arXiv subjects

Bao Wang

Publications and source records attributed to Bao Wang.

At least 73 records · Page 4Linked to original sources

arXiv2020

MomentumRNN: Integrating Momentum into Recurrent Neural Networks

Designing deep neural networks is an art that often involves an expensive search over candidate architectures. To overcome this for recurrent neural nets (RNNs), we establish a connection between the hidden state dynamics in an RNN and gradient descent (GD). We then integrate momentum into this framework and propose a new family of RNNs, called {\em MomentumRNNs}. We theoretically prove and numerically demonstrate that MomentumRNNs alleviate the vanishing gradient issue in training RNNs. We study the momentum long-short term memory (MomentumLSTM) and verify its advantages in convergence speed and accuracy over its LSTM counterpart across a variety of benchmarks. We also demonstrate that MomentumRNN is applicable to many types of recurrent cells, including those in the state-of-the-art orthogonal RNNs. Finally, we show that other advanced momentum-based optimization methods, such as Adam and Nesterov accelerated gradients with a restart, can be easily incorporated into the MomentumRNN framework for designing new recurrent cells with even better performance. The code is available at https://github.com/minhtannguyen/MomentumRNN.

cs.LG↗

arXiv2020

A Deterministic Gradient-Based Approach to Avoid Saddle Points

Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning models efficiently. First-order methods such as gradient descent are usually the methods of choice for training machine learning models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent [Osher et al., arXiv:1806.06317], called modified Laplacian smoothing gradient descent (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region's dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is floor((n-1)/2), and hence it is significantly smaller than that of the gradient descent whose dimension is n-1.

cs.LG↗

arXiv2020

Geometrical formulation for adjoint-symmetries of partial differential equations

A geometrical formulation for adjoint-symmetries as 1-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. Additionally, for systems of evolution equations, adjoint-symmetries are shown to have another geometrical formulation given by 1-forms that are invariant under the flow generated by the system on the solution space. This result is generalized to systems of evolution equations with spatial constraints, where adjoint-symmetry 1-forms are shown to be invariant up to a functional multiplier of a normal 1-form associated to the constraint equations. All of the results are applicable to the PDE systems of interest in applied mathematics and mathematical physics.

math-ph↗

arXiv2020

Reflections in the Sky: Joint Trajectory and Passive Beamforming Design for Secure UAV Networks with Reconfigurable Intelligent Surface

This paper investigates the problem of secure energy efficiency maximization for a reconfigurable intelligent surface (RIS) assisted uplink wireless communication system, where an unmanned aerial vehicle (UAV) equipped with an RIS works as a mobile relay between the base station (BS) and a group of users. We focus on maximizing the secure energy efficiency of the system via jointly optimizing the UAV's trajectory, the RIS's phase shift, users' association and transmit power. To tackle this problem, we divide the original problem into three sub-problems, and propose an efficient iterative algorithm. In particular, the successive convex approximation method (SCA) is applied to solve the nonconvex UAV trajectory, the RIS's phase shift, and transmit power optimization sub-problems. We further provide two schemes to simplify the solution of phase and trajectory sub-problem. Simulation results demonstrate that the proposed algorithm converges fast, and the proposed design can enhance the secure energy efficiency by up to 38\% gains, as compared to the traditional schemes without any RIS.

eess.SP↗

arXiv2020

Adversarial Defense via Data Dependent Activation Function and Total Variation Minimization

We improve the robustness of Deep Neural Net (DNN) to adversarial attacks by using an interpolating function as the output activation. This data-dependent activation remarkably improves both the generalization and robustness of DNN. In the CIFAR10 benchmark, we raise the robust accuracy of the adversarially trained ResNet20 from $\sim 46\%$ to $\sim 69\%$ under the state-of-the-art Iterative Fast Gradient Sign Method (IFGSM) based adversarial attack. When we combine this data-dependent activation with total variation minimization on adversarial images and training data augmentation, we achieve an improvement in robust accuracy by 38.9$\%$ for ResNet56 under the strongest IFGSM attack. Furthermore, We provide an intuitive explanation of our defense by analyzing the geometry of the feature space.

cs.LG↗

arXiv2020

Scheduled Restart Momentum for Accelerated Stochastic Gradient Descent

Stochastic gradient descent (SGD) with constant momentum and its variants such as Adam are the optimization algorithms of choice for training deep neural networks (DNNs). Since DNN training is incredibly computationally expensive, there is great interest in speeding up the convergence. Nesterov accelerated gradient (NAG) improves the convergence rate of gradient descent (GD) for convex optimization using a specially designed momentum; however, it accumulates error when an inexact gradient is used (such as in SGD), slowing convergence at best and diverging at worst. In this paper, we propose Scheduled Restart SGD (SRSGD), a new NAG-style scheme for training DNNs. SRSGD replaces the constant momentum in SGD by the increasing momentum in NAG but stabilizes the iterations by resetting the momentum to zero according to a schedule. Using a variety of models and benchmarks for image classification, we demonstrate that, in training DNNs, SRSGD significantly improves convergence and generalization; for instance in training ResNet200 for ImageNet classification, SRSGD achieves an error rate of 20.93% vs. the benchmark of 22.13%. These improvements become more significant as the network grows deeper. Furthermore, on both CIFAR and ImageNet, SRSGD reaches similar or even better error rates with significantly fewer training epochs compared to the SGD baseline.

cs.LG↗

arXiv2020

Sparsity Meets Robustness: Channel Pruning for the Feynman-Kac Formalism Principled Robust Deep Neural Nets

Deep neural nets (DNNs) compression is crucial for adaptation to mobile devices. Though many successful algorithms exist to compress naturally trained DNNs, developing efficient and stable compression algorithms for robustly trained DNNs remains widely open. In this paper, we focus on a co-design of efficient DNN compression algorithms and sparse neural architectures for robust and accurate deep learning. Such a co-design enables us to advance the goal of accommodating both sparsity and robustness. With this objective in mind, we leverage the relaxed augmented Lagrangian based algorithms to prune the weights of adversarially trained DNNs, at both structured and unstructured levels. Using a Feynman-Kac formalism principled robust and sparse DNNs, we can at least double the channel sparsity of the adversarially trained ResNet20 for CIFAR10 classification, meanwhile, improve the natural accuracy by $8.69$\% and the robust accuracy under the benchmark $20$ iterations of IFGSM attack by $5.42$\%. The code is available at \url{https://github.com/BaoWangMath/rvsm-rgsm-admm}.

cs.LG↗

arXiv2019

DP-LSSGD: A Stochastic Optimization Method to Lift the Utility in Privacy-Preserving ERM

Machine learning (ML) models trained by differentially private stochastic gradient descent (DP-SGD) have much lower utility than the non-private ones. To mitigate this degradation, we propose a DP Laplacian smoothing SGD (DP-LSSGD) to train ML models with differential privacy (DP) guarantees. At the core of DP-LSSGD is the Laplacian smoothing, which smooths out the Gaussian noise used in the Gaussian mechanism. Under the same amount of noise used in the Gaussian mechanism, DP-LSSGD attains the same DP guarantee, but in practice, DP-LSSGD makes training both convex and nonconvex ML models more stable and enables the trained models to generalize better. The proposed algorithm is simple to implement and the extra computational complexity and memory overhead compared with DP-SGD are negligible. DP-LSSGD is applicable to train a large variety of ML models, including DNNs. The code is available at \url{https://github.com/BaoWangMath/DP-LSSGD}.

cs.LG↗

arXiv2019

Laplacian Smoothing Stochastic Gradient Markov Chain Monte Carlo

As an important Markov Chain Monte Carlo (MCMC) method, stochastic gradient Langevin dynamics (SGLD) algorithm has achieved great success in Bayesian learning and posterior sampling. However, SGLD typically suffers from slow convergence rate due to its large variance caused by the stochastic gradient. In order to alleviate these drawbacks, we leverage the recently developed Laplacian Smoothing (LS) technique and propose a Laplacian smoothing stochastic gradient Langevin dynamics (LS-SGLD) algorithm. We prove that for sampling from both log-concave and non-log-concave densities, LS-SGLD achieves strictly smaller discretization error in $2$-Wasserstein distance, although its mixing rate can be slightly slower. Experiments on both synthetic and real datasets verify our theoretical results, and demonstrate the superior performance of LS-SGLD on different machine learning tasks including posterior sampling, Bayesian logistic regression and training Bayesian convolutional neural networks. The code is available at \url{https://github.com/BaoWangMath/LS-MCMC}.

cs.LG↗

arXiv2019

Graph Interpolating Activation Improves Both Natural and Robust Accuracies in Data-Efficient Deep Learning

Improving the accuracy and robustness of deep neural nets (DNNs) and adapting them to small training data are primary tasks in deep learning research. In this paper, we replace the output activation function of DNNs, typically the data-agnostic softmax function, with a graph Laplacian-based high dimensional interpolating function which, in the continuum limit, converges to the solution of a Laplace-Beltrami equation on a high dimensional manifold. Furthermore, we propose end-to-end training and testing algorithms for this new architecture. The proposed DNN with graph interpolating activation integrates the advantages of both deep learning and manifold learning. Compared to the conventional DNNs with the softmax function as output activation, the new framework demonstrates the following major advantages: First, it is better applicable to data-efficient learning in which we train high capacity DNNs without using a large number of training data. Second, it remarkably improves both natural accuracy on the clean images and robust accuracy on the adversarial images crafted by both white-box and black-box adversarial attacks. Third, it is a natural choice for semi-supervised learning. For reproducibility, the code is available at \url{https://github.com/BaoWangMath/DNN-DataDependentActivation}.

cs.LG↗

arXiv2019

ResNets Ensemble via the Feynman-Kac Formalism to Improve Natural and Robust Accuracies

Empirical adversarial risk minimization (EARM) is a widely used mathematical framework to robustly train deep neural nets (DNNs) that are resistant to adversarial attacks. However, both natural and robust accuracies, in classifying clean and adversarial images, respectively, of the trained robust models are far from satisfactory. In this work, we unify the theory of optimal control of transport equations with the practice of training and testing of ResNets. Based on this unified viewpoint, we propose a simple yet effective ResNets ensemble algorithm to boost the accuracy of the robustly trained model on both clean and adversarial images. The proposed algorithm consists of two components: First, we modify the base ResNets by injecting a variance specified Gaussian noise to the output of each residual mapping. Second, we average over the production of multiple jointly trained modified ResNets to get the final prediction. These two steps give an approximation to the Feynman-Kac formula for representing the solution of a transport equation with viscosity, or a convection-diffusion equation. For the CIFAR10 benchmark, this simple algorithm leads to a robust model with a natural accuracy of {\bf 85.62}\% on clean images and a robust accuracy of ${\bf 57.94 \%}$ under the 20 iterations of the IFGSM attack, which outperforms the current state-of-the-art in defending against IFGSM attack on the CIFAR10. Both natural and robust accuracies of the proposed ResNets ensemble can be improved dynamically as the building block ResNet advances. The code is available at: \url{https://github.com/BaoWangMath/EnResNet}.

cs.LG↗

arXiv2019

Laplacian Smoothing Gradient Descent

We propose a class of very simple modifications of gradient descent and stochastic gradient descent. We show that when applied to a large variety of machine learning problems, ranging from logistic regression to deep neural nets, the proposed surrogates can dramatically reduce the variance, allow to take a larger step size, and improve the generalization accuracy. The methods only involve multiplying the usual (stochastic) gradient by the inverse of a positive definitive matrix (which can be computed efficiently by FFT) with a low condition number coming from a one-dimensional discrete Laplacian or its high order generalizations. It also preserves the mean and increases the smallest component and decreases the largest component. The theory of Hamilton-Jacobi partial differential equations demonstrates that the implicit version of the new algorithm is almost the same as doing gradient descent on a new function which (i) has the same global minima as the original function and (ii) is ``more convex". Moreover, we show that optimization algorithms with these surrogates converge uniformly in the discrete Sobolev $H_σ^p$ sense and reduce the optimality gap for convex optimization problems. The code is available at: \url{https://github.com/BaoWangMath/LaplacianSmoothing-GradientDescent}

cs.LG↗

arXiv2019

A Study on Graph-Structured Recurrent Neural Networks and Sparsification with Application to Epidemic Forecasting

We study epidemic forecasting on real-world health data by a graph-structured recurrent neural network (GSRNN). We achieve state-of-the-art forecasting accuracy on the benchmark CDC dataset. To improve model efficiency, we sparsify the network weights via transformed-$\ell_1$ penalty and maintain prediction accuracy at the same level with 70% of the network weights being zero.

cs.LG↗

arXiv2018

Phase-mediated magnon chaos-order transition in cavity optomagnonics

Magnon as a quantized spin wave has attracted extensive attentions in various fields of physics, such as magnon spintronics, microwave photonics, and cavity quantum electrodynamics. Here, we explore theoretically magnon chaos-order transition in cavity optomagnonics, which is still remains largely unexplored in this emerging field. We find that the evolution of the magnon experiences the transition from order to period-doubling bifurcation and finally enter chaos by adjusting the microwave driving power. Different from the normal chaos, the magnon chaos-order transition proposed here is phase-mediated. Beyond their fundamental scientific significance, our results will contribute to the comprehending of nonlinear phenomena and chaos in optomagnonical system, and may find applications in chaos-based secure communication.

cond-mat.mes-hall↗

arXiv2018

Mathematical Analysis of Adversarial Attacks

In this paper, we analyze efficacy of the fast gradient sign method (FGSM) and the Carlini-Wagner's L2 (CW-L2) attack. We prove that, within a certain regime, the untargeted FGSM can fool any convolutional neural nets (CNNs) with ReLU activation; the targeted FGSM can mislead any CNNs with ReLU activation to classify any given image into any prescribed class. For a special two-layer neural network: a linear layer followed by the softmax output activation, we show that the CW-L2 attack increases the ratio of the classification probability between the target and ground truth classes. Moreover, we provide numerical results to verify all our theoretical results.

cs.LG↗

arXiv2018

Stop memorizing: A data-dependent regularization framework for intrinsic pattern learning

Deep neural networks (DNNs) typically have enough capacity to fit random data by brute force even when conventional data-dependent regularizations focusing on the geometry of the features are imposed. We find out that the reason for this is the inconsistency between the enforced geometry and the standard softmax cross entropy loss. To resolve this, we propose a new framework for data-dependent DNN regularization, the Geometrically-Regularized-Self-Validating neural Networks (GRSVNet). During training, the geometry enforced on one batch of features is simultaneously validated on a separate batch using a validation loss consistent with the geometry. We study a particular case of GRSVNet, the Orthogonal-Low-rank Embedding (OLE)-GRSVNet, which is capable of producing highly discriminative features residing in orthogonal low-rank subspaces. Numerical experiments show that OLE-GRSVNet outperforms DNNs with conventional regularization when trained on real data. More importantly, unlike conventional DNNs, OLE-GRSVNet refuses to memorize random data or random labels, suggesting it only learns intrinsic patterns by reducing the memorizing capacity of the baseline DNN.

cs.CV↗

arXiv2018

Error estimation of weighted nonlocal Laplacian on random point cloud

We analyze the convergence of the weighted nonlocal Laplacian (WNLL) on high dimensional randomly distributed data. The analysis reveals the importance of the scaling weight $μ\sim P|/|S|$ with $|P|$ and $|S|$ be the number of entire and labeled data, respectively. The result gives a theoretical foundation of WNLL for high dimensional data interpolation.

math.NA↗

arXiv2018

Magnon-induced high-order sideband generation

Magnon Kerr nonlinearity plays crucial roles in the study of cavity optomagnonics system and may bring many novel physical phenomena and important applications. In this work, we report the investigation of high- order sideband generation induced by magnon Kerr nonlinearity in a cavity-magnon system, which is critically lacking in this emerging research field. We uncover that the driving field plays a significant role in controlling the generation and amplification of the higher-order sidebands and the sideband spacing can be adjusted by regulating the beat frequency between the pump laser and the probe laser, which is extremely eventful for the spacing modulation of the sideband spectrum. Based on the recent experimental progress, our results deepen our cognition into optomagnonics nonlinearity and may find interesting applications in optical frequency metrology and optical communications.

physics.optics↗