $Z^*$ and show it has an error of $(1+o(1))\frac{σ^2 d(d-1)}{2np}$ when initialized by spectral methods. A matching minimax lower bound is further established which leads to the optimality of the proposed algorithm as it achieves the exact minimax risk.

Chao Gao, Anderson Y. Zhang

We study the phase synchronization problem with noisy measurements $Y=z^*z^{*H}+σW\in\mathbb{C}^{n\times n}$, where $z^*$ is an $n$-dimensional complex unit-modulus vector and $W$ is a complex-valued Gaussian random matrix. It is assumed that each entry $Y_{jk}$ is observed with probability $p$. We prove that an SDP relaxation of the MLE achieves the error bound $(1+o(1))\frac{σ^2}{2np}$ under a normalized squared $\ell_2$ loss. This result matches the minimax lower bound of the problem, and even the leading constant is sharp. The analysis of the SDP is based on an equivalent non-convex programming whose solution can be characterized as a fixed point of the generalized power iteration lifted to a higher dimensional space. This viewpoint unifies the proofs of the statistical optimality of three different methods: MLE, SDP, and generalized power method. The technique is also applied to the analysis of the SDP for $\mathbb{Z}_2$ synchronization, and we achieve the minimax optimal error $\exp\left(-(1-o(1))\frac{np}{2σ^2}\right)$ with a sharp constant in the exponent.

Pinhan Chen, Chao Gao, Anderson Y. Zhang

Given partially observed pairwise comparison data generated by the Bradley-Terry-Luce (BTL) model, we study the problem of top-$k$ ranking. That is, to optimally identify the set of top-$k$ players. We derive the minimax rate with respect to a normalized Hamming loss. This provides the first result in the literature that characterizes the partial recovery error in terms of the proportion of mistakes for top-$k$ ranking. We also derive the optimal signal to noise ratio condition for the exact recovery of the top-$k$ set. The maximum likelihood estimator (MLE) is shown to achieve both optimal partial recovery and optimal exact recovery. On the other hand, we show another popular algorithm, the spectral method, is in general sub-optimal. Our results complement the recent work by Chen et al. (2019) that shows both the MLE and the spectral method achieve the optimal sample complexity for exact recovery. It turns out the leading constants of the sample complexity are different for the two algorithms. Another contribution that may be of independent interest is the analysis of the MLE without any penalty or regularization for the BTL model. This closes an important gap between theory and practice in the literature of ranking.

Chao Gao, Anderson Y. Zhang

We propose a general modeling and algorithmic framework for discrete structure recovery that can be applied to a wide range of problems. Under this framework, we are able to study the recovery of clustering labels, ranks of players, signs of regression coefficients, cyclic shifts, and even group elements from a unified perspective. A simple iterative algorithm is proposed for discrete structure recovery, which generalizes methods including Lloyd's algorithm and the power method. A linear convergence result for the proposed algorithm is established in this paper under appropriate abstract conditions on stochastic errors and initialization. We illustrate our general theory by applying it on several representative problems: (1) clustering in Gaussian mixture model, (2) approximate ranking, (3) sign recovery in compressed sensing, (4) multireference alignment, and (5) group synchronization, and show that minimax rate is achieved in each case.