arXiv subjects

Cong Fang

Publications and source records attributed to Cong Fang.

69 records · Page 4Linked to original sources

arXiv2021

Exploring Deep Neural Networks via Layer-Peeled Model: Minority Collapse in Imbalanced Training

In this paper, we introduce the \textit{Layer-Peeled Model}, a nonconvex yet analytically tractable optimization program, in a quest to better understand deep neural networks that are trained for a sufficiently long time. As the name suggests, this new model is derived by isolating the topmost layer from the remainder of the neural network, followed by imposing certain constraints separately on the two parts of the network. We demonstrate that the Layer-Peeled Model, albeit simple, inherits many characteristics of well-trained neural networks, thereby offering an effective tool for explaining and predicting common empirical patterns of deep learning training. First, when working on class-balanced datasets, we prove that any solution to this model forms a simplex equiangular tight frame, which in part explains the recently discovered phenomenon of neural collapse \cite{papyan2020prevalence}. More importantly, when moving to the imbalanced case, our analysis of the Layer-Peeled Model reveals a hitherto unknown phenomenon that we term \textit{Minority Collapse}, which fundamentally limits the performance of deep learning models on the minority classes. In addition, we use the Layer-Peeled Model to gain insights into how to mitigate Minority Collapse. Interestingly, this phenomenon is first predicted by the Layer-Peeled Model before being confirmed by our computational experiments.

cs.LG↗

arXiv2020

Mathematical Models of Overparameterized Neural Networks

Deep learning has received considerable empirical successes in recent years. However, while many ad hoc tricks have been discovered by practitioners, until recently, there has been a lack of theoretical understanding for tricks invented in the deep learning literature. Known by practitioners that overparameterized neural networks are easy to learn, in the past few years there have been important theoretical developments in the analysis of overparameterized neural networks. In particular, it was shown that such systems behave like convex systems under various restricted settings, such as for two-layer NNs, and when learning is restricted locally in the so-called neural tangent kernel space around specialized initializations. This paper discusses some of these recent progresses leading to significant better understanding of neural networks. We will focus on the analysis of two-layer neural networks, and explain the key mathematical models, with their algorithmic implications. We will then discuss challenges in understanding deep neural networks and some current research directions.

cs.LG↗

arXiv2020

Improved Analysis of Clipping Algorithms for Non-convex Optimization

Gradient clipping is commonly used in training deep neural networks partly due to its practicability in relieving the exploding gradient problem. Recently, \citet{zhang2019gradient} show that clipped (stochastic) Gradient Descent (GD) converges faster than vanilla GD/SGD via introducing a new assumption called $(L_0, L_1)$-smoothness, which characterizes the violent fluctuation of gradients typically encountered in deep neural networks. However, their iteration complexities on the problem-dependent parameters are rather pessimistic, and theoretical justification of clipping combined with other crucial techniques, e.g. momentum acceleration, are still lacking. In this paper, we bridge the gap by presenting a general framework to study the clipping algorithms, which also takes momentum methods into consideration. We provide convergence analysis of the framework in both deterministic and stochastic setting, and demonstrate the tightness of our results by comparing them with existing lower bounds. Our results imply that the efficiency of clipping methods will not degenerate even in highly non-smooth regions of the landscape. Experiments confirm the superiority of clipping-based methods in deep learning tasks.

cs.LG↗

arXiv2020

Decentralized Accelerated Gradient Methods With Increasing Penalty Parameters

In this paper, we study the communication and (sub)gradient computation costs in distributed optimization and give a sharp complexity analysis for the proposed distributed accelerated gradient methods. We present two algorithms based on the framework of the accelerated penalty method with increasing penalty parameters. Our first algorithm is for smooth distributed optimization and it obtains the near optimal $O\left(\sqrt{\frac{L}{ε(1-σ_2(W))}}\log\frac{1}ε\right)$ communication complexity and the optimal $O\left(\sqrt{\frac{L}ε}\right)$ gradient computation complexity for $L$-smooth convex problems, where $σ_2(W)$ denotes the second largest singular value of the weight matrix $W$ associated to the network and $ε$ is the target accuracy. When the problem is $μ$-strongly convex and $L$-smooth, our algorithm has the near optimal $O\left(\sqrt{\frac{L}{μ(1-σ_2(W))}}\log^2\frac{1}ε\right)$ complexity for communications and the optimal $O\left(\sqrt{\frac{L}μ}\log\frac{1}ε\right)$ complexity for gradient computations. Our communication complexities are only worse by a factor of $\left(\log\frac{1}ε\right)$ than the lower bounds for the smooth distributed optimization. %As far as we know, our method is the first to achieve both communication and gradient computation lower bounds up to an extra logarithm factor for smooth distributed optimization. Our second algorithm is designed for non-smooth distributed optimization and it achieves both the optimal $O\left(\frac{1}{ε\sqrt{1-σ_2(W)}}\right)$ communication complexity and $O\left(\frac{1}{ε^2}\right)$ subgradient computation complexity, which match the communication and subgradient computation complexity lower bounds for non-smooth distributed optimization.

math.OC↗

arXiv2020

Modeling from Features: a Mean-field Framework for Over-parameterized Deep Neural Networks

This paper proposes a new mean-field framework for over-parameterized deep neural networks (DNNs), which can be used to analyze neural network training. In this framework, a DNN is represented by probability measures and functions over its features (that is, the function values of the hidden units over the training data) in the continuous limit, instead of the neural network parameters as most existing studies have done. This new representation overcomes the degenerate situation where all the hidden units essentially have only one meaningful hidden unit in each middle layer, and further leads to a simpler representation of DNNs, for which the training objective can be reformulated as a convex optimization problem via suitable re-parameterization. Moreover, we construct a non-linear dynamics called neural feature flow, which captures the evolution of an over-parameterized DNN trained by Gradient Descent. We illustrate the framework via the standard DNN and the Residual Network (Res-Net) architectures. Furthermore, we show, for Res-Net, when the neural feature flow process converges, it reaches a global minimal solution under suitable conditions. Our analysis leads to the first global convergence proof for over-parameterized neural network training with more than $3$ layers in the mean-field regime.

stat.ML↗

arXiv2019

Convex Formulation of Overparameterized Deep Neural Networks

Analysis of over-parameterized neural networks has drawn significant attention in recentyears. It was shown that such systems behave like convex systems under various restrictedsettings, such as for two-level neural networks, and when learning is only restricted locally inthe so-called neural tangent kernel space around specialized initializations. However, there areno theoretical techniques that can analyze fully trained deep neural networks encountered inpractice. This paper solves this fundamental problem by investigating such overparameterizeddeep neural networks when fully trained. We generalize a new technique called neural feature repopulation, originally introduced in (Fang et al., 2019a) for two-level neural networks, to analyze deep neural networks. It is shown that under suitable representations, overparameterized deep neural networks are inherently convex, and when optimized, the system can learn effective features suitable for the underlying learning task under mild conditions. This new analysis is consistent with empirical observations that deep neural networks are capable of learning efficient feature representations. Therefore, the highly unexpected result of this paper can satisfactorily explain the practical success of deep neural networks. Empirical studies confirm that predictions of our theory are consistent with results observed in practice.

cs.LG↗

arXiv2019

Over Parameterized Two-level Neural Networks Can Learn Near Optimal Feature Representations

Recently, over-parameterized neural networks have been extensively analyzed in the literature. However, the previous studies cannot satisfactorily explain why fully trained neural networks are successful in practice. In this paper, we present a new theoretical framework for analyzing over-parameterized neural networks which we call neural feature repopulation. Our analysis can satisfactorily explain the empirical success of two level neural networks that are trained by standard learning algorithms. Our key theoretical result is that in the limit of infinite number of hidden neurons, over-parameterized two-level neural networks trained via the standard (noisy) gradient descent learns a well-defined feature distribution (population), and the limiting feature distribution is nearly optimal for the underlying learning task under certain conditions. Empirical studies confirm that predictions of our theory are consistent with the results observed in real practice.

cs.LG↗

arXiv2019

Sharp Analysis for Nonconvex SGD Escaping from Saddle Points

In this paper, we give a sharp analysis for Stochastic Gradient Descent (SGD) and prove that SGD is able to efficiently escape from saddle points and find an $(ε, O(ε^{0.5}))$-approximate second-order stationary point in $\tilde{O}(ε^{-3.5})$ stochastic gradient computations for generic nonconvex optimization problems, when the objective function satisfies gradient-Lipschitz, Hessian-Lipschitz, and dispersive noise assumptions. This result subverts the classical belief that SGD requires at least $O(ε^{-4})$ stochastic gradient computations for obtaining an $(ε,O(ε^{0.5}))$-approximate second-order stationary point. Such SGD rate matches, up to a polylogarithmic factor of problem-dependent parameters, the rate of most accelerated nonconvex stochastic optimization algorithms that adopt additional techniques, such as Nesterov's momentum acceleration, negative curvature search, as well as quadratic and cubic regularization tricks. Our novel analysis gives new insights into nonconvex SGD and can be potentially generalized to a broad class of stochastic optimization algorithms.

math.OC↗

arXiv2019

Hessian-Aware Zeroth-Order Optimization for Black-Box Adversarial Attack

Zeroth-order optimization is an important research topic in machine learning. In recent years, it has become a key tool in black-box adversarial attack to neural network based image classifiers. However, existing zeroth-order optimization algorithms rarely extract second-order information of the model function. In this paper, we utilize the second-order information of the objective function and propose a novel \textit{Hessian-aware zeroth-order algorithm} called \texttt{ZO-HessAware}. Our theoretical result shows that \texttt{ZO-HessAware} has an improved zeroth-order convergence rate and query complexity under structured Hessian approximation, where we propose a few approximation methods for estimating Hessian. Our empirical studies on the black-box adversarial attack problem validate that our algorithm can achieve improved success rates with a lower query complexity.

cs.LG↗

arXiv2019

A Stochastic Trust Region Method for Non-convex Minimization

We target the problem of finding a local minimum in non-convex finite-sum minimization. Towards this goal, we first prove that the trust region method with inexact gradient and Hessian estimation can achieve a convergence rate of order $\mathcal{O}(1/{k^{2/3}})$ as long as those differential estimations are sufficiently accurate. Combining such result with a novel Hessian estimator, we propose the sample-efficient stochastic trust region (STR) algorithm which finds an $(ε, \sqrtε)$-approximate local minimum within $\mathcal{O}({\sqrt{n}}/{ε^{1.5}})$ stochastic Hessian oracle queries. This improves state-of-the-art result by $\mathcal{O}(n^{1/6})$. Experiments verify theoretical conclusions and the efficiency of STR.

math.OC↗

arXiv2018

Lifted Proximal Operator Machines

We propose a new optimization method for training feed-forward neural networks. By rewriting the activation function as an equivalent proximal operator, we approximate a feed-forward neural network by adding the proximal operators to the objective function as penalties, hence we call the lifted proximal operator machine (LPOM). LPOM is block multi-convex in all layer-wise weights and activations. This allows us to use block coordinate descent to update the layer-wise weights and activations in parallel. Most notably, we only use the mapping of the activation function itself, rather than its derivatives, thus avoiding the gradient vanishing or blow-up issues in gradient based training methods. So our method is applicable to various non-decreasing Lipschitz continuous activation functions, which can be saturating and non-differentiable. LPOM does not require more auxiliary variables than the layer-wise activations, thus using roughly the same amount of memory as stochastic gradient descent (SGD) does. We further prove the convergence of updating the layer-wise weights and activations. Experiments on MNIST and CIFAR-10 datasets testify to the advantages of LPOM.

cs.LG↗

arXiv2018

SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator

In this paper, we propose a new technique named \textit{Stochastic Path-Integrated Differential EstimatoR} (SPIDER), which can be used to track many deterministic quantities of interest with significantly reduced computational cost. We apply SPIDER to two tasks, namely the stochastic first-order and zeroth-order methods. For stochastic first-order method, combining SPIDER with normalized gradient descent, we propose two new algorithms, namely SPIDER-SFO and SPIDER-SFO\textsuperscript{+}, that solve non-convex stochastic optimization problems using stochastic gradients only. We provide sharp error-bound results on their convergence rates. In special, we prove that the SPIDER-SFO and SPIDER-SFO\textsuperscript{+} algorithms achieve a record-breaking gradient computation cost of $\mathcal{O}\left( \min( n^{1/2} ε^{-2}, ε^{-3} ) \right)$ for finding an $ε$-approximate first-order and $\tilde{\mathcal{O}}\left( \min( n^{1/2} ε^{-2}+ε^{-2.5}, ε^{-3} ) \right)$ for finding an $(ε, \mathcal{O}(ε^{0.5}))$-approximate second-order stationary point, respectively. In addition, we prove that SPIDER-SFO nearly matches the algorithmic lower bound for finding approximate first-order stationary points under the gradient Lipschitz assumption in the finite-sum setting. For stochastic zeroth-order method, we prove a cost of $\mathcal{O}( d \min( n^{1/2} ε^{-2}, ε^{-3}) )$ which outperforms all existing results.

math.OC↗

arXiv2018

Accelerating Asynchronous Algorithms for Convex Optimization by Momentum Compensation

Asynchronous algorithms have attracted much attention recently due to the crucial demands on solving large-scale optimization problems. However, the accelerated versions of asynchronous algorithms are rarely studied. In this paper, we propose the "momentum compensation" technique to accelerate asynchronous algorithms for convex problems. Specifically, we first accelerate the plain Asynchronous Gradient Descent, which achieves a faster $O(1/\sqrtε)$ (v.s. $O(1/ε)$) convergence rate for non-strongly convex functions, and $O(\sqrtκ\log(1/ε))$ (v.s. $O(κ\log(1/ε))$) for strongly convex functions to reach an $ε$- approximate minimizer with the condition number $κ$. We further apply the technique to accelerate modern stochastic asynchronous algorithms such as Asynchronous Stochastic Coordinate Descent and Asynchronous Stochastic Gradient Descent. Both of the resultant practical algorithms are faster than existing ones by order. To the best of our knowledge, we are the first to consider accelerated algorithms that allow updating by delayed gradients and are the first to propose truly accelerated asynchronous algorithms. Finally, the experimental results on a shared memory system show that acceleration can lead to significant performance gains on ill-conditioned problems.

math.OC↗

arXiv2017

Convergence Rates Analysis of The Quadratic Penalty Method and Its Applications to Decentralized Distributed Optimization

In this paper, we study a variant of the quadratic penalty method for linearly constrained convex problems, which has already been widely used but actually lacks theoretical justification. Namely, the penalty parameter steadily increases and the penalized objective function is minimized inexactly rather than exactly, e.g., with only one step of the proximal gradient descent. For such a variant of the quadratic penalty method, we give counterexamples to show that it may not give a solution to the original constrained problem. By choosing special penalty parameters, we ensure the convergence and further establish the convergence rates of $O\left(\frac{1}{\sqrt{K}}\right)$ for the generally convex problems and $O\left(\frac{1}{K}\right)$ for strongly convex ones, where $K$ is the number of iterations. Furthermore, by adopting Nesterov's extrapolation we show that the convergence rates can be improved to $O\left(\frac{1}{K}\right)$ for the generally convex problems and $O\left(\frac{1}{K^2}\right)$ for strongly convex ones. When applied to the decentralized distributed optimization, the penalty methods studied in this paper become the widely used distributed gradient method and the fast distributed gradient method. However, due to the totally different analysis framework, we can improve their $O\left(\frac{\log K}{\sqrt{K}}\right)$ and $O\left(\frac{\log K}{K}\right)$ convergence rates to $O\left(\frac{1}{\sqrt{K}}\right)$ and $O\left(\frac{1}{K}\right)$ with fewer assumptions on the network topology for general convex problems. Using our analysis framework, we also extend the fast distributed gradient method to a communication efficient version, i.e., finding an $\varepsilon$ solution in $O\left(\frac{1}{\varepsilon}\right)$ communications and $O\left(\frac{1}{\varepsilon^{2+δ}}\right)$ computations for the non-smooth problems, where $δ$ is a small constant.

math.NA↗

arXiv2017

Faster and Non-ergodic O(1/K) Stochastic Alternating Direction Method of Multipliers

We study stochastic convex optimization subjected to linear equality constraints. Traditional Stochastic Alternating Direction Method of Multipliers and its Nesterov's acceleration scheme can only achieve ergodic O(1/\sqrt{K}) convergence rates, where K is the number of iteration. By introducing Variance Reduction (VR) techniques, the convergence rates improve to ergodic O(1/K). In this paper, we propose a new stochastic ADMM which elaborately integrates Nesterov's extrapolation and VR techniques. We prove that our algorithm can achieve a non-ergodic O(1/K) convergence rate which is optimal for separable linearly constrained non-smooth convex problems, while the convergence rates of VR based ADMM methods are actually tight O(1/\sqrt{K}) in non-ergodic sense. To the best of our knowledge, this is the first work that achieves a truly accelerated, stochastic convergence rate for constrained convex problems. The experimental results demonstrate that our algorithm is significantly faster than the existing state-of-the-art stochastic ADMM methods.

math.OC↗