arXiv subjects

Guang Cheng

Publications and source records attributed to Guang Cheng.

At least 127 records · Page 7Linked to original sources

arXiv2022

Attention Enables Zero Approximation Error

Deep learning models have been widely applied in various aspects of daily life. Many variant models based on deep learning structures have achieved even better performances. Attention-based architectures have become almost ubiquitous in deep learning structures. Especially, the transformer model has now defeated the convolutional neural network in image classification tasks to become the most widely used tool. However, the theoretical properties of attention-based models are seldom considered. In this work, we show that with suitable adaptations, the single-head self-attention transformer with a fixed number of transformer encoder blocks and free parameters is able to generate any desired polynomial of the input with no error. The number of transformer encoder blocks is the same as the degree of the target polynomial. Even more exciting, we find that these transformer encoder blocks in this model do not need to be trained. As a direct consequence, we show that the single-head self-attention transformer with increasing numbers of free parameters is universal. These surprising theoretical results clearly explain the outstanding performances of the transformer model and may shed light on future modifications in real applications. We also provide some experiments to verify our theoretical result.

cs.LG↗

arXiv2022

Benefit of Interpolation in Nearest Neighbor Algorithms

In some studies \citep[e.g.,][]{zhang2016understanding} of deep learning, it is observed that over-parametrized deep neural networks achieve a small testing error even when the training error is almost zero. Despite numerous works towards understanding this so-called "double descent" phenomenon \citep[e.g.,][]{belkin2018reconciling,belkin2019two}, in this paper, we turn into another way to enforce zero training error (without over-parametrization) through a data interpolation mechanism. Specifically, we consider a class of interpolated weighting schemes in the nearest neighbors (NN) algorithms. By carefully characterizing the multiplicative constant in the statistical risk, we reveal a U-shaped performance curve for the level of data interpolation in both classification and regression setups. This sharpens the existing result \citep{belkin2018does} that zero training error does not necessarily jeopardize predictive performances and claims a counter-intuitive result that a mild degree of data interpolation actually {\em strictly} improve the prediction performance and statistical stability over those of the (un-interpolated) $k$-NN algorithm. In the end, the universality of our results, such as change of distance measure and corrupted testing data, will also be discussed.

stat.ML↗

arXiv2022

Unlabeled Data Help: Minimax Analysis and Adversarial Robustness

The recent proposed self-supervised learning (SSL) approaches successfully demonstrate the great potential of supplementing learning algorithms with additional unlabeled data. However, it is still unclear whether the existing SSL algorithms can fully utilize the information of both labelled and unlabeled data. This paper gives an affirmative answer for the reconstruction-based SSL algorithm \citep{lee2020predicting} under several statistical models. While existing literature only focuses on establishing the upper bound of the convergence rate, we provide a rigorous minimax analysis, and successfully justify the rate-optimality of the reconstruction-based SSL algorithm under different data generation models. Furthermore, we incorporate the reconstruction-based SSL into the existing adversarial training algorithms and show that learning from unlabeled data helps improve the robustness.

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arXiv2021

Regularization Matters: A Nonparametric Perspective on Overparametrized Neural Network

Overparametrized neural networks trained by gradient descent (GD) can provably overfit any training data. However, the generalization guarantee may not hold for noisy data. From a nonparametric perspective, this paper studies how well overparametrized neural networks can recover the true target function in the presence of random noises. We establish a lower bound on the $L_2$ estimation error with respect to the GD iterations, which is away from zero without a delicate scheme of early stopping. In turn, through a comprehensive analysis of $\ell_2$-regularized GD trajectories, we prove that for overparametrized one-hidden-layer ReLU neural network with the $\ell_2$ regularization: (1) the output is close to that of the kernel ridge regression with the corresponding neural tangent kernel; (2) minimax {optimal} rate of $L_2$ estimation error can be achieved. Numerical experiments confirm our theory and further demonstrate that the $\ell_2$ regularization approach improves the training robustness and works for a wider range of neural networks.

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arXiv2021

Variance Reduction on General Adaptive Stochastic Mirror Descent

In this work, we investigate the idea of variance reduction by studying its properties with general adaptive mirror descent algorithms in nonsmooth nonconvex finite-sum optimization problems. We propose a simple yet generalized framework for variance reduced adaptive mirror descent algorithms named SVRAMD and provide its convergence analysis in both the nonsmooth nonconvex problem and the P-L conditioned problem. We prove that variance reduction reduces the SFO complexity of adaptive mirror descent algorithms and thus accelerates their convergence. In particular, our general theory implies that variance reduction can be applied to algorithms using time-varying step sizes and self-adaptive algorithms such as AdaGrad and RMSProp. Moreover, the convergence rates of SVRAMD recover the best existing rates of non-adaptive variance reduced mirror descent algorithms without complicated algorithmic components. Extensive experiments in deep learning validate our theoretical findings.

stat.ML↗

arXiv2021

A Quantitative Framework for Network Resilience Evaluation using Dynamic Bayesian Network

Measuring and evaluating network resilience has become an important aspect since the network is vulnerable to both uncertain disturbances and malicious attacks. Networked systems are often composed of many dynamic components and change over time, which makes it difficult for existing methods to access the changeable situation of network resilience. This paper establishes a novel quantitative framework for evaluating network resilience using the Dynamic Bayesian Network. The proposed framework can be used to evaluate the network's multi-stage resilience processes when suffering various attacks and recoveries. First, we define the dynamic capacities of network components and establish the network's five core resilience capabilities to describe the resilient networking stages including preparation, resistance, adaptation, recovery, and evolution; the five core resilience capabilities consist of rapid response capability, sustained resistance capability, continuous running capability, rapid convergence capability, and dynamic evolution capability. Then, we employ a two-time slices approach based on the Dynamic Bayesian Network to quantify five crucial performances of network resilience based on core capabilities proposed above. The proposed approach can ensure the time continuity of resilience evaluation in time-varying networks. Finally, our proposed evaluation framework is applied to different attacks and recovery conditions in typical simulations and real-world network topology. Results and comparisons with extant studies indicate that the proposed method can achieve a more accurate and comprehensive evaluation and can be applied to network scenarios under various attack and recovery intensities.

cs.NI↗

arXiv2021

On the Generalization Properties of Adversarial Training

Modern machine learning and deep learning models are shown to be vulnerable when testing data are slightly perturbed. Existing theoretical studies of adversarial training algorithms mostly focus on either adversarial training losses or local convergence properties. In contrast, this paper studies the generalization performance of a generic adversarial training algorithm. Specifically, we consider linear regression models and two-layer neural networks (with lazy training) using squared loss under low-dimensional and high-dimensional regimes. In the former regime, after overcoming the non-smoothness of adversarial training, the adversarial risk of the trained models can converge to the minimal adversarial risk. In the latter regime, we discover that data interpolation prevents the adversarially robust estimator from being consistent. Therefore, inspired by successes of the least absolute shrinkage and selection operator (LASSO), we incorporate the L1 penalty in the high dimensional adversarial learning and show that it leads to consistent adversarially robust estimation. A series of numerical studies are conducted to demonstrate how the smoothness and L1 penalization help improve the adversarial robustness of DNN models.

stat.ML↗

arXiv2021

Stein Neural Sampler

We propose two novel samplers to generate high-quality samples from a given (un-normalized) probability density. Motivated by the success of generative adversarial networks, we construct our samplers using deep neural networks that transform a reference distribution to the target distribution. Training schemes are developed to minimize two variations of the Stein discrepancy, which is designed to work with un-normalized densities. Once trained, our samplers are able to generate samples instantaneously. We show that the proposed methods are theoretically sound and experience fewer convergence issues compared with traditional sampling approaches according to our empirical studies.

stat.ML↗

arXiv2020

Power Iteration for Tensor PCA

In this paper, we study the power iteration algorithm for the spiked tensor model, as introduced in [44]. We give necessary and sufficient conditions for the convergence of the power iteration algorithm. When the power iteration algorithm converges, for the rank one spiked tensor model, we show the estimators for the spike strength and linear functionals of the signal are asymptotically Gaussian; for the multi-rank spiked tensor model, we show the estimators are asymptotically mixtures of Gaussian. This new phenomenon is different from the spiked matrix model. Using these asymptotic results of our estimators, we construct valid and efficient confidence intervals for spike strengths and linear functionals of the signals.

math.ST↗

arXiv2020

Adversarially Robust Estimate and Risk Analysis in Linear Regression

Adversarially robust learning aims to design algorithms that are robust to small adversarial perturbations on input variables. Beyond the existing studies on the predictive performance to adversarial samples, our goal is to understand statistical properties of adversarially robust estimates and analyze adversarial risk in the setup of linear regression models. By discovering the statistical minimax rate of convergence of adversarially robust estimators, we emphasize the importance of incorporating model information, e.g., sparsity, in adversarially robust learning. Further, we reveal an explicit connection of adversarial and standard estimates, and propose a straightforward two-stage adversarial learning framework, which facilitates to utilize model structure information to improve adversarial robustness. In theory, the consistency of the adversarially robust estimator is proven and its Bahadur representation is also developed for the statistical inference purpose. The proposed estimator converges in a sharp rate under either low-dimensional or sparse scenario. Moreover, our theory confirms two phenomena in adversarially robust learning: adversarial robustness hurts generalization, and unlabeled data help improve the generalization. In the end, we conduct numerical simulations to verify our theory.

stat.ML↗

arXiv2020

Machine Learning in/for Blockchain: Future and Challenges

Machine learning and blockchain are two of the most noticeable technologies in recent years. The first one is the foundation of artificial intelligence and big data, and the second one has significantly disrupted the financial industry. Both technologies are data-driven, and thus there are rapidly growing interests in integrating them for more secure and efficient data sharing and analysis. In this paper, we review the research on combining blockchain and machine learning technologies and demonstrate that they can collaborate efficiently and effectively. In the end, we point out some future directions and expect more researches on deeper integration of the two promising technologies.

cs.CR↗

arXiv2020

Online Forgetting Process for Linear Regression Models

Motivated by the EU's "Right To Be Forgotten" regulation, we initiate a study of statistical data deletion problems where users' data are accessible only for a limited period of time. This setting is formulated as an online supervised learning task with \textit{constant memory limit}. We propose a deletion-aware algorithm \texttt{FIFD-OLS} for the low dimensional case, and witness a catastrophic rank swinging phenomenon due to the data deletion operation, which leads to statistical inefficiency. As a remedy, we propose the \texttt{FIFD-Adaptive Ridge} algorithm with a novel online regularization scheme, that effectively offsets the uncertainty from deletion. In theory, we provide the cumulative regret upper bound for both online forgetting algorithms. In the experiment, we showed \texttt{FIFD-Adaptive Ridge} outperforms the ridge regression algorithm with fixed regularization level, and hopefully sheds some light on more complex statistical models.

stat.ML↗

arXiv2020

Efficient Variational Inference for Sparse Deep Learning with Theoretical Guarantee

Sparse deep learning aims to address the challenge of huge storage consumption by deep neural networks, and to recover the sparse structure of target functions. Although tremendous empirical successes have been achieved, most sparse deep learning algorithms are lacking of theoretical support. On the other hand, another line of works have proposed theoretical frameworks that are computationally infeasible. In this paper, we train sparse deep neural networks with a fully Bayesian treatment under spike-and-slab priors, and develop a set of computationally efficient variational inferences via continuous relaxation of Bernoulli distribution. The variational posterior contraction rate is provided, which justifies the consistency of the proposed variational Bayes method. Notably, our empirical results demonstrate that this variational procedure provides uncertainty quantification in terms of Bayesian predictive distribution and is also capable to accomplish consistent variable selection by training a sparse multi-layer neural network.

stat.ML↗

arXiv2020

Nearly Optimal Variational Inference for High Dimensional Regression with Shrinkage Priors

We propose a variational Bayesian (VB) procedure for high-dimensional linear model inferences with heavy tail shrinkage priors, such as student-t prior. Theoretically, we establish the consistency of the proposed VB method and prove that under the proper choice of prior specifications, the contraction rate of the VB posterior is nearly optimal. It justifies the validity of VB inference as an alternative of Markov Chain Monte Carlo (MCMC) sampling. Meanwhile, comparing to conventional MCMC methods, the VB procedure achieves much higher computational efficiency, which greatly alleviates the computing burden for modern machine learning applications such as massive data analysis. Through numerical studies, we demonstrate that the proposed VB method leads to shorter computing time, higher estimation accuracy, and lower variable selection error than competitive sparse Bayesian methods.

stat.ML↗

arXiv2020

Directional Pruning of Deep Neural Networks

In the light of the fact that the stochastic gradient descent (SGD) often finds a flat minimum valley in the training loss, we propose a novel directional pruning method which searches for a sparse minimizer in or close to that flat region. The proposed pruning method does not require retraining or the expert knowledge on the sparsity level. To overcome the computational formidability of estimating the flat directions, we propose to use a carefully tuned $\ell_1$ proximal gradient algorithm which can provably achieve the directional pruning with a small learning rate after sufficient training. The empirical results demonstrate the promising results of our solution in highly sparse regime (92% sparsity) among many existing pruning methods on the ResNet50 with the ImageNet, while using only a slightly higher wall time and memory footprint than the SGD. Using the VGG16 and the wide ResNet 28x10 on the CIFAR-10 and CIFAR-100, we demonstrate that our solution reaches the same minima valley as the SGD, and the minima found by our solution and the SGD do not deviate in directions that impact the training loss. The code that reproduces the results of this paper is available at https://github.com/donlan2710/gRDA-Optimizer/tree/master/directional_pruning.

cs.LG↗

arXiv2020

Sparse Confidence Sets for Normal Mean Models

In this paper, we propose a new framework to construct confidence sets for a $d$-dimensional unknown sparse parameter $θ$ under the normal mean model $X\sim N(θ,σ^2I)$. A key feature of the proposed confidence set is its capability to account for the sparsity of $θ$, thus named as {\em sparse} confidence set. This is in sharp contrast with the classical methods, such as Bonferroni confidence intervals and other resampling based procedures, where the sparsity of $θ$ is often ignored. Specifically, we require the desired sparse confidence set to satisfy the following two conditions: (i) uniformly over the parameter space, the coverage probability for $θ$ is above a pre-specified level; (ii) there exists a random subset $S$ of $\{1,...,d\}$ such that $S$ guarantees the pre-specified true negative rate (TNR) for detecting nonzero $θ_j$'s. To exploit the sparsity of $θ$, we define that the confidence interval for $θ_j$ degenerates to a single point 0 for any $j\notin S$. Under this new framework, we first consider whether there exist sparse confidence sets that satisfy the above two conditions. To address this question, we establish a non-asymptotic minimax lower bound for the non-coverage probability over a suitable class of sparse confidence sets. The lower bound deciphers the role of sparsity and minimum signal-to-noise ratio (SNR) in the construction of sparse confidence sets. Furthermore, under suitable conditions on the SNR, a two-stage procedure is proposed to construct a sparse confidence set. To evaluate the optimality, the proposed sparse confidence set is shown to attain a minimax lower bound of some properly defined risk function up to a constant factor. Finally, we develop an adaptive procedure to the unknown sparsity and SNR. Numerical studies are conducted to verify the theoretical results.

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arXiv2020

Adaptive Variational Bayesian Inference for Sparse Deep Neural Network

In this work, we focus on variational Bayesian inference on the sparse Deep Neural Network (DNN) modeled under a class of spike-and-slab priors. Given a pre-specified sparse DNN structure, the corresponding variational posterior contraction rate is characterized that reveals a trade-off between the variational error and the approximation error, which are both determined by the network structural complexity (i.e., depth, width and sparsity). However, the optimal network structure, which strikes the balance of the aforementioned trade-off and yields the best rate, is generally unknown in reality. Therefore, our work further develops an {\em adaptive} variational inference procedure that can automatically select a reasonably good (data-dependent) network structure that achieves the best contraction rate, without knowing the optimal network structure. In particular, when the true function is H{ö}lder smooth, the adaptive variational inference is capable to attain (near-)optimal rate without the knowledge of smoothness level. The above rate still suffers from the curse of dimensionality, and thus motivates the teacher-student setup, i.e., the true function is a sparse DNN model, under which the rate only logarithmically depends on the input dimension.

math.ST↗

arXiv2020

Moderate-Dimensional Inferences on Quadratic Functionals in Ordinary Least Squares

Statistical inferences for quadratic functionals of linear regression parameter have found wide applications including signal detection, global testing, inferences of error variance and fraction of variance explained. Classical theory based on ordinary least squares estimator works perfectly in the low-dimensional regime, but fails when the parameter dimension $p_n$ grows proportionally to the sample size $n$. In some cases, its performance is not satisfactory even when $n\ge 5p_n$. The main contribution of this paper is to develop {\em dimension-adaptive} inferences for quadratic functionals when $\lim_{n\to \infty} p_n/n=τ\in[0,1)$. We propose a bias-and-variance-corrected test statistic and demonstrate that its theoretical validity (such as consistency and asymptotic normality) is adaptive to both low dimension with $τ= 0$ and moderate dimension with $τ\in(0, 1)$. Our general theory holds, in particular, without Gaussian design/error or structural parameter assumption, and applies to a broad class of quadratic functionals covering all aforementioned applications. As a by-product, we find that the classical fixed-dimensional results continue to hold {\em if and only if} the signal-to-noise ratio is large enough, say when $p_n$ diverges but slower than $n$. Extensive numerical results demonstrate the satisfactory performance of the proposed methodology even when $p_n\ge 0.9n$ in some extreme cases. The mathematical arguments are based on the random matrix theory and leave-one-observation-out method.

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