arXiv subjects

Pengkun Yang

Publications and source records attributed to Pengkun Yang.

31 records · Page 2Linked to original sources

arXiv2023

Age Optimal Sampling Under Unknown Delay Statistics

This paper revisits the problem of sampling and transmitting status updates through a channel with random delay under a sampling frequency constraint \cite{sun_17_tit}. We use the Age of Information (AoI) to characterize the status information freshness at the receiver. The goal is to design a sampling policy that can minimize the average AoI when the statistics of delay is unknown. We reformulate the problem as the optimization of a renewal-reward process, and propose an online sampling strategy based on the Robbins-Monro algorithm. We prove that the proposed algorithm satisfies the sampling frequency constraint. Moreover, when the transmission delay is bounded and its distribution is absolutely continuous, the average AoI obtained by the proposed algorithm converges to the minimum AoI when the number of samples $K$ goes to infinity with probability 1. We show that the optimality gap decays with rate $\mathcal{O}\left(\ln K/K\right)$, and the proposed algorithm is minimax rate optimal. Simulation results validate the performance of our proposed algorithm.

cs.IT↗

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 $\tildeΩ(k)$ 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↗

arXiv2022

A Non-parametric View of FedAvg and FedProx: Beyond Stationary Points

Federated Learning (FL) is a promising decentralized learning framework and has great potentials in privacy preservation and in lowering the computation load at the cloud. Recent work showed that FedAvg and FedProx - the two widely-adopted FL algorithms - fail to reach the stationary points of the global optimization objective even for homogeneous linear regression problems. Further, it is concerned that the common model learned might not generalize well locally at all in the presence of heterogeneity. In this paper, we analyze the convergence and statistical efficiency of FedAvg and FedProx, addressing the above two concerns. Our analysis is based on the standard non-parametric regression in a reproducing kernel Hilbert space (RKHS), and allows for heterogeneous local data distributions and unbalanced local datasets. We prove that the estimation errors, measured in either the empirical norm or the RKHS norm, decay with a rate of 1/t in general and exponentially for finite-rank kernels. In certain heterogeneous settings, these upper bounds also imply that both FedAvg and FedProx achieve the optimal error rate. To further analytically quantify the impact of the heterogeneity at each client, we propose and characterize a novel notion-federation gain, defined as the reduction of the estimation error for a client to join the FL. We discover that when the data heterogeneity is moderate, a client with limited local data can benefit from a common model with a large federation gain. Numerical experiments further corroborate our theoretical findings.

stat.ML↗

arXiv2022

Boosting Active Learning via Improving Test Performance

Central to active learning (AL) is what data should be selected for annotation. Existing works attempt to select highly uncertain or informative data for annotation. Nevertheless, it remains unclear how selected data impacts the test performance of the task model used in AL. In this work, we explore such an impact by theoretically proving that selecting unlabeled data of higher gradient norm leads to a lower upper-bound of test loss, resulting in better test performance. However, due to the lack of label information, directly computing gradient norm for unlabeled data is infeasible. To address this challenge, we propose two schemes, namely expected-gradnorm and entropy-gradnorm. The former computes the gradient norm by constructing an expected empirical loss while the latter constructs an unsupervised loss with entropy. Furthermore, we integrate the two schemes in a universal AL framework. We evaluate our method on classical image classification and semantic segmentation tasks. To demonstrate its competency in domain applications and its robustness to noise, we also validate our method on a cellular imaging analysis task, namely cryo-Electron Tomography subtomogram classification. Results demonstrate that our method achieves superior performance against the state of the art. Our source code is available at https://github.com/xulabs/aitom/blob/master/doc/projects/al_gradnorm.md.

cs.LG↗

arXiv2021

Optimal estimation of high-dimensional location Gaussian mixtures

This paper studies the optimal rate of estimation in a finite Gaussian location mixture model in high dimensions without separation conditions. We assume that the number of components $k$ is bounded and that the centers lie in a ball of bounded radius, while allowing the dimension $d$ to be as large as the sample size $n$. Extending the one-dimensional result of Heinrich and Kahn \cite{HK2015}, we show that the minimax rate of estimating the mixing distribution in Wasserstein distance is $Θ((d/n)^{1/4} + n^{-1/(4k-2)})$, achieved by an estimator computable in time $O(nd^2+n^{5/4})$. Furthermore, we show that the mixture density can be estimated at the optimal parametric rate $Θ(\sqrt{d/n})$ in Hellinger distance and provide a computationally efficient algorithm to achieve this rate in the special case of $k=2$. Both the theoretical and methodological development rely on a careful application of the method of moments. Central to our results is the observation that the information geometry of finite Gaussian mixtures is characterized by the moment tensors of the mixing distribution, whose low-rank structure can be exploited to obtain a sharp local entropy bound.

math.ST↗

arXiv2021

Polynomial methods in statistical inference: theory and practice

This survey provides an exposition of a suite of techniques based on the theory of polynomials, collectively referred to as polynomial methods, which have recently been applied to address several challenging problems in statistical inference successfully. Topics including polynomial approximation, polynomial interpolation and majorization, moment space and positive polynomials, orthogonal polynomials and Gaussian quadrature are discussed, with their major probabilistic and statistical applications in property estimation on large domains and learning mixture models. These techniques provide useful tools not only for the design of highly practical algorithms with provable optimality, but also for establishing the fundamental limits of the inference problems through the method of moment matching. The effectiveness of the polynomial method is demonstrated in concrete problems such as entropy and support size estimation, distinct elements problem, and learning Gaussian mixture models.

math.ST↗

arXiv2020

Modeling from Features: a Mean-field Framework for Over-parameterized Deep Neural Networks

This paper proposes a new mean-field framework for over-parameterized deep neural networks (DNNs), which can be used to analyze neural network training. In this framework, a DNN is represented by probability measures and functions over its features (that is, the function values of the hidden units over the training data) in the continuous limit, instead of the neural network parameters as most existing studies have done. This new representation overcomes the degenerate situation where all the hidden units essentially have only one meaningful hidden unit in each middle layer, and further leads to a simpler representation of DNNs, for which the training objective can be reformulated as a convex optimization problem via suitable re-parameterization. Moreover, we construct a non-linear dynamics called neural feature flow, which captures the evolution of an over-parameterized DNN trained by Gradient Descent. We illustrate the framework via the standard DNN and the Residual Network (Res-Net) architectures. Furthermore, we show, for Res-Net, when the neural feature flow process converges, it reaches a global minimal solution under suitable conditions. Our analysis leads to the first global convergence proof for over-parameterized neural network training with more than $3$ layers in the mean-field regime.

stat.ML↗

arXiv2019

On Learning Over-parameterized Neural Networks: A Functional Approximation Perspective

We consider training over-parameterized two-layer neural networks with Rectified Linear Unit (ReLU) using gradient descent (GD) method. Inspired by a recent line of work, we study the evolutions of network prediction errors across GD iterations, which can be neatly described in a matrix form. When the network is sufficiently over-parameterized, these matrices individually approximate {\em an} integral operator which is determined by the feature vector distribution $ρ$ only. Consequently, GD method can be viewed as {\em approximately} applying the powers of this integral operator on the underlying/target function $f^*$ that generates the responses/labels. We show that if $f^*$ admits a low-rank approximation with respect to the eigenspaces of this integral operator, then the empirical risk decreases to this low-rank approximation error at a linear rate which is determined by $f^*$ and $ρ$ only, i.e., the rate is independent of the sample size $n$. Furthermore, if $f^*$ has zero low-rank approximation error, then, as long as the width of the neural network is $Ω(n\log n)$, the empirical risk decreases to $Θ(1/\sqrt{n})$. To the best of our knowledge, this is the first result showing the sufficiency of nearly-linear network over-parameterization. We provide an application of our general results to the setting where $ρ$ is the uniform distribution on the spheres and $f^*$ is a polynomial. Throughout this paper, we consider the scenario where the input dimension $d$ is fixed.

cs.LG↗

arXiv2019

Optimal estimation of Gaussian mixtures via denoised method of moments

The Method of Moments [Pea94] is one of the most widely used methods in statistics for parameter estimation, by means of solving the system of equations that match the population and estimated moments. However, in practice and especially for the important case of mixture models, one frequently needs to contend with the difficulties of non-existence or non-uniqueness of statistically meaningful solutions, as well as the high computational cost of solving large polynomial systems. Moreover, theoretical analysis of the method of moments are mainly confined to asymptotic normality style of results established under strong assumptions. This paper considers estimating a $k$-component Gaussian location mixture with a common (possibly unknown) variance parameter. To overcome the aforementioned theoretic and algorithmic hurdles, a crucial step is to denoise the moment estimates by projecting to the truncated moment space (via semidefinite programming) before solving the method of moments equations. Not only does this regularization ensures existence and uniqueness of solutions, it also yields fast solvers by means of Gauss quadrature. Furthermore, by proving new moment comparison theorems in the Wasserstein distance via polynomial interpolation and majorization techniques, we establish the statistical guarantees and adaptive optimality of the proposed procedure, as well as oracle inequality in misspecified models. These results can also be viewed as provable algorithms for Generalized Method of Moments [Han82] which involves non-convex optimization and lacks theoretical guarantees.

math.ST↗

arXiv2018

Sample complexity of the distinct elements problem

We consider the distinct elements problem, where the goal is to estimate the number of distinct colors in an urn containing $ k $ balls based on $n$ samples drawn with replacements. Based on discrete polynomial approximation and interpolation, we propose an estimator with additive error guarantee that achieves the optimal sample complexity within $O(\log\log k)$ factors, and in fact within constant factors for most cases. The estimator can be computed in $O(n)$ time for an accurate estimation. The result also applies to sampling without replacement provided the sample size is a vanishing fraction of the urn size. One of the key auxiliary results is a sharp bound on the minimum singular values of a real rectangular Vandermonde matrix, which might be of independent interest.

math.ST↗

arXiv2016

Chebyshev polynomials, moment matching, and optimal estimation of the unseen

We consider the problem of estimating the support size of a discrete distribution whose minimum non-zero mass is at least $ \frac{1}{k}$. Under the independent sampling model, we show that the sample complexity, i.e., the minimal sample size to achieve an additive error of $εk$ with probability at least 0.1 is within universal constant factors of $ \frac{k}{\log k}\log^2\frac{1}ε $, which improves the state-of-the-art result of $ \frac{k}{ε^2 \log k} $ in \cite{VV13}. Similar characterization of the minimax risk is also obtained. Our procedure is a linear estimator based on the Chebyshev polynomial and its approximation-theoretic properties, which can be evaluated in $O(n+\log^2 k)$ time and attains the sample complexity within a factor of six asymptotically. The superiority of the proposed estimator in terms of accuracy, computational efficiency and scalability is demonstrated in a variety of synthetic and real datasets.

math.ST↗

arXiv2016

Minimax rates of entropy estimation on large alphabets via best polynomial approximation

Consider the problem of estimating the Shannon entropy of a distribution over $k$ elements from $n$ independent samples. We show that the minimax mean-square error is within universal multiplicative constant factors of $$\Big(\frac{k }{n \log k}\Big)^2 + \frac{\log^2 k}{n}$$ if $n$ exceeds a constant factor of $\frac{k}{\log k}$; otherwise there exists no consistent estimator. This refines the recent result of Valiant-Valiant \cite{VV11} that the minimal sample size for consistent entropy estimation scales according to $Θ(\frac{k}{\log k})$. The apparatus of best polynomial approximation plays a key role in both the construction of optimal estimators and, via a duality argument, the minimax lower bound.

cs.IT↗

arXiv2013

Software Defined Radio Implementation of Signaling Splitting in Hyper-Cellular Network

This paper presents the design and implementation of signaling splitting scheme in hyper-cellular network on a software defined radio platform. Hyper-cellular network is a novel architecture of future mobile communication systems in which signaling and data are decoupled at the air interface to mitigate the signaling overhead and allow energy efficient operation of base stations. On an open source software defined radio platform, OpenBTS, we investigate the feasibility of signaling splitting for GSM protocol and implement a novel system which can prove the proposed concept. Standard GSM handsets can camp on the network with the help of signaling base station, and data base station will be appointed to handle phone calls on demand. Our work initiates the systematic approach to study hyper-cellular concept in real wireless environment with both software and hardware implementations.

cs.NI↗