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Yingbin Liang

Publications and source records attributed to Yingbin Liang.

At least 163 records · Page 9Linked to original sources
arXiv2017

Convergence Analysis of Proximal Gradient with Momentum for Nonconvex Optimization

In many modern machine learning applications, structures of underlying mathematical models often yield nonconvex optimization problems. Due to the intractability of nonconvexity, there is a rising need to develop efficient methods for solving general nonconvex problems with certain performance guarantee. In this work, we investigate the accelerated proximal gradient method for nonconvex programming (APGnc). The method compares between a usual proximal gradient step and a linear extrapolation step, and accepts the one that has a lower function value to achieve a monotonic decrease. In specific, under a general nonsmooth and nonconvex setting, we provide a rigorous argument to show that the limit points of the sequence generated by APGnc are critical points of the objective function. Then, by exploiting the Kurdyka-Łojasiewicz (\KL) property for a broad class of functions, we establish the linear and sub-linear convergence rates of the function value sequence generated by APGnc. We further propose a stochastic variance reduced APGnc (SVRG-APGnc), and establish its linear convergence under a special case of the \KL property. We also extend the analysis to the inexact version of these methods and develop an adaptive momentum strategy that improves the numerical performance.

cs.LG↗
arXiv2017

State-Dependent Gaussian Multiple Access Channels: New Outer Bounds and Capacity Results

This paper studies a two-user state-dependent Gaussian multiple-access channel (MAC) with state noncausally known at one encoder. Two scenarios are considered: i) each user wishes to communicate an independent message to the common receiver, and ii) the two encoders send a common message to the receiver and the non-cognitive encoder (i.e., the encoder that does not know the state) sends an independent individual message (this model is also known as the MAC with degraded message sets). For both scenarios, new outer bounds on the capacity region are derived, which improve uniformly over the best known outer bounds. In the first scenario, the two corner points of the capacity region as well as the sum rate capacity are established, and it is shown that a single-letter solution is adequate to achieve both the corner points and the sum rate capacity. Furthermore, the full capacity region is characterized in situations in which the sum rate capacity is equal to the capacity of the helper problem. The proof exploits the optimal-transportation idea of Polyanskiy and Wu (which was used previously to establish an outer bound on the capacity region of the interference channel) and the worst-case Gaussian noise result for the case in which the input and the noise are dependent.

cs.IT↗
arXiv2017

Distributed Proximal Gradient Algorithm for Partially Asynchronous Computer Clusters

With ever growing data volume and model size, an error-tolerant, communication efficient, yet versatile distributed algorithm has become vital for the success of many large-scale machine learning applications. In this work we propose m-PAPG, an implementation of the flexible proximal gradient algorithm in model parallel systems equipped with the partially asynchronous communication protocol. The worker machines communicate asynchronously with a controlled staleness bound $s$ and operate at different frequencies. We characterize various convergence properties of m-PAPG: 1) Under a general non-smooth and non-convex setting, we prove that every limit point of the sequence generated by m-PAPG is a critical point of the objective function; 2) Under an error bound condition, we prove that the function value decays linearly for every $s$ steps; 3) Under the Kurdyka-$Ł$ojasiewicz inequality, we prove that the sequences generated by m-PAPG converge to the same critical point, provided that a proximal Lipschitz condition is satisfied.

math.OC↗
arXiv2016

Nonparametric Detection of Anomalous Data Streams

A nonparametric anomalous hypothesis testing problem is investigated, in which there are totally n sequences with s anomalous sequences to be detected. Each typical sequence contains m independent and identically distributed (i.i.d.) samples drawn from a distribution p, whereas each anomalous sequence contains m i.i.d. samples drawn from a distribution q that is distinct from p. The distributions p and q are assumed to be unknown in advance. Distribution-free tests are constructed using maximum mean discrepancy as the metric, which is based on mean embeddings of distributions into a reproducing kernel Hilbert space. The probability of error is bounded as a function of the sample size m, the number s of anomalous sequences and the number n of sequences. It is then shown that with s known, the constructed test is exponentially consistent if m is greater than a constant factor of log n, for any p and q, whereas with s unknown, m should has an order strictly greater than log n. Furthermore, it is shown that no test can be consistent for arbitrary p and q if m is less than a constant factor of log n, thus the order-level optimality of the proposed test is established. Numerical results are provided to demonstrate that our tests outperform (or perform as well as) the tests based on other competitive approaches under various cases.

cs.LG↗
arXiv2016

Reshaped Wirtinger Flow and Incremental Algorithm for Solving Quadratic System of Equations

We study the phase retrieval problem, which solves quadratic system of equations, i.e., recovers a vector $\boldsymbol{x}\in \mathbb{R}^n$ from its magnitude measurements $y_i=|\langle \boldsymbol{a}_i, \boldsymbol{x}\rangle|, i=1,..., m$. We develop a gradient-like algorithm (referred to as RWF representing reshaped Wirtinger flow) by minimizing a nonconvex nonsmooth loss function. In comparison with existing nonconvex Wirtinger flow (WF) algorithm \cite{candes2015phase}, although the loss function becomes nonsmooth, it involves only the second power of variable and hence reduces the complexity. We show that for random Gaussian measurements, RWF enjoys geometric convergence to a global optimal point as long as the number $m$ of measurements is on the order of $n$, the dimension of the unknown $\boldsymbol{x}$. This improves the sample complexity of WF, and achieves the same sample complexity as truncated Wirtinger flow (TWF) \cite{chen2015solving}, but without truncation in gradient loop. Furthermore, RWF costs less computationally than WF, and runs faster numerically than both WF and TWF. We further develop the incremental (stochastic) reshaped Wirtinger flow (IRWF) and show that IRWF converges linearly to the true signal. We further establish performance guarantee of an existing Kaczmarz method for the phase retrieval problem based on its connection to IRWF. We also empirically demonstrate that IRWF outperforms existing ITWF algorithm (stochastic version of TWF) as well as other batch algorithms.

stat.ML↗
arXiv2016

A Kernel-Based Nonparametric Test for Anomaly Detection over Line Networks

The nonparametric problem of detecting existence of an anomalous interval over a one dimensional line network is studied. Nodes corresponding to an anomalous interval (if exists) receive samples generated by a distribution q, which is different from the distribution p that generates samples for other nodes. If anomalous interval does not exist, then all nodes receive samples generated by p. It is assumed that the distributions p and q are arbitrary, and are unknown. In order to detect whether an anomalous interval exists, a test is built based on mean embeddings of distributions into a reproducing kernel Hilbert space (RKHS) and the metric of maximummean discrepancy (MMD). It is shown that as the network size n goes to infinity, if the minimum length of candidate anomalous intervals is larger than a threshold which has the order O(log n), the proposed test is asymptotically successful, i.e., the probability of detection error approaches zero asymptotically. An efficient algorithm to perform the test with substantial computational complexity reduction is proposed, and is shown to be asymptotically successful if the condition on the minimum length of candidate anomalous interval is satisfied. Numerical results are provided, which are consistent with the theoretical results.

cs.IT↗
arXiv2016

Nonparametric Detection of Geometric Structures over Networks

Nonparametric detection of existence of an anomalous structure over a network is investigated. Nodes corresponding to the anomalous structure (if one exists) receive samples generated by a distribution q, which is different from a distribution p generating samples for other nodes. If an anomalous structure does not exist, all nodes receive samples generated by p. It is assumed that the distributions p and q are arbitrary and unknown. The goal is to design statistically consistent tests with probability of errors converging to zero as the network size becomes asymptotically large. Kernel-based tests are proposed based on maximum mean discrepancy that measures the distance between mean embeddings of distributions into a reproducing kernel Hilbert space. Detection of an anomalous interval over a line network is first studied. Sufficient conditions on minimum and maximum sizes of candidate anomalous intervals are characterized in order to guarantee the proposed test to be consistent. It is also shown that certain necessary conditions must hold to guarantee any test to be universally consistent. Comparison of sufficient and necessary conditions yields that the proposed test is order-level optimal and nearly optimal respectively in terms of minimum and maximum sizes of candidate anomalous intervals. Generalization of the results to other networks is further developed. Numerical results are provided to demonstrate the performance of the proposed tests.

stat.ML↗
arXiv2015

Universal Outlying sequence detection For Continuous Observations

The following detection problem is studied, in which there are $M$ sequences of samples out of which one outlier sequence needs to be detected. Each typical sequence contains $n$ independent and identically distributed (i.i.d.) continuous observations from a known distribution $π$, and the outlier sequence contains $n$ i.i.d. observations from an outlier distribution $μ$, which is distinct from $π$, but otherwise unknown. A universal test based on KL divergence is built to approximate the maximum likelihood test, with known $π$ and unknown $μ$. A data-dependent partitions based KL divergence estimator is employed. Such a KL divergence estimator is further shown to converge to its true value exponentially fast when the density ratio satisfies $0<K_1\leq \frac{dμ}{dπ}\leq K_2$, where $K_1$ and $K_2$ are positive constants, and this further implies that the test is exponentially consistent. The performance of the test is compared with that of a recently introduced test for this problem based on the machine learning approach of maximum mean discrepancy (MMD). We identify regimes in which the KL divergence based test is better than the MMD based test.

cs.IT↗
arXiv2014

The Capacity Region of the Source-Type Model for Secret Key and Private Key Generation

The problem of simultaneously generating a secret key (SK) and private key (PK) pair among three terminals via public discussion is investigated, in which each terminal observes a component of correlated sources. All three terminals are required to generate a common secret key concealed from an eavesdropper that has access to public discussion, while two designated terminals are required to generate an extra private key concealed from both the eavesdropper and the remaining terminal. An outer bound on the SK-PK capacity region was established in [1], and was shown to be achievable for one case. In this paper, achievable schemes are designed to achieve the outer bound for the remaining two cases, and hence the SK-PK capacity region is established in general. The main technique lies in the novel design of a random binning-joint decoding scheme that achieves the existing outer bound.

cs.IT↗
arXiv2014

Parallel Gaussian Networks with a Common State-Cognitive Helper

A class of state-dependent parallel networks with a common state-cognitive helper, in which $K$ transmitters wish to send $K$ messages to their corresponding receivers over $K$ state-corrupted parallel channels, and a helper who knows the state information noncausally wishes to assist these receivers to cancel state interference. Furthermore, the helper also has its own message to be sent simultaneously to its corresponding receiver. Since the state information is known only to the helper, but not to the corresponding transmitters $1,\dots,K$, transmitter-side state cognition and receiver-side state interference are mismatched. Our focus is on the high state power regime, i.e., the state power goes to infinity. Three (sub)models are studied. Model I serves as a basic model, which consists of only one transmitter-receiver (with state corruption) pair in addition to a helper that assists the receiver to cancel state in addition to transmitting its own message. Model II consists of two transmitter-receiver pairs in addition to a helper, and only one receiver is interfered by a state sequence. Model III generalizes model I include multiple transmitter-receiver pairs with each receiver corrupted by independent state. For all models, inner and outer bounds on the capacity region are derived, and comparison of the two bounds leads to characterization of either full or partial boundary of the capacity region under various channel parameters.

cs.IT↗
arXiv2014

An Information Theoretic Approach to Secret Sharing

A novel information theoretic approach is proposed to solve the secret sharing problem, in which a dealer distributes one or multiple secrets among a set of participants that for each secret only qualified sets of users can recover it by pooling their shares together while non-qualified sets of users obtain no information about the secret even if they pool their shares together. While existing secret sharing systems (implicitly) assume that communications between the dealer and participants are noiseless, this paper takes a more practical assumption that the dealer delivers shares to the participants via a noisy broadcast channel. An information theoretic approach is proposed, which exploits the channel as additional resources to achieve secret sharing requirements. In this way, secret sharing problems can be reformulated as equivalent secure communication problems via wiretap channels, and can be solved by employing powerful information theoretic security techniques. This approach is first developed for the classic secret sharing problem, in which only one secret is to be shared. This classic problem is shown to be equivalent to a communication problem over a compound wiretap channel. The lower and upper bounds on the secrecy capacity of the compound channel provide the corresponding bounds on the secret sharing rate. The power of the approach is further demonstrated by a more general layered multi-secret sharing problem, which is shown to be equivalent to the degraded broadcast multiple-input multiple-output (MIMO) channel with layered decoding and secrecy constraints. The secrecy capacity region for the degraded MIMO broadcast channel is characterized, which provides the secret sharing capacity region. Furthermore, these secure encoding schemes that achieve the secrecy capacity region provide an information theoretic scheme for sharing the secrets.

cs.IT↗
arXiv2013

Sharp Threshold for Multivariate Multi-Response Linear Regression via Block Regularized Lasso

In this paper, we investigate a multivariate multi-response (MVMR) linear regression problem, which contains multiple linear regression models with differently distributed design matrices, and different regression and output vectors. The goal is to recover the support union of all regression vectors using $l_1/l_2$-regularized Lasso. We characterize sufficient and necessary conditions on sample complexity \emph{as a sharp threshold} to guarantee successful recovery of the support union. Namely, if the sample size is above the threshold, then $l_1/l_2$-regularized Lasso correctly recovers the support union; and if the sample size is below the threshold, $l_1/l_2$-regularized Lasso fails to recover the support union. In particular, the threshold precisely captures the impact of the sparsity of regression vectors and the statistical properties of the design matrices on sample complexity. Therefore, the threshold function also captures the advantages of joint support union recovery using multi-task Lasso over individual support recovery using single-task Lasso.

cs.LG↗
arXiv2009

Fading Cognitive Multiple-Access Channels With Confidential Messages

The fading cognitive multiple-access channel with confidential messages (CMAC-CM) is investigated, in which two users attempt to transmit common information to a destination and user 1 also has confidential information intended for the destination. User 1 views user 2 as an eavesdropper and wishes to keep its confidential information as secret as possible from user 2. The multiple-access channel (both the user-to-user channel and the user-to-destination channel) is corrupted by multiplicative fading gain coefficients in addition to additive white Gaussian noise. The channel state information (CSI) is assumed to be known at both the users and the destination. A parallel CMAC-CM with independent subchannels is first studied. The secrecy capacity region of the parallel CMAC-CM is established, which yields the secrecy capacity region of the parallel CMAC-CM with degraded subchannels. Next, the secrecy capacity region is established for the parallel Gaussian CMAC-CM, which is used to study the fading CMAC-CM. When both users know the CSI, they can dynamically change their transmission powers with the channel realization to achieve the optimal performance. The closed-form power allocation function that achieves every boundary point of the secrecy capacity region is derived.

cs.IT↗
arXiv2009

Multiple-Input Multiple-Output Gaussian Broadcast Channels with Common and Confidential Messages

This paper considers the problem of the multiple-input multiple-output (MIMO) Gaussian broadcast channel with two receivers (receivers 1 and 2) and two messages: a common message intended for both receivers and a confidential message intended only for receiver 1 but needing to be kept asymptotically perfectly secure from receiver 2. A matrix characterization of the secrecy capacity region is established via a channel enhancement argument. The enhanced channel is constructed by first splitting receiver 1 into two virtual receivers and then enhancing only the virtual receiver that decodes the confidential message. The secrecy capacity region of the enhanced channel is characterized using an extremal entropy inequality previously established for characterizing the capacity region of a degraded compound MIMO Gaussian broadcast channel.

cs.IT↗
arXiv2009

On the Compound MIMO Broadcast Channels with Confidential Messages

We study the compound multi-input multi-output (MIMO) broadcast channel with confidential messages (BCC), where one transmitter sends a common message to two receivers and two confidential messages respectively to each receiver. The channel state may take one of a finite set of states, and the transmitter knows the state set but does not know the realization of the state. We study achievable rates with perfect secrecy in the high SNR regime by characterizing an achievable secrecy degree of freedom (s.d.o.f.) region for two models, the Gaussian MIMO-BCC and the ergodic fading multi-input single-output (MISO)-BCC without a common message. We show that by exploiting an additional temporal dimension due to state variation in the ergodic fading model, the achievable s.d.o.f. region can be significantly improved compared to the Gaussian model with a constant state, although at the price of a larger delay.

cs.IT↗
arXiv2009

Opportunistic Communications in Fading Multiaccess Relay Channels

The problem of optimal resource allocation is studied for ergodic fading orthogonal multiaccess relay channels (MARCs) in which the users (sources) communicate with a destination with the aid of a half-duplex relay that transmits on a channel orthogonal to that used by the transmitting sources. Under the assumption that the instantaneous fading state information is available at all nodes, the maximum sum-rate and the optimal user and relay power allocations (policies) are developed for a decode-and-forward (DF) relay. With the observation that a DF relay results in two multiaccess channels, one at the relay and the other at the destination, a single known lemma on the sum-rate of two intersecting polymatroids is used to determine the DF sum-rate and the optimal user and relay policies. The lemma also enables a broad topological classification of fading MARCs into one of three types. The first type is the set of partially clustered MARCs where a user is clustered either with the relay or with the destination such that the users waterfill on their bottle-neck links to the distant receiver. The second type is the set of clustered MARCs where all users are either proximal to the relay or to the destination such that opportunistic multiuser scheduling to one of the receivers is optimal. The third type consists of arbitrarily clustered MARCs which are a combination of the first two types, and for this type it is shown that the optimal policies are opportunistic non-waterfilling solutions. The analysis is extended to develop the rate region of a K-user orthogonal half-duplex MARC. Finally, cutset outer bounds are used to show that DF achieves the capacity region for a class of clustered orthogonal half-duplex MARCs.

cs.IT↗