arXiv subjects

Tuo Zhao

Publications and source records attributed to Tuo Zhao.

At least 145 records · Page 8Linked to original sources

arXiv2019

Inductive Bias of Gradient Descent based Adversarial Training on Separable Data

Adversarial training is a principled approach for training robust neural networks. Despite of tremendous successes in practice, its theoretical properties still remain largely unexplored. In this paper, we provide new theoretical insights of gradient descent based adversarial training by studying its computational properties, specifically on its inductive bias. We take the binary classification task on linearly separable data as an illustrative example, where the loss asymptotically attains its infimum as the parameter diverges to infinity along certain directions. Specifically, we show that when the adversarial perturbation during training has bounded $\ell_2$-norm, the classifier learned by gradient descent based adversarial training converges in direction to the maximum $\ell_2$-norm margin classifier at the rate of $\tilde{\mathcal{O}}(1/\sqrt{T})$, significantly faster than the rate $\mathcal{O}(1/\log T)$ of training with clean data. In addition, when the adversarial perturbation during training has bounded $\ell_q$-norm for some $q\ge 1$, the resulting classifier converges in direction to a maximum mixed-norm margin classifier, which has a natural interpretation of robustness, as being the maximum $\ell_2$-norm margin classifier under worst-case $\ell_q$-norm perturbation to the data. Our findings provide theoretical backups for adversarial training that it indeed promotes robustness against adversarial perturbation.

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arXiv2019

On Tighter Generalization Bound for Deep Neural Networks: CNNs, ResNets, and Beyond

We establish a margin based data dependent generalization error bound for a general family of deep neural networks in terms of the depth and width, as well as the Jacobian of the networks. Through introducing a new characterization of the Lipschitz properties of neural network family, we achieve significantly tighter generalization bounds than existing results. Moreover, we show that the generalization bound can be further improved for bounded losses. Aside from the general feedforward deep neural networks, our results can be applied to derive new bounds for popular architectures, including convolutional neural networks (CNNs) and residual networks (ResNets). When achieving same generalization errors with previous arts, our bounds allow for the choice of larger parameter spaces of weight matrices, inducing potentially stronger expressive ability for neural networks. Numerical evaluation is also provided to support our theory.

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arXiv2019

On Scalable and Efficient Computation of Large Scale Optimal Transport

Optimal Transport (OT) naturally arises in many machine learning applications, yet the heavy computational burden limits its wide-spread uses. To address the scalability issue, we propose an implicit generative learning-based framework called SPOT (Scalable Push-forward of Optimal Transport). Specifically, we approximate the optimal transport plan by a pushforward of a reference distribution, and cast the optimal transport problem into a minimax problem. We then can solve OT problems efficiently using primal dual stochastic gradient-type algorithms. We also show that we can recover the density of the optimal transport plan using neural ordinary differential equations. Numerical experiments on both synthetic and real datasets illustrate that SPOT is robust and has favorable convergence behavior. SPOT also allows us to efficiently sample from the optimal transport plan, which benefits downstream applications such as domain adaptation.

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arXiv2019

On Fast Convergence of Proximal Algorithms for SQRT-Lasso Optimization: Don't Worry About Its Nonsmooth Loss Function

Many machine learning techniques sacrifice convenient computational structures to gain estimation robustness and modeling flexibility. However, by exploring the modeling structures, we find these "sacrifices" do not always require more computational efforts. To shed light on such a "free-lunch" phenomenon, we study the square-root-Lasso (SQRT-Lasso) type regression problem. Specifically, we show that the nonsmooth loss functions of SQRT-Lasso type regression ease tuning effort and gain adaptivity to inhomogeneous noise, but is not necessarily more challenging than Lasso in computation. We can directly apply proximal algorithms (e.g. proximal gradient descent, proximal Newton, and proximal Quasi-Newton algorithms) without worrying the nonsmoothness of the loss function. Theoretically, we prove that the proximal algorithms combined with the pathwise optimization scheme enjoy fast convergence guarantees with high probability. Numerical results are provided to support our theory.

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arXiv2019

On Computation and Generalization of GANs with Spectrum Control

Generative Adversarial Networks (GANs), though powerful, is hard to train. Several recent works (brock2016neural,miyato2018spectral) suggest that controlling the spectra of weight matrices in the discriminator can significantly improve the training of GANs. Motivated by their discovery, we propose a new framework for training GANs, which allows more flexible spectrum control (e.g., making the weight matrices of the discriminator have slow singular value decays). Specifically, we propose a new reparameterization approach for the weight matrices of the discriminator in GANs, which allows us to directly manipulate the spectra of the weight matrices through various regularizers and constraints, without intensively computing singular value decompositions. Theoretically, we further show that the spectrum control improves the generalization ability of GANs. Our experiments on CIFAR-10, STL-10, and ImageNet datasets confirm that compared to other methods, our proposed method is capable of generating images with competitive quality by utilizing spectral normalization and encouraging the slow singular value decay.

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arXiv2018

The Physical Systems Behind Optimization Algorithms

We use differential equations based approaches to provide some {\it \textbf{physics}} insights into analyzing the dynamics of popular optimization algorithms in machine learning. In particular, we study gradient descent, proximal gradient descent, coordinate gradient descent, proximal coordinate gradient, and Newton's methods as well as their Nesterov's accelerated variants in a unified framework motivated by a natural connection of optimization algorithms to physical systems. Our analysis is applicable to more general algorithms and optimization problems {\it \textbf{beyond}} convexity and strong convexity, e.g. Polyak-Łojasiewicz and error bound conditions (possibly nonconvex).

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arXiv2018

Provable Gaussian Embedding with One Observation

The success of machine learning methods heavily relies on having an appropriate representation for data at hand. Traditionally, machine learning approaches relied on user-defined heuristics to extract features encoding structural information about data. However, recently there has been a surge in approaches that learn how to encode the data automatically in a low dimensional space. Exponential family embedding provides a probabilistic framework for learning low-dimensional representation for various types of high-dimensional data. Though successful in practice, theoretical underpinnings for exponential family embeddings have not been established. In this paper, we study the Gaussian embedding model and develop the first theoretical results for exponential family embedding models. First, we show that, under mild condition, the embedding structure can be learned from one observation by leveraging the parameter sharing between different contexts even though the data are dependent with each other. Second, we study properties of two algorithms used for learning the embedding structure and establish convergence results for each of them. The first algorithm is based on a convex relaxation, while the other solved the non-convex formulation of the problem directly. Experiments demonstrate the effectiveness of our approach.

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arXiv2018

Dimensionality Reduction for Stationary Time Series via Stochastic Nonconvex Optimization

Stochastic optimization naturally arises in machine learning. Efficient algorithms with provable guarantees, however, are still largely missing, when the objective function is nonconvex and the data points are dependent. This paper studies this fundamental challenge through a streaming PCA problem for stationary time series data. Specifically, our goal is to estimate the principle component of time series data with respect to the covariance matrix of the stationary distribution. Computationally, we propose a variant of Oja's algorithm combined with downsampling to control the bias of the stochastic gradient caused by the data dependency. Theoretically, we quantify the uncertainty of our proposed stochastic algorithm based on diffusion approximations. This allows us to prove the asymptotic rate of convergence and further implies near optimal asymptotic sample complexity. Numerical experiments are provided to support our analysis.

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arXiv2018

Detecting Nonlinear Causality in Multivariate Time Series with Sparse Additive Models

We propose a nonparametric method for detecting nonlinear causal relationship within a set of multidimensional discrete time series, by using sparse additive models (SpAMs). We show that, when the input to the SpAM is a $β$-mixing time series, the model can be fitted by first approximating each unknown function with a linear combination of a set of B-spline bases, and then solving a group-lasso-type optimization problem with nonconvex regularization. Theoretically, we characterize the oracle statistical properties of the proposed sparse estimator in function estimation and model selection. Numerically, we propose an efficient pathwise iterative shrinkage thresholding algorithm (PISTA), which tames the nonconvexity and guarantees linear convergence towards the desired sparse estimator with high probability.

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arXiv2018

On Quadratic Convergence of DC Proximal Newton Algorithm for Nonconvex Sparse Learning in High Dimensions

We propose a DC proximal Newton algorithm for solving nonconvex regularized sparse learning problems in high dimensions. Our proposed algorithm integrates the proximal Newton algorithm with multi-stage convex relaxation based on the difference of convex (DC) programming, and enjoys both strong computational and statistical guarantees. Specifically, by leveraging a sophisticated characterization of sparse modeling structures/assumptions (i.e., local restricted strong convexity and Hessian smoothness), we prove that within each stage of convex relaxation, our proposed algorithm achieves (local) quadratic convergence, and eventually obtains a sparse approximate local optimum with optimal statistical properties after only a few convex relaxations. Numerical experiments are provided to support our theory.

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arXiv2018

Deep Hyperspherical Learning

Convolution as inner product has been the founding basis of convolutional neural networks (CNNs) and the key to end-to-end visual representation learning. Benefiting from deeper architectures, recent CNNs have demonstrated increasingly strong representation abilities. Despite such improvement, the increased depth and larger parameter space have also led to challenges in properly training a network. In light of such challenges, we propose hyperspherical convolution (SphereConv), a novel learning framework that gives angular representations on hyperspheres. We introduce SphereNet, deep hyperspherical convolution networks that are distinct from conventional inner product based convolutional networks. In particular, SphereNet adopts SphereConv as its basic convolution operator and is supervised by generalized angular softmax loss - a natural loss formulation under SphereConv. We show that SphereNet can effectively encode discriminative representation and alleviate training difficulty, leading to easier optimization, faster convergence and comparable (even better) classification accuracy over convolutional counterparts. We also provide some theoretical insights for the advantages of learning on hyperspheres. In addition, we introduce the learnable SphereConv, i.e., a natural improvement over prefixed SphereConv, and SphereNorm, i.e., hyperspherical learning as a normalization method. Experiments have verified our conclusions.

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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.

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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.

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arXiv2017

Misspecified Nonconvex Statistical Optimization for Phase Retrieval

Existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying "true" statistical models. To address this issue, we take a first step towards taming model misspecification by studying the high-dimensional sparse phase retrieval problem with misspecified link functions. In particular, we propose a simple variant of the thresholded Wirtinger flow algorithm that, given a proper initialization, linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions. We further provide extensive numerical experiments to support our theoretical findings.

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arXiv2017

Online Factorization and Partition of Complex Networks From Random Walks

Finding the reduced-dimensional structure is critical to understanding complex networks. Existing approaches such as spectral clustering are applicable only when the full network is explicitly observed. In this paper, we focus on the online factorization and partition of implicit large-scale networks based on observations from an associated random walk. We formulate this into a nonconvex stochastic factorization problem and propose an efficient and scalable stochastic generalized Hebbian algorithm. The algorithm is able to process dependent state-transition data dynamically generated by the underlying network and learn a low-dimensional representation for each vertex. By applying a diffusion approximation analysis, we show that the continuous-time limiting process of the stochastic algorithm converges globally to the "principal components" of the Markov chain and achieves a nearly optimal sample complexity. Once given the learned low-dimensional representations, we further apply clustering techniques to recover the network partition. We show that when the associated Markov process is lumpable, one can recover the partition exactly with high probability. We apply the proposed approach to model the traffic flow of Manhattan as city-wide random walks. By using our algorithm to analyze the taxi trip data, we discover a latent partition of the Manhattan city that closely matches the traffic dynamics.

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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.

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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.

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arXiv2017

Pathwise Coordinate Optimization for Sparse Learning: Algorithm and Theory

The pathwise coordinate optimization is one of the most important computational frameworks for high dimensional convex and nonconvex sparse learning problems. It differs from the classical coordinate optimization algorithms in three salient features: {\it warm start initialization}, {\it active set updating}, and {\it strong rule for coordinate preselection}. Such a complex algorithmic structure grants superior empirical performance, but also poses significant challenge to theoretical analysis. To tackle this long lasting problem, we develop a new theory showing that these three features play pivotal roles in guaranteeing the outstanding statistical and computational performance of the pathwise coordinate optimization framework. Particularly, we analyze the existing pathwise coordinate optimization algorithms and provide new theoretical insights into them. The obtained insights further motivate the development of several modifications to improve the pathwise coordinate optimization framework, which guarantees linear convergence to a unique sparse local optimum with optimal statistical properties in parameter estimation and support recovery. This is the first result on the computational and statistical guarantees of the pathwise coordinate optimization framework in high dimensions. Thorough numerical experiments are provided to support our theory.

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