arXiv subjects

Huishuai Zhang

Publications and source records attributed to Huishuai Zhang.

83 records · Page 5Linked to original sources

arXiv2020

Convergence of Distributed Stochastic Variance Reduced Methods without Sampling Extra Data

Stochastic variance reduced methods have gained a lot of interest recently for empirical risk minimization due to its appealing run time complexity. When the data size is large and disjointly stored on different machines, it becomes imperative to distribute the implementation of such variance reduced methods. In this paper, we consider a general framework that directly distributes popular stochastic variance reduced methods in the master/slave model, by assigning outer loops to the parameter server, and inner loops to worker machines. This framework is natural and friendly to implement, but its theoretical convergence is not well understood. We obtain a comprehensive understanding of algorithmic convergence with respect to data homogeneity by measuring the smoothness of the discrepancy between the local and global loss functions. We establish the linear convergence of distributed versions of a family of stochastic variance reduced algorithms, including those using accelerated and recursive gradient updates, for minimizing strongly convex losses. Our theory captures how the convergence of distributed algorithms behaves as the number of machines and the size of local data vary. Furthermore, we show that when the data are less balanced, regularization can be used to ensure convergence at a slower rate. We also demonstrate that our analysis can be further extended to handle nonconvex loss functions.

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arXiv2020

On Layer Normalization in the Transformer Architecture

The Transformer is widely used in natural language processing tasks. To train a Transformer however, one usually needs a carefully designed learning rate warm-up stage, which is shown to be crucial to the final performance but will slow down the optimization and bring more hyper-parameter tunings. In this paper, we first study theoretically why the learning rate warm-up stage is essential and show that the location of layer normalization matters. Specifically, we prove with mean field theory that at initialization, for the original-designed Post-LN Transformer, which places the layer normalization between the residual blocks, the expected gradients of the parameters near the output layer are large. Therefore, using a large learning rate on those gradients makes the training unstable. The warm-up stage is practically helpful for avoiding this problem. On the other hand, our theory also shows that if the layer normalization is put inside the residual blocks (recently proposed as Pre-LN Transformer), the gradients are well-behaved at initialization. This motivates us to remove the warm-up stage for the training of Pre-LN Transformers. We show in our experiments that Pre-LN Transformers without the warm-up stage can reach comparable results with baselines while requiring significantly less training time and hyper-parameter tuning on a wide range of applications.

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arXiv2019

Generalization Error Bounds with Probabilistic Guarantee for SGD in Nonconvex Optimization

The success of deep learning has led to a rising interest in the generalization property of the stochastic gradient descent (SGD) method, and stability is one popular approach to study it. Existing works based on stability have studied nonconvex loss functions, but only considered the generalization error of the SGD in expectation. In this paper, we establish various generalization error bounds with probabilistic guarantee for the SGD. Specifically, for both general nonconvex loss functions and gradient dominant loss functions, we characterize the on-average stability of the iterates generated by SGD in terms of the on-average variance of the stochastic gradients. Such characterization leads to improved bounds for the generalization error for SGD. We then study the regularized risk minimization problem with strongly convex regularizers, and obtain improved generalization error bounds for proximal SGD. With strongly convex regularizers, we further establish the generalization error bounds for nonconvex loss functions under proximal SGD with high-probability guarantee, i.e., exponential concentration in probability.

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arXiv2019

SGD Converges to Global Minimum in Deep Learning via Star-convex Path

Stochastic gradient descent (SGD) has been found to be surprisingly effective in training a variety of deep neural networks. However, there is still a lack of understanding on how and why SGD can train these complex networks towards a global minimum. In this study, we establish the convergence of SGD to a global minimum for nonconvex optimization problems that are commonly encountered in neural network training. Our argument exploits the following two important properties: 1) the training loss can achieve zero value (approximately), which has been widely observed in deep learning; 2) SGD follows a star-convex path, which is verified by various experiments in this paper. In such a context, our analysis shows that SGD, although has long been considered as a randomized algorithm, converges in an intrinsically deterministic manner to a global minimum.

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arXiv2018

Capacity Control of ReLU Neural Networks by Basis-path Norm

Recently, path norm was proposed as a new capacity measure for neural networks with Rectified Linear Unit (ReLU) activation function, which takes the rescaling-invariant property of ReLU into account. It has been shown that the generalization error bound in terms of the path norm explains the empirical generalization behaviors of the ReLU neural networks better than that of other capacity measures. Moreover, optimization algorithms which take path norm as the regularization term to the loss function, like Path-SGD, have been shown to achieve better generalization performance. However, the path norm counts the values of all paths, and hence the capacity measure based on path norm could be improperly influenced by the dependency among different paths. It is also known that each path of a ReLU network can be represented by a small group of linearly independent basis paths with multiplication and division operation, which indicates that the generalization behavior of the network only depends on only a few basis paths. Motivated by this, we propose a new norm \emph{Basis-path Norm} based on a group of linearly independent paths to measure the capacity of neural networks more accurately. We establish a generalization error bound based on this basis path norm, and show it explains the generalization behaviors of ReLU networks more accurately than previous capacity measures via extensive experiments. In addition, we develop optimization algorithms which minimize the empirical risk regularized by the basis-path norm. Our experiments on benchmark datasets demonstrate that the proposed regularization method achieves clearly better performance on the test set than the previous regularization approaches.

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arXiv2018

Train Feedfoward Neural Network with Layer-wise Adaptive Rate via Approximating Back-matching Propagation

Stochastic gradient descent (SGD) has achieved great success in training deep neural network, where the gradient is computed through back-propagation. However, the back-propagated values of different layers vary dramatically. This inconsistence of gradient magnitude across different layers renders optimization of deep neural network with a single learning rate problematic. We introduce the back-matching propagation which computes the backward values on the layer's parameter and the input by matching backward values on the layer's output. This leads to solving a bunch of least-squares problems, which requires high computational cost. We then reduce the back-matching propagation with approximations and propose an algorithm that turns to be the regular SGD with a layer-wise adaptive learning rate strategy. This allows an easy implementation of our algorithm in current machine learning frameworks equipped with auto-differentiation. We apply our algorithm in training modern deep neural networks and achieve favorable results over SGD.

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arXiv2017

Block-diagonal Hessian-free Optimization for Training Neural Networks

Second-order methods for neural network optimization have several advantages over methods based on first-order gradient descent, including better scaling to large mini-batch sizes and fewer updates needed for convergence. But they are rarely applied to deep learning in practice because of high computational cost and the need for model-dependent algorithmic variations. We introduce a variant of the Hessian-free method that leverages a block-diagonal approximation of the generalized Gauss-Newton matrix. Our method computes the curvature approximation matrix only for pairs of parameters from the same layer or block of the neural network and performs conjugate gradient updates independently for each block. Experiments on deep autoencoders, deep convolutional networks, and multilayer LSTMs demonstrate better convergence and generalization compared to the original Hessian-free approach and the Adam method.

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arXiv2017

Nonconvex Low-Rank Matrix Recovery with Arbitrary Outliers via Median-Truncated Gradient Descent

Recent work has demonstrated the effectiveness of gradient descent for directly recovering the factors of low-rank matrices from random linear measurements in a globally convergent manner when initialized properly. However, the performance of existing algorithms is highly sensitive in the presence of outliers that may take arbitrary values. In this paper, we propose a truncated gradient descent algorithm to improve the robustness against outliers, where the truncation is performed to rule out the contributions of samples that deviate significantly from the {\em sample median} of measurement residuals adaptively in each iteration. We demonstrate that, when initialized in a basin of attraction close to the ground truth, the proposed algorithm converges to the ground truth at a linear rate for the Gaussian measurement model with a near-optimal number of measurements, even when a constant fraction of the measurements are arbitrarily corrupted. In addition, we propose a new truncated spectral method that ensures an initialization in the basin of attraction at slightly higher requirements. We finally provide numerical experiments to validate the superior performance of the proposed approach.

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arXiv2017

Median-Truncated Nonconvex Approach for Phase Retrieval with Outliers

This paper investigates the phase retrieval problem, which aims to recover a signal from the magnitudes of its linear measurements. We develop statistically and computationally efficient algorithms for the situation when the measurements are corrupted by sparse outliers that can take arbitrary values. We propose a novel approach to robustify the gradient descent algorithm by using the sample median as a guide for pruning spurious samples in initialization and local search. Adopting the Poisson loss and the reshaped quadratic loss respectively, we obtain two algorithms termed median-TWF and median-RWF, both of which provably recover the signal from a near-optimal number of measurements when the measurement vectors are composed of i.i.d. Gaussian entries, up to a logarithmic factor, even when a constant fraction of the measurements are adversarially corrupted. We further show that both algorithms are stable in the presence of additional dense bounded noise. Our analysis is accomplished by developing non-trivial concentration results of median-related quantities, which may be of independent interest. We provide numerical experiments to demonstrate the effectiveness of our approach.

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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.

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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.

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