arXiv subjects

Pengkun Yang

Publications and source records attributed to Pengkun Yang.

At least 19 recordsLinked to original sources

arXiv2026

Prompt Perturbation for Reliable LLM Evaluation over Comparison Graphs

Evaluating large language models (LLMs) is important for understanding their capabilities, comparing competing systems, and supporting the deployment of reliable models in practice. For open-ended tasks, pairwise evaluation has become a popular paradigm, in which two responses to the same prompt are compared and the resulting judgments are aggregated into an overall ranking. A central challenge of this paradigm is intransitivity: the induced comparison outcomes may fail to support any coherent global ranking. For example, one may observe cyclic preferences such as $A \succ B \succ C \succ A$, or inconsistencies involving ties such as $A \equiv B\equiv C\neq A$. Such contradictions make the resulting leaderboard unstable and challenging to interpret. In this paper, we propose a prompt perturbation framework for improving the consistency of pairwise LLM evaluation. Our approach generates perturbed variants of each prompt, uses the resulting comparison graphs to identify and filter out structurally inconsistent comparison patterns, and then applies standard ranking methods to the filtered comparisons. A key feature of the proposed framework is that graph-level structural consistency is incorporated explicitly into the evaluation pipeline before ranking aggregation. This provides a simple and principled way to reduce cyclic inconsistencies and improve the reliability of LLM rankings.

cs.CL↗

arXiv2026

Collaborative Yet Personalized Policy Training: Single-Timescale Federated Actor-Critic

Despite the popularity of the actor-critic method and the practical needs of collaborative policy training, existing works typically either overlook environmental heterogeneity or give up personalization altogether by training a single shared policy across all agents. We consider a federated actor-critic framework in which agents share a common linear subspace representation while maintaining personalized local policy components, and agents iteratively estimate the common subspace, local critic heads, and local policies (i.e., actors). Under canonical single-timescale updates with Markovian sampling, we establish finite-time convergence via a novel joint linear approximation framework. Specifically, we show that the critic error converges to zero at the rate of $\tilde{\mathcal{O}}(1/((1-\gamma)^4\sqrt{TK}))$, and the policy gradient norm converges to zero at the rate of $\tilde{\mathcal{O}}(1/((1-\gamma)^6\sqrt{TK}))$, where $T$ is the number of rounds, $K$ is the number of agents, and $\gamma\in (0,1)$ is the discount factor. These results demonstrate linear speedup with respect to the number of agents $K$, despite heterogeneous Markovian trajectories under distinct transition kernels and coupled learning dynamics. To address these challenges, we develop a new perturbation analysis for the projected subspace updates and QR decomposition steps, together with conditional mixing arguments for heterogeneous Markovian noise. Furthermore, to handle the additional complications induced by policy updates and temporal dependence, we establish fine-grained characterizations of the discrepancies between function evaluations under Markovian sampling and under temporally frozen policies. Experiments instantiate the framework within PPO on federated \texttt{Hopper-v5} action-map heterogeneity, showing gains over Single PPO and FedAvg PPO and downstream transfer from the learned shared trunk.

cs.LG↗

arXiv2026

Attributed Network Alignment: Statistical Limits and Efficient Algorithm

This paper studies the problem of recovering a hidden vertex correspondence between two correlated graphs when both edge weights and node features are observed. While most existing work on graph alignment relies primarily on edge information, many real-world applications provide informative node features in addition to graph topology. To capture this setting, we introduce the featured correlated Gaussian Wigner model, where two graphs are coupled through an unknown vertex permutation, and the node features are correlated under the same permutation. We characterize the optimal information-theoretic thresholds for exact recovery and partial recovery of the latent mapping. On the algorithmic side, we propose QPAlign, an algorithm based on a quadratic programming relaxation, and demonstrate its strong empirical performance on both synthetic and real datasets. Moreover, we also derive theoretical guarantees for the proposed procedure, supporting its reliability and providing convergence guarantees.

math.ST↗

arXiv2026

Personalized Multi-Agent Average Reward TD-Learning via Joint Linear Approximation

We study personalized multi-agent average reward TD learning, in which a collection of agents interacts with different environments and jointly learns their respective value functions. We focus on the setting where there exists a shared linear representation, and the agents' optimal weights collectively lie in an unknown linear subspace. Inspired by the recent success of personalized federated learning (PFL), we study the convergence of cooperative single-timescale TD learning in which agents iteratively estimate the common subspace and local heads. We showed that this decomposition can filter out conflicting signals, effectively mitigating the negative impacts of ``misaligned'' signals, and achieving linear speedup. The main technical challenges lie in the heterogeneity, the Markovian sampling, and their intricate interplay in shaping error evolutions. Specifically, not only are the error dynamics of multiple variables closely interconnected, but there is also no direct contraction for the principal angle distance between the optimal subspace and the estimated subspace. We hope our analytical techniques can be useful to inspire research on deeper exploration into leveraging common structures. Experiments are provided to show the benefits of learning via a shared structure to the more general control problem.

cs.LG↗

arXiv2026

Sample Complexity of Correlation Detection in Erd\H{o}s-R\'enyi graphs

In this paper, we study correlation detection problem between a pair of Erd\H{o}s-R\'enyi graphs. Under the null hypothesis, the two graphs are independent, whereas under the alternative, they are edge-correlated through a latent permutation. We consider the partial-observation setting in which only an induced subgraph of each graph is sampled, and determine the sharp information-theoretic threshold for the sample size, including its leading constant. We also establish a general conditional second moment principle and identify a continuous transition in the sharp threshold constant across two different regimes.

math.ST↗

arXiv2025

Nonparametric Inference on Unlabeled Histograms

Statistical inference on histograms and frequency counts plays a central role in categorical data analysis. Moving beyond classical methods that directly analyze labeled frequencies, we introduce a framework that models the multiset of unlabeled histograms via a mixture distribution to better capture unseen domain elements in large-alphabet regime. We study the nonparametric maximum likelihood estimator (NPMLE) under this framework, and establish its optimal convergence rate under the Poisson setting. The NPMLE also immediately yields flexible and efficient plug-in estimators for functional estimation problems, where a localized variant further achieves the optimal sample complexity for a wide range of symmetric functionals. Extensive experiments on synthetic, real-world datasets, and large language models highlight the practical benefits of the proposed method.

math.ST↗

arXiv2025

Testing Correlation in Graphs by Counting Bounded Degree Motifs

We investigate the problem of detecting correlation between two Erd\H{o}s-R\'enyi graphs $G(n,p)$, formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are independent, while under the alternative hypothesis, they are correlated through a latent bijective mapping between their vertex sets. We develop a polynomial-time test by counting bounded degree motifs and prove its effectiveness for any constant correlation coefficient $\rho$ when the edge connecting probability satisfies $p\ge n^{-1+\delta}$ for some constant $\delta>0$. In particular, our guarantee improves the constrain of motif-counting methods from $\rho\ge \sqrt{\alpha}$ to any constant $\rho = \Omega(1)$, where $\alpha\approx 0.338$ is the Otter's constant.

cs.SI↗

arXiv2025

Identifiability and Estimation in High-Dimensional Nonparametric Latent Structure Models

This paper studies the problems of identifiability and estimation in high-dimensional nonparametric latent structure models. We introduce an identifiability theorem that generalizes existing conditions, establishing a unified framework applicable to diverse statistical settings. Our results rigorously demonstrate how increased dimensionality, coupled with diversity in variables, inherently facilitates identifiability. For the estimation problem, we establish near-optimal minimax rate bounds for the high-dimensional nonparametric density estimation under latent structures with smooth marginals. Contrary to the conventional curse of dimensionality, our sample complexity scales only polynomially with the dimension. Additionally, we develop a perturbation theory for component recovery and propose a recovery procedure based on simultaneous diagonalization.

math.ST↗

arXiv2025

Sample Complexity of Correlation Detection in the Gaussian Wigner Model

Correlation analysis is a fundamental step in uncovering meaningful insights from complex datasets. In this paper, we study the problem of detecting correlations between two random graphs following the Gaussian Wigner model with unlabeled vertices. Specifically, the task is formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are independent, while under the alternative hypothesis, they are edge-correlated through a latent vertex permutation, yet maintain the same marginal distributions as under the null. We focus on the scenario where two induced subgraphs, each with a fixed number of vertices, are sampled. We determine the optimal rate for the sample size required for correlation detection, derived through an analysis of the conditional second moment. Additionally, we propose an efficient approximate algorithm that significantly reduces running time.

math.ST↗

arXiv2024

Learning with Shared Representations: Statistical Rates and Efficient Algorithms

Collaborative learning through latent shared feature representations enables heterogeneous clients to train personalized models with improved performance and reduced sample complexity. Despite empirical success and extensive study, the theoretical understanding of such methods remains incomplete, even for representations restricted to low-dimensional linear subspaces. In this work, we establish new upper and lower bounds on the statistical error in learning low-dimensional shared representations across clients. Our analysis captures both statistical heterogeneity (including covariate and concept shifts) and variation in local dataset sizes, aspects often overlooked in prior work. We further extend these results to nonlinear models including logistic regression and one-hidden-layer ReLU networks. Specifically, we design a spectral estimator that leverages independent replicas of local averages to approximate the non-convex least-squares solution and derive a nearly matching minimax lower bound. Our estimator achieves the optimal statistical rate when the shared representation is well covered across clients -- i.e., when no direction is severely underrepresented. Our results reveal two distinct phases of the optimal rate: a standard parameter-counting regime and a penalized regime when the number of clients is large or local datasets are small. These findings precisely characterize when collaboration benefits the overall system or individual clients in transfer learning and private fine-tuning.

cs.LG↗

arXiv2024

On the Convergence Rates of Federated Q-Learning across Heterogeneous Environments

Large-scale multi-agent systems are often deployed across wide geographic areas, where agents interact with heterogeneous environments. There is an emerging interest in understanding the role of heterogeneity in the performance of the federated versions of classic reinforcement learning algorithms. In this paper, we study synchronous federated Q-learning, which aims to learn an optimal Q-function by having $K$ agents average their local Q-estimates per $E$ iterations. We observe an interesting phenomenon on the convergence speeds in terms of $K$ and $E$. Similar to the homogeneous environment settings, there is a linear speed-up concerning $K$ in reducing the errors that arise from sampling randomness. Yet, in sharp contrast to the homogeneous settings, $E>1$ leads to significant performance degradation. Specifically, we provide a fine-grained characterization of the error evolution in the presence of environmental heterogeneity, which decay to zero as the number of iterations $T$ increases. The slow convergence of having $E>1$ turns out to be fundamental rather than an artifact of our analysis. We prove that, for a wide range of stepsizes, the $\ell_{\infty}$ norm of the error cannot decay faster than $\Theta (E/T)$. In addition, our experiments demonstrate that the convergence exhibits an interesting two-phase phenomenon. For any given stepsize, there is a sharp phase-transition of the convergence: the error decays rapidly in the beginning yet later bounces up and stabilizes. Provided that the phase-transition time can be estimated, choosing different stepsizes for the two phases leads to faster overall convergence.

cs.LG↗

arXiv2024

Minimax estimation of functionals in sparse vector model with correlated observations

We consider the observations of an unknown $s$-sparse vector ${\boldsymbol \theta}$ corrupted by Gaussian noise with zero mean and unknown covariance matrix ${\boldsymbol \Sigma}$. We propose minimax optimal methods of estimating the $\ell_2$ norm of ${\boldsymbol \theta}$ and testing the hypothesis $H_0: {\boldsymbol \theta}=0$ against sparse alternatives when only partial information about ${\boldsymbol \Sigma}$ is available, such as an upper bound on its Frobenius norm and the values of its diagonal entries to within an unknown scaling factor. We show that the minimax rates of the estimation and testing are leveraged not by the dimension of the problem but by the value of the Frobenius norm of ${\boldsymbol \Sigma}$.

math.ST↗

arXiv2024

Information-Theoretic Thresholds for the Alignments of Partially Correlated Graphs

This paper studies the problem of recovering the hidden vertex correspondence between two correlated random graphs. We propose the partially correlated Erd\H{o}s-R\'enyi graphs model, wherein a pair of induced subgraphs with a certain number are correlated. We investigate the information-theoretic thresholds for recovering the latent correlated subgraphs and the hidden vertex correspondence. We prove that there exists an optimal rate for partial recovery for the number of correlated nodes, above which one can correctly match a fraction of vertices and below which correctly matching any positive fraction is impossible, and we also derive an optimal rate for exact recovery. In the proof of possibility results, we propose correlated functional digraphs, which partition the edges of the intersection graph into two types of components, and bound the error probability by lower-order cumulant generating functions. The proof of impossibility results build upon the generalized Fano's inequality and the recovery thresholds settled in correlated Erd\H{o}s-R\'enyi graphs model.

cs.IT↗

arXiv2024

On the best approximation by finite Gaussian mixtures

We consider the problem of approximating a general Gaussian location mixture by finite mixtures. The minimum order of finite mixtures that achieve a prescribed accuracy (measured by various $f$-divergences) is determined within constant factors for the family of mixing distributions with compactly support or appropriate assumptions on the tail probability including subgaussian and subexponential. While the upper bound is achieved using the technique of local moment matching, the lower bound is established by relating the best approximation error to the low-rank approximation of certain trigonometric moment matrices, followed by a refined spectral analysis of their minimum eigenvalue. In the case of Gaussian mixing distributions, this result corrects a previous lower bound in [Allerton Conference 48 (2010) 620-628].

math.ST↗

arXiv2023

Sampling for Remote Estimation of an Ornstein-Uhlenbeck Process through Channel with Unknown Delay Statistics

In this paper, we consider sampling an Ornstein-Uhlenbeck (OU) process through a channel for remote estimation. The goal is to minimize the mean square error (MSE) at the estimator under a sampling frequency constraint when the channel delay statistics is unknown. Sampling for MSE minimization is reformulated into an optimal stopping problem. By revisiting the threshold structure of the optimal stopping policy when the delay statistics is known, we propose an online sampling algorithm to learn the optimum threshold using stochastic approximation algorithm and the virtual queue method. We prove that with probability 1, the MSE of the proposed online algorithm converges to the minimum MSE that is achieved when the channel delay statistics is known. The cumulative MSE gap of our proposed algorithm compared with the minimum MSE up to the $(k+1)$-th sample grows with rate at most $\mathcal{O}(\ln k)$. Our proposed online algorithm can satisfy the sampling frequency constraint theoretically. Finally, simulation results are provided to demonstrate the performance of the proposed algorithm.

cs.IT↗

arXiv2023

Fast and Multiphase Rates for Nearest Neighbor Classifiers

We study the scaling of classification error rates with respect to the size of the training dataset. In contrast to classical results where rates are minimax optimal for a problem class, this work starts with the empirical observation that, even for a fixed data distribution, the error scaling can have \emph{diverse} rates across different ranges of sample size. To understand when and why the error rate is non-uniform, we theoretically analyze nearest neighbor classifiers. We show that an error scaling law can have fine-grained rates: in the early phase, the test error depends polynomially on the data dimension and decreases fast; whereas in the later phase, the error depends exponentially on the data dimension and decreases slowly. Our analysis highlights the complexity of the data distribution in determining the test error. When the data are distributed benignly, we show that the generalization error of nearest neighbor classifier can depend polynomially, instead of exponentially, on the data dimension.

stat.ML↗

arXiv2023

Federated Learning in the Presence of Adversarial Client Unavailability

Federated learning is a decentralized machine learning framework that enables collaborative model training without revealing raw data. Due to the diverse hardware and software limitations, a client may not always be available for the computation requests from the parameter server. An emerging line of research is devoted to tackling arbitrary client unavailability. However, existing work still imposes structural assumptions on the unavailability patterns, impeding their applicability in challenging scenarios wherein the unavailability patterns are beyond the control of the parameter server. Moreover, in harsh environments like battlefields, adversaries can selectively and adaptively silence specific clients. In this paper, we relax the structural assumptions and consider adversarial client unavailability. To quantify the degrees of client unavailability, we use the notion of $\epsilon$-adversary dropout fraction. We show that simple variants of FedAvg or FedProx, albeit completely agnostic to $\epsilon$, converge to an estimation error on the order of $\epsilon (G^2 + \sigma^2)$ for non-convex global objectives and $\epsilon(G^2 + \sigma^2)/\mu^2$ for $\mu$ strongly convex global objectives, where $G$ is a heterogeneity parameter and $\sigma^2$ is the noise level. Conversely, we prove that any algorithm has to suffer an estimation error of at least $\epsilon (G^2 + \sigma^2)/8$ and $\epsilon(G^2 + \sigma^2)/(8\mu^2)$ for non-convex global objectives and $\mu$-strongly convex global objectives. Furthermore, the convergence speeds of the FedAvg or FedProx variants are $O(1/\sqrt{T})$ for non-convex objectives and $O(1/T)$ for strongly-convex objectives, both of which are the best possible for any first-order method that only has access to noisy gradients.

cs.LG↗

arXiv2022

Global Convergence of Federated Learning for Mixed Regression

This paper studies the problem of model training under Federated Learning when clients exhibit cluster structure. We contextualize this problem in mixed regression, where each client has limited local data generated from one of $k$ unknown regression models. We design an algorithm that achieves global convergence from any initialization, and works even when local data volume is highly unbalanced -- there could exist clients that contain $O(1)$ data points only. Our algorithm first runs moment descent on a few anchor clients (each with $\tilde{\Omega}(k)$ data points) to obtain coarse model estimates. Then each client alternately estimates its cluster labels and refines the model estimates based on FedAvg or FedProx. A key innovation in our analysis is a uniform estimate on the clustering errors, which we prove by bounding the VC dimension of general polynomial concept classes based on the theory of algebraic geometry.

cs.LG↗