arXiv subjects

Yudong Chen

Publications and source records attributed to Yudong Chen.

At least 91 records · Page 5Linked to original sources

arXiv2020

Solitary beam propagation in a nonlinear optical resonator enables high-efficiency pulse compression and mode self-cleaning

Generating intense ultrashort pulses with high-quality spatial modes is crucial for ultrafast and strong-field science. This can be accomplished by controlling propagation of femtosecond pulses under the influence of Kerr nonlinearity and achieving stable propagation with high intensity. In this work, we propose that the generation of spatial solitons in periodic layered Kerr media can provide an optimum condition for supercontinuum generation and pulse compression using multiple thin plates. With both the experimental and theoretical investigations, we successfully identify these solitary modes and reveal a universal relationship between the beam size and the critical nonlinear phase. Space-time coupling is shown to strongly influence the spectral, spatial and temporal profiles of femtosecond pulses. Taking advantage of the unique characters of these solitary modes, we demonstrate single-stage supercontinuum generation and compression of femtosecond pulses from initially 170 fs down to 22 fs with an efficiency ~90%. We also provide evidence of efficient mode self-cleaning which suggests rich spatial-temporal self-organization processes of laser beams in a nonlinear resonator.

physics.optics↗

arXiv2020

Defending Against Saddle Point Attack in Byzantine-Robust Distributed Learning

We study robust distributed learning that involves minimizing a non-convex loss function with saddle points. We consider the Byzantine setting where some worker machines have abnormal or even arbitrary and adversarial behavior. In this setting, the Byzantine machines may create fake local minima near a saddle point that is far away from any true local minimum, even when robust gradient estimators are used. We develop ByzantinePGD, a robust first-order algorithm that can provably escape saddle points and fake local minima, and converge to an approximate true local minimizer with low iteration complexity. As a by-product, we give a simpler algorithm and analysis for escaping saddle points in the usual non-Byzantine setting. We further discuss three robust gradient estimators that can be used in ByzantinePGD, including median, trimmed mean, and iterative filtering. We characterize their performance in concrete statistical settings, and argue for their near-optimality in low and high dimensional regimes.

cs.LG↗

arXiv2020

Learning Zero-Sum Simultaneous-Move Markov Games Using Function Approximation and Correlated Equilibrium

We develop provably efficient reinforcement learning algorithms for two-player zero-sum finite-horizon Markov games with simultaneous moves. To incorporate function approximation, we consider a family of Markov games where the reward function and transition kernel possess a linear structure. Both the offline and online settings of the problems are considered. In the offline setting, we control both players and aim to find the Nash Equilibrium by minimizing the duality gap. In the online setting, we control a single player playing against an arbitrary opponent and aim to minimize the regret. For both settings, we propose an optimistic variant of the least-squares minimax value iteration algorithm. We show that our algorithm is computationally efficient and provably achieves an $\tilde O(\sqrt{d^3 H^3 T} )$ upper bound on the duality gap and regret, where $d$ is the linear dimension, $H$ the horizon and $T$ the total number of timesteps. Our results do not require additional assumptions on the sampling model. Our setting requires overcoming several new challenges that are absent in Markov decision processes or turn-based Markov games. In particular, to achieve optimism with simultaneous moves, we construct both upper and lower confidence bounds of the value function, and then compute the optimistic policy by solving a general-sum matrix game with these bounds as the payoff matrices. As finding the Nash Equilibrium of a general-sum game is computationally hard, our algorithm instead solves for a Coarse Correlated Equilibrium (CCE), which can be obtained efficiently. To our best knowledge, such a CCE-based scheme for optimism has not appeared in the literature and might be of interest in its own right.

cs.LG↗

arXiv2020

Risk-Sensitive Reinforcement Learning: Near-Optimal Risk-Sample Tradeoff in Regret

We study risk-sensitive reinforcement learning in episodic Markov decision processes with unknown transition kernels, where the goal is to optimize the total reward under the risk measure of exponential utility. We propose two provably efficient model-free algorithms, Risk-Sensitive Value Iteration (RSVI) and Risk-Sensitive Q-learning (RSQ). These algorithms implement a form of risk-sensitive optimism in the face of uncertainty, which adapts to both risk-seeking and risk-averse modes of exploration. We prove that RSVI attains an $\tilde{O}\big(λ(|β| H^2) \cdot \sqrt{H^{3} S^{2}AT} \big)$ regret, while RSQ attains an $\tilde{O}\big(λ(|β| H^2) \cdot \sqrt{H^{4} SAT} \big)$ regret, where $λ(u) = (e^{3u}-1)/u$ for $u>0$. In the above, $β$ is the risk parameter of the exponential utility function, $S$ the number of states, $A$ the number of actions, $T$ the total number of timesteps, and $H$ the episode length. On the flip side, we establish a regret lower bound showing that the exponential dependence on $|β|$ and $H$ is unavoidable for any algorithm with an $\tilde{O}(\sqrt{T})$ regret (even when the risk objective is on the same scale as the original reward), thus certifying the near-optimality of the proposed algorithms. Our results demonstrate that incorporating risk awareness into reinforcement learning necessitates an exponential cost in $|β|$ and $H$, which quantifies the fundamental tradeoff between risk sensitivity (related to aleatoric uncertainty) and sample efficiency (related to epistemic uncertainty). To the best of our knowledge, this is the first regret analysis of risk-sensitive reinforcement learning with the exponential utility.

cs.LG↗

arXiv2020

Leave-one-out Approach for Matrix Completion: Primal and Dual Analysis

In this paper, we introduce a powerful technique based on Leave-one-out analysis to the study of low-rank matrix completion problems. Using this technique, we develop a general approach for obtaining fine-grained, entrywise bounds for iterative stochastic procedures in the presence of probabilistic dependency. We demonstrate the power of this approach in analyzing two of the most important algorithms for matrix completion: (i) the non-convex approach based on Projected Gradient Descent (PGD) for a rank-constrained formulation, also known as the Singular Value Projection algorithm, and (ii) the convex relaxation approach based on nuclear norm minimization (NNM). Using this approach, we establish the first convergence guarantee for the original form of PGD without regularization or sample splitting}, and in particular shows that it converges linearly in the infinity norm. For NNM, we use this approach to study a fictitious iterative procedure that arises in the dual analysis. Our results show that \NNM recovers an $ d $-by-$ d $ rank-$ r $ matrix with $\mathcal{O}(μr \log(μr) d \log d )$ observed entries. This bound has optimal dependence on the matrix dimension and is independent of the condition number. To the best of our knowledge, this is the first sample complexity result for a tractable matrix completion algorithm that satisfies these two properties simultaneously.

stat.ML↗

arXiv2020

Structures of Spurious Local Minima in $k$-means

$k$-means clustering is a fundamental problem in unsupervised learning. The problem concerns finding a partition of the data points into $k$ clusters such that the within-cluster variation is minimized. Despite its importance and wide applicability, a theoretical understanding of the $k$-means problem has not been completely satisfactory. Existing algorithms with theoretical performance guarantees often rely on sophisticated (sometimes artificial) algorithmic techniques and restricted assumptions on the data. The main challenge lies in the non-convex nature of the problem; in particular, there exist additional local solutions other than the global optimum. Moreover, the simplest and most popular algorithm for $k$-means, namely Lloyd's algorithm, generally converges to such spurious local solutions both in theory and in practice. In this paper, we approach the $k$-means problem from a new perspective, by investigating the structures of these spurious local solutions under a probabilistic generative model with $k$ ground truth clusters. As soon as $k=3$, spurious local minima provably exist, even for well-separated and balanced clusters. One such local minimum puts two centers at one true cluster, and the third center in the middle of the other two true clusters. For general $k$, one local minimum puts multiple centers at a true cluster, and one center in the middle of multiple true clusters. Perhaps surprisingly, we prove that this is essentially the only type of spurious local minima under a separation condition. Our results pertain to the $k$-means formulation for mixtures of Gaussians or bounded distributions. Our theoretical results corroborate existing empirical observations and provide justification for several improved algorithms for $k$-means clustering.

stat.ML↗

arXiv2019

Random Fourier Features via Fast Surrogate Leverage Weighted Sampling

In this paper, we propose a fast surrogate leverage weighted sampling strategy to generate refined random Fourier features for kernel approximation. Compared to the current state-of-the-art method that uses the leverage weighted scheme [Li-ICML2019], our new strategy is simpler and more effective. It uses kernel alignment to guide the sampling process and it can avoid the matrix inversion operator when we compute the leverage function. Given n observations and s random features, our strategy can reduce the time complexity from O(ns^2+s^3) to O(ns^2), while achieving comparable (or even slightly better) prediction performance when applied to kernel ridge regression (KRR). In addition, we provide theoretical guarantees on the generalization performance of our approach, and in particular characterize the number of random features required to achieve statistical guarantees in KRR. Experiments on several benchmark datasets demonstrate that our algorithm achieves comparable prediction performance and takes less time cost when compared to [Li-ICML2019].

cs.LG↗

arXiv2019

Factor Group-Sparse Regularization for Efficient Low-Rank Matrix Recovery

This paper develops a new class of nonconvex regularizers for low-rank matrix recovery. Many regularizers are motivated as convex relaxations of the matrix rank function. Our new factor group-sparse regularizers are motivated as a relaxation of the number of nonzero columns in a factorization of the matrix. These nonconvex regularizers are sharper than the nuclear norm; indeed, we show they are related to Schatten-$p$ norms with arbitrarily small $0 < p \leq 1$. Moreover, these factor group-sparse regularizers can be written in a factored form that enables efficient and effective nonconvex optimization; notably, the method does not use singular value decomposition. We provide generalization error bounds for low-rank matrix completion which show improved upper bounds for Schatten-$p$ norm reglarization as $p$ decreases. Compared to the max norm and the factored formulation of the nuclear norm, factor group-sparse regularizers are more efficient, accurate, and robust to the initial guess of rank. Experiments show promising performance of factor group-sparse regularization for low-rank matrix completion and robust principal component analysis.

cs.LG↗

arXiv2019

Global Convergence of Least Squares EM for Demixing Two Log-Concave Densities

This work studies the location estimation problem for a mixture of two rotation invariant log-concave densities. We demonstrate that Least Squares EM, a variant of the EM algorithm, converges to the true location parameter from a randomly initialized point. We establish the explicit convergence rates and sample complexity bounds, revealing their dependence on the signal-to-noise ratio and the tail property of the log-concave distribution. Moreover, we show that this global convergence property is robust under model mis-specification. Our analysis generalizes previous techniques for proving the convergence results for Gaussian mixtures. In particular, we make use of an angle-decreasing property for establishing global convergence of Least Squares EM beyond Gaussian settings, as $\ell_2$ distance contraction no longer holds globally for general log-concave mixtures.

stat.ML↗

arXiv2019

Clustering Degree-Corrected Stochastic Block Model with Outliers

For the degree corrected stochastic block model in the presence of arbitrary or even adversarial outliers, we develop a convex-optimization-based clustering algorithm that includes a penalization term depending on the positive deviation of a node from the expected number of edges to other inliers. We prove that under mild conditions, this method achieves exact recovery of the underlying clusters. Our synthetic experiments show that our algorithm performs well on heterogeneous networks, and in particular those with Pareto degree distributions, for which outliers have a broad range of possible degrees that may enhance their adversarial power. We also demonstrate that our method allows for recovery with significantly lower error rates compared to existing algorithms.

cs.LG↗

arXiv2019

Global Convergence of EM Algorithm for Mixtures of Two Component Linear Regression

The Expectation-Maximization algorithm is perhaps the most broadly used algorithm for inference of latent variable problems. A theoretical understanding of its performance, however, largely remains lacking. Recent results established that EM enjoys global convergence for Gaussian Mixture Models. For Mixed Linear Regression, however, only local convergence results have been established, and those only for the high SNR regime. We show here that EM converges for mixed linear regression with two components (it is known that it may fail to converge for three or more), and moreover that this convergence holds for random initialization. Our analysis reveals that EM exhibits very different behavior in Mixed Linear Regression from its behavior in Gaussian Mixture Models, and hence our proofs require the development of several new ideas.

stat.ML↗

arXiv2019

Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence

The task of recovering a low-rank matrix from its noisy linear measurements plays a central role in computational science. Smooth formulations of the problem often exhibit an undesirable phenomenon: the condition number, classically defined, scales poorly with the dimension of the ambient space. In contrast, we here show that in a variety of concrete circumstances, nonsmooth penalty formulations do not suffer from the same type of ill-conditioning. Consequently, standard algorithms for nonsmooth optimization, such as subgradient and prox-linear methods, converge at a rapid dimension-independent rate when initialized within constant relative error of the solution. Moreover, nonsmooth formulations are naturally robust against outliers. Our framework subsumes such important computational tasks as phase retrieval, blind deconvolution, quadratic sensing, matrix completion, and robust PCA. Numerical experiments on these problems illustrate the benefits of the proposed approach.

math.OC↗

arXiv2019

Achieving the Bayes Error Rate in Synchronization and Block Models by SDP, Robustly

We study the statistical performance of semidefinite programming (SDP) relaxations for clustering under random graph models. Under the $\mathbb{Z}_{2}$ Synchronization model, Censored Block Model and Stochastic Block Model, we show that SDP achieves an error rate of the form \[ \exp\Big[-\big(1-o(1)\big)\bar{n} I^* \Big]. \] Here $\bar{n}$ is an appropriate multiple of the number of nodes and $I^*$ is an information-theoretic measure of the signal-to-noise ratio. We provide matching lower bounds on the Bayes error for each model and therefore demonstrate that the SDP approach is Bayes optimal. As a corollary, our results imply that SDP achieves the optimal exact recovery threshold under each model. Furthermore, we show that SDP is robust: the above bound remains valid under semirandom versions of the models in which the observed graph is modified by a monotone adversary. Our proof is based on a novel primal-dual analysis of SDP under a unified framework for all three models, and the analysis shows that SDP tightly approximates a joint majority voting procedure.

math.ST↗

arXiv2019

Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm

In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product). Induced by the t-product, we first rigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor average rank, and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm. These definitions, their relationships and properties are consistent with matrix cases. Equipped with the new tensor nuclear norm, we then solve the TRPCA problem by solving a convex program and provide the theoretical guarantee for the exact recovery. Our TRPCA model and recovery guarantee include matrix RPCA as a special case. Numerical experiments verify our results, and the applications to image recovery and background modeling problems demonstrate the effectiveness of our method.

stat.ML↗

arXiv2018

Convex Relaxation Methods for Community Detection

This paper surveys recent theoretical advances in convex optimization approaches for community detection. We introduce some important theoretical techniques and results for establishing the consistency of convex community detection under various statistical models. In particular, we discuss the basic techniques based on the primal and dual analysis. We also present results that demonstrate several distinctive advantages of convex community detection, including robustness against outlier nodes, consistency under weak assortativity, and adaptivity to heterogeneous degrees. This survey is not intended to be a complete overview of the vast literature on this fast-growing topic. Instead, we aim to provide a big picture of the remarkable recent development in this area and to make the survey accessible to a broad audience. We hope that this expository article can serve as an introductory guide for readers who are interested in using, designing, and analyzing convex relaxation methods in network analysis.

math.ST↗

arXiv2018

Learning Mixtures of Sparse Linear Regressions Using Sparse Graph Codes

In this paper, we consider the mixture of sparse linear regressions model. Let $β^{(1)},\ldots,β^{(L)}\in\mathbb{C}^n$ be $ L $ unknown sparse parameter vectors with a total of $ K $ non-zero coefficients. Noisy linear measurements are obtained in the form $y_i={x}_i^H β^{(\ell_i)} + w_i$, each of which is generated randomly from one of the sparse vectors with the label $ \ell_i $ unknown. The goal is to estimate the parameter vectors efficiently with low sample and computational costs. This problem presents significant challenges as one needs to simultaneously solve the demixing problem of recovering the labels $ \ell_i $ as well as the estimation problem of recovering the sparse vectors $ β^{(\ell)} $. Our solution to the problem leverages the connection between modern coding theory and statistical inference. We introduce a new algorithm, Mixed-Coloring, which samples the mixture strategically using query vectors $ {x}_i $ constructed based on ideas from sparse graph codes. Our novel code design allows for both efficient demixing and parameter estimation. In the noiseless setting, for a constant number of sparse parameter vectors, our algorithm achieves the order-optimal sample and time complexities of $Θ(K)$. In the presence of Gaussian noise, for the problem with two parameter vectors (i.e., $L=2$), we show that the Robust Mixed-Coloring algorithm achieves near-optimal $Θ(K polylog(n))$ sample and time complexities. When $K=O(n^α)$ for some constant $α\in(0,1)$ (i.e., $K$ is sublinear in $n$), we can achieve sample and time complexities both sublinear in the ambient dimension. In one of our experiments, to recover a mixture of two regressions with dimension $n=500$ and sparsity $K=50$, our algorithm is more than $300$ times faster than EM algorithm, with about one third of its sample cost.

cs.IT↗

arXiv2018

Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization

This paper studies the Tensor Robust Principal Component (TRPCA) problem which extends the known Robust PCA (Candes et al. 2011) to the tensor case. Our model is based on a new tensor Singular Value Decomposition (t-SVD) (Kilmer and Martin 2011) and its induced tensor tubal rank and tensor nuclear norm. Consider that we have a 3-way tensor ${\mathcal{X}}\in\mathbb{R}^{n_1\times n_2\times n_3}$ such that ${\mathcal{X}}={\mathcal{L}}_0+{\mathcal{E}}_0$, where ${\mathcal{L}}_0$ has low tubal rank and ${\mathcal{E}}_0$ is sparse. Is that possible to recover both components? In this work, we prove that under certain suitable assumptions, we can recover both the low-rank and the sparse components exactly by simply solving a convex program whose objective is a weighted combination of the tensor nuclear norm and the $\ell_1$-norm, i.e., $\min_{\mathcal{L},\ {\mathcal{E}}} \ \|{\mathcal{L}}\|_*+λ\|{\mathcal{E}}\|_1, \ \text{s.t.} \ {\mathcal{X}}={\mathcal{L}}+{\mathcal{E}}$, where $λ= {1}/{\sqrt{\max(n_1,n_2)n_3}}$. Interestingly, TRPCA involves RPCA as a special case when $n_3=1$ and thus it is a simple and elegant tensor extension of RPCA. Also numerical experiments verify our theory and the application for the image denoising demonstrates the effectiveness of our method.

cs.CV↗

arXiv2018

Harnessing Structures in Big Data via Guaranteed Low-Rank Matrix Estimation

Low-rank modeling plays a pivotal role in signal processing and machine learning, with applications ranging from collaborative filtering, video surveillance, medical imaging, to dimensionality reduction and adaptive filtering. Many modern high-dimensional data and interactions thereof can be modeled as lying approximately in a low-dimensional subspace or manifold, possibly with additional structures, and its proper exploitations lead to significant reduction of costs in sensing, computation and storage. In recent years, there is a plethora of progress in understanding how to exploit low-rank structures using computationally efficient procedures in a provable manner, including both convex and nonconvex approaches. On one side, convex relaxations such as nuclear norm minimization often lead to statistically optimal procedures for estimating low-rank matrices, where first-order methods are developed to address the computational challenges; on the other side, there is emerging evidence that properly designed nonconvex procedures, such as projected gradient descent, often provide globally optimal solutions with a much lower computational cost in many problems. This survey article will provide a unified overview of these recent advances on low-rank matrix estimation from incomplete measurements. Attention is paid to rigorous characterization of the performance of these algorithms, and to problems where the low-rank matrix have additional structural properties that require new algorithmic designs and theoretical analysis.

stat.ML↗