arXiv subjects

Han Liu

Publications and source records attributed to Han Liu.

At least 343 records · Page 19Linked to original sources

arXiv2018

A Unified Framework for Testing High Dimensional Parameters: A Data-Adaptive Approach

High dimensional hypothesis test deals with models in which the number of parameters is significantly larger than the sample size. Existing literature develops a variety of individual tests. Some of them are sensitive to the dense and small disturbance, and others are sensitive to the sparse and large disturbance. Hence, the powers of these tests depend on the assumption of the alternative scenario. This paper provides a unified framework for developing new tests which are adaptive to a large variety of alternative scenarios in high dimensions. In particular, our framework includes arbitrary hypotheses which can be tested using high dimensional $U$-statistic based vectors. Under this framework, we first develop a broad family of tests based on a novel variant of the $L_p$-norm with $p\in \{1,\dots,\infty\}$. We then combine these tests to construct a data-adaptive test that is simultaneously powerful under various alternative scenarios. To obtain the asymptotic distributions of these tests, we utilize the multiplier bootstrap for $U$-statistics. In addition, we consider the computational aspect of the bootstrap method and propose a novel low-cost scheme. We prove the optimality of the proposed tests. Thorough numerical results on simulated and real datasets are provided to support our theory.

math.ST↗

arXiv2018

Property Testing in High Dimensional Ising models

This paper explores the information-theoretic limitations of graph property testing in zero-field Ising models. Instead of learning the entire graph structure, sometimes testing a basic graph property such as connectivity, cycle presence or maximum clique size is a more relevant and attainable objective. Since property testing is more fundamental than graph recovery, any necessary conditions for property testing imply corresponding conditions for graph recovery, while custom property tests can be statistically and/or computationally more efficient than graph recovery based algorithms. Understanding the statistical complexity of property testing requires the distinction of ferromagnetic (i.e., positive interactions only) and general Ising models. Using combinatorial constructs such as graph packing and strong monotonicity, we characterize how target properties affect the corresponding minimax upper and lower bounds within the realm of ferromagnets. On the other hand, by studying the detection of an antiferromagnetic (i.e., negative interactions only) Curie-Weiss model buried in Rademacher noise, we show that property testing is strictly more challenging over general Ising models. In terms of methodological development, we propose two types of correlation based tests: computationally efficient screening for ferromagnets, and score type tests for general models, including a fast cycle presence test. Our correlation screening tests match the information-theoretic bounds for property testing in ferromagnets.

math.ST↗

arXiv2018

On Stein's Identity and Near-Optimal Estimation in High-dimensional Index Models

We consider estimating the parametric components of semi-parametric multiple index models in a high-dimensional and non-Gaussian setting. Such models form a rich class of non-linear models with applications to signal processing, machine learning and statistics. Our estimators leverage the score function based first and second-order Stein's identities and do not require the covariates to satisfy Gaussian or elliptical symmetry assumptions common in the literature. Moreover, to handle score functions and responses that are heavy-tailed, our estimators are constructed via carefully thresholding their empirical counterparts. We show that our estimator achieves near-optimal statistical rate of convergence in several settings. We supplement our theoretical results via simulation experiments that confirm the theory.

math.ST↗

arXiv2018

The design of ultra-strong laser with one-dimensional function photonic crystal

With the optical kerr effect, the conventional photonic crystal can be turned into the function photonic crystal under the action of pump light. In the paper, we have designed the ultra-strong light source and laser with one-dimensional function photonic crystal. When the incident light is the ordinary light, the output is ultra-strong light source, and when the incident light is the low power laser, the output is ultra-strong laser, the maximum magnification can be reached $10^{80}$ and even more. Otherwise, we analyzed the effect of period number, medium refractive index and thickness, incident angle, the pump light irradiation way and pump light intensity on the magnification, these results shall help to optimal design ultra-strong light source and laser.

physics.gen-ph↗

arXiv2018

Low modes regularity criterion for a chemotaxis-Navier-Stokes system

In this paper we study the regularity problem of a three dimensional chemotaxis-Navier-Stokes system on a periodic domain. A new regularity criterion in terms of only low modes of the oxygen concentration and the fluid velocity is obtained via a wavenumber splitting approach. The result improves many existing criteria in the literature.

math.AP↗

arXiv2018

Feedback-Based Tree Search for Reinforcement Learning

Inspired by recent successes of Monte-Carlo tree search (MCTS) in a number of artificial intelligence (AI) application domains, we propose a model-based reinforcement learning (RL) technique that iteratively applies MCTS on batches of small, finite-horizon versions of the original infinite-horizon Markov decision process. The terminal condition of the finite-horizon problems, or the leaf-node evaluator of the decision tree generated by MCTS, is specified using a combination of an estimated value function and an estimated policy function. The recommendations generated by the MCTS procedure are then provided as feedback in order to refine, through classification and regression, the leaf-node evaluator for the next iteration. We provide the first sample complexity bounds for a tree search-based RL algorithm. In addition, we show that a deep neural network implementation of the technique can create a competitive AI agent for the popular multi-player online battle arena (MOBA) game King of Glory.

cs.AI↗

arXiv2018

An Extreme-Value Approach for Testing the Equality of Large U-Statistic Based Correlation Matrices

There has been an increasing interest in testing the equality of large Pearson's correlation matrices. However, in many applications it is more important to test the equality of large rank-based correlation matrices since they are more robust to outliers and nonlinearity. Unlike the Pearson's case, testing the equality of large rank-based statistics has not been well explored and requires us to develop new methods and theory. In this paper, we provide a framework for testing the equality of two large U-statistic based correlation matrices, which include the rank-based correlation matrices as special cases. Our approach exploits extreme value statistics and the Jackknife estimator for uncertainty assessment and is valid under a fully nonparametric model. Theoretically, we develop a theory for testing the equality of U-statistic based correlation matrices. We then apply this theory to study the problem of testing large Kendall's tau correlation matrices and demonstrate its optimality. For proving this optimality, a novel construction of least favourable distributions is developed for the correlation matrix comparison.

math.ST↗

arXiv2018

The Enemy Among Us: Detecting Hate Speech with Threats Based 'Othering' Language Embeddings

Offensive or antagonistic language targeted at individuals and social groups based on their personal characteristics (also known as cyber hate speech or cyberhate) has been frequently posted and widely circulated viathe World Wide Web. This can be considered as a key risk factor for individual and societal tension linked toregional instability. Automated Web-based cyberhate detection is important for observing and understandingcommunity and regional societal tension - especially in online social networks where posts can be rapidlyand widely viewed and disseminated. While previous work has involved using lexicons, bags-of-words orprobabilistic language parsing approaches, they often suffer from a similar issue which is that cyberhate can besubtle and indirect - thus depending on the occurrence of individual words or phrases can lead to a significantnumber of false negatives, providing inaccurate representation of the trends in cyberhate. This problemmotivated us to challenge thinking around the representation of subtle language use, such as references toperceived threats from "the other" including immigration or job prosperity in a hateful context. We propose anovel framework that utilises language use around the concept of "othering" and intergroup threat theory toidentify these subtleties and we implement a novel classification method using embedding learning to computesemantic distances between parts of speech considered to be part of an "othering" narrative. To validate ourapproach we conduct several experiments on different types of cyberhate, namely religion, disability, race andsexual orientation, with F-measure scores for classifying hateful instances obtained through applying ourmodel of 0.93, 0.86, 0.97 and 0.98 respectively, providing a significant improvement in classifier accuracy overthe state-of-the-art

cs.CL↗

arXiv2018

Fully Decentralized Multi-Agent Reinforcement Learning with Networked Agents

We consider the problem of \emph{fully decentralized} multi-agent reinforcement learning (MARL), where the agents are located at the nodes of a time-varying communication network. Specifically, we assume that the reward functions of the agents might correspond to different tasks, and are only known to the corresponding agent. Moreover, each agent makes individual decisions based on both the information observed locally and the messages received from its neighbors over the network. Within this setting, the collective goal of the agents is to maximize the globally averaged return over the network through exchanging information with their neighbors. To this end, we propose two decentralized actor-critic algorithms with function approximation, which are applicable to large-scale MARL problems where both the number of states and the number of agents are massively large. Under the decentralized structure, the actor step is performed individually by each agent with no need to infer the policies of others. For the critic step, we propose a consensus update via communication over the network. Our algorithms are fully incremental and can be implemented in an online fashion. Convergence analyses of the algorithms are provided when the value functions are approximated within the class of linear functions. Extensive simulation results with both linear and nonlinear function approximations are presented to validate the proposed algorithms. Our work appears to be the first study of fully decentralized MARL algorithms for networked agents with function approximation, with provable convergence guarantees.

cs.LG↗

arXiv2018

Combinatorial Inference for Graphical Models

We propose a new family of combinatorial inference problems for graphical models. Unlike classical statistical inference where the main interest is point estimation or parameter testing, combinatorial inference aims at testing the global structure of the underlying graph. Examples include testing the graph connectivity, the presence of a cycle of certain size, or the maximum degree of the graph. To begin with, we develop a unified theory for the fundamental limits of a large family of combinatorial inference problems. We propose new concepts including structural packing and buffer entropies to characterize how the complexity of combinatorial graph structures impacts the corresponding minimax lower bounds. On the other hand, we propose a family of novel and practical structural testing algorithms to match the lower bounds. We provide thorough numerical results on both synthetic graphical models and brain networks to illustrate the usefulness of these proposed methods.

math.ST↗

arXiv2018

Kernel Meets Sieve: Post-Regularization Confidence Bands for Sparse Additive Model

We develop a novel procedure for constructing confidence bands for components of a sparse additive model. Our procedure is based on a new kernel-sieve hybrid estimator that combines two most popular nonparametric estimation methods in the literature, the kernel regression and the spline method, and is of interest in its own right. Existing methods for fitting sparse additive model are primarily based on sieve estimators, while the literature on confidence bands for nonparametric models are primarily based upon kernel or local polynomial estimators. Our kernel-sieve hybrid estimator combines the best of both worlds and allows us to provide a simple procedure for constructing confidence bands in high-dimensional sparse additive models. We prove that the confidence bands are asymptotically honest by studying approximation with a Gaussian process. Thorough numerical results on both synthetic data and real-world neuroscience data are provided to demonstrate the efficacy of the theory.

stat.ML↗

arXiv2018

Post-Regularization Inference for Time-Varying Nonparanormal Graphical Models

We propose a novel class of time-varying nonparanormal graphical models, which allows us to model high dimensional heavy-tailed systems and the evolution of their latent network structures. Under this model, we develop statistical tests for presence of edges both locally at a fixed index value and globally over a range of values. The tests are developed for a high-dimensional regime, are robust to model selection mistakes and do not require commonly assumed minimum signal strength. The testing procedures are based on a high dimensional, debiasing-free moment estimator, which uses a novel kernel smoothed Kendall's tau correlation matrix as an input statistic. The estimator consistently estimates the latent inverse Pearson correlation matrix uniformly in both the index variable and kernel bandwidth. Its rate of convergence is shown to be minimax optimal. Our method is supported by thorough numerical simulations and an application to a neural imaging data set.

stat.ML↗

arXiv2018

Symmetry, Saddle Points, and Global Optimization Landscape of Nonconvex Matrix Factorization

We propose a general theory for studying the \xl{landscape} of nonconvex \xl{optimization} with underlying symmetric structures \tz{for a class of machine learning problems (e.g., low-rank matrix factorization, phase retrieval, and deep linear neural networks)}. In specific, we characterize the locations of stationary points and the null space of Hessian matrices \xl{of the objective function} via the lens of invariant groups\removed{for associated optimization problems, including low-rank matrix factorization, phase retrieval, and deep linear neural networks}. As a major motivating example, we apply the proposed general theory to characterize the global \xl{landscape} of the \xl{nonconvex optimization in} low-rank matrix factorization problem. In particular, we illustrate how the rotational symmetry group gives rise to infinitely many nonisolated strict saddle points and equivalent global minima of the objective function. By explicitly identifying all stationary points, we divide the entire parameter space into three regions: ($\cR_1$) the region containing the neighborhoods of all strict saddle points, where the objective has negative curvatures; ($\cR_2$) the region containing neighborhoods of all global minima, where the objective enjoys strong convexity along certain directions; and ($\cR_3$) the complement of the above regions, where the gradient has sufficiently large magnitudes. We further extend our result to the matrix sensing problem. Such global landscape implies strong global convergence guarantees for popular iterative algorithms with arbitrary initial solutions.

cs.LG↗

arXiv2017

Nonconvex Sparse Learning via Stochastic Optimization with Progressive Variance Reduction

We propose a stochastic variance reduced optimization algorithm for solving sparse learning problems with cardinality constraints. Sufficient conditions are provided, under which the proposed algorithm enjoys strong linear convergence guarantees and optimal estimation accuracy in high dimensions. We further extend the proposed algorithm to an asynchronous parallel variant with a near linear speedup. Numerical experiments demonstrate the efficiency of our algorithm in terms of both parameter estimation and computational performance.

cs.LG↗

arXiv2017

Homotopy Parametric Simplex Method for Sparse Learning

High dimensional sparse learning has imposed a great computational challenge to large scale data analysis. In this paper, we are interested in a broad class of sparse learning approaches formulated as linear programs parametrized by a {\em regularization factor}, and solve them by the parametric simplex method (PSM). Our parametric simplex method offers significant advantages over other competing methods: (1) PSM naturally obtains the complete solution path for all values of the regularization parameter; (2) PSM provides a high precision dual certificate stopping criterion; (3) PSM yields sparse solutions through very few iterations, and the solution sparsity significantly reduces the computational cost per iteration. Particularly, we demonstrate the superiority of PSM over various sparse learning approaches, including Dantzig selector for sparse linear regression, LAD-Lasso for sparse robust linear regression, CLIME for sparse precision matrix estimation, sparse differential network estimation, and sparse Linear Programming Discriminant (LPD) analysis. We then provide sufficient conditions under which PSM always outputs sparse solutions such that its computational performance can be significantly boosted. Thorough numerical experiments are provided to demonstrate the outstanding performance of the PSM method.

cs.LG↗

arXiv2017

On Faster Convergence of Cyclic Block Coordinate Descent-type Methods for Strongly Convex Minimization

The cyclic block coordinate descent-type (CBCD-type) methods, which performs iterative updates for a few coordinates (a block) simultaneously throughout the procedure, have shown remarkable computational performance for solving strongly convex minimization problems. Typical applications include many popular statistical machine learning methods such as elastic-net regression, ridge penalized logistic regression, and sparse additive regression. Existing optimization literature has shown that for strongly convex minimization, the CBCD-type methods attain iteration complexity of $\mathcal{O}(p\log(1/ε))$, where $ε$ is a pre-specified accuracy of the objective value, and $p$ is the number of blocks. However, such iteration complexity explicitly depends on $p$, and therefore is at least $p$ times worse than the complexity $\mathcal{O}(\log(1/ε))$ of gradient descent (GD) methods. To bridge this theoretical gap, we propose an improved convergence analysis for the CBCD-type methods. In particular, we first show that for a family of quadratic minimization problems, the iteration complexity $\mathcal{O}(\log^2(p)\cdot\log(1/ε))$ of the CBCD-type methods matches that of the GD methods in term of dependency on $p$, up to a $\log^2 p$ factor. Thus our complexity bounds are sharper than the existing bounds by at least a factor of $p/\log^2(p)$. We also provide a lower bound to confirm that our improved complexity bounds are tight (up to a $\log^2 (p)$ factor), under the assumption that the largest and smallest eigenvalues of the Hessian matrix do not scale with $p$. Finally, we generalize our analysis to other strongly convex minimization problems beyond quadratic ones.

math.OC↗

arXiv2017

A New Perspective on Robust $M$-Estimation: Finite Sample Theory and Applications to Dependence-Adjusted Multiple Testing

Heavy-tailed errors impair the accuracy of the least squares estimate, which can be spoiled by a single grossly outlying observation. As argued in the seminal work of Peter Huber in 1973 [{\it Ann. Statist.} {\bf 1} (1973) 799--821], robust alternatives to the method of least squares are sorely needed. To achieve robustness against heavy-tailed sampling distributions, we revisit the Huber estimator from a new perspective by letting the tuning parameter involved diverge with the sample size. In this paper, we develop nonasymptotic concentration results for such an adaptive Huber estimator, namely, the Huber estimator with the tuning parameter adapted to sample size, dimension, and the variance of the noise. Specifically, we obtain a sub-Gaussian-type deviation inequality and a nonasymptotic Bahadur representation when noise variables only have finite second moments. The nonasymptotic results further yield two conventional normal approximation results that are of independent interest, the Berry-Esseen inequality and Cramér-type moderate deviation. As an important application to large-scale simultaneous inference, we apply these robust normal approximation results to analyze a dependence-adjusted multiple testing procedure for moderately heavy-tailed data. It is shown that the robust dependence-adjusted procedure asymptotically controls the overall false discovery proportion at the nominal level under mild moment conditions. Thorough numerical results on both simulated and real datasets are also provided to back up our theory.

math.ST↗

arXiv2017

Second-Order Methods with Cubic Regularization Under Inexact Information

In this paper, we generalize (accelerated) Newton's method with cubic regularization under inexact second-order information for (strongly) convex optimization problems. Under mild assumptions, we provide global rate of convergence of these methods and show the explicit dependence of the rate of convergence on the problem parameters. While the complexity bounds of our presented algorithms are theoretically worse than those of their exact counterparts, they are at least as good as those of the optimal first-order methods. Our numerical experiments also show that using inexact Hessians can significantly speed up the algorithms in practice.

math.OC↗