Bao Wang

Tan Nguyen, Richard G. Baraniuk, Robert M. Kirby et al.

Transformers have achieved remarkable success in sequence modeling and beyond but suffer from quadratic computational and memory complexities with respect to the length of the input sequence. Leveraging techniques include sparse and linear attention and hashing tricks; efficient transformers have been proposed to reduce the quadratic complexity of transformers but significantly degrade the accuracy. In response, we first interpret the linear attention and residual connections in computing the attention map as gradient descent steps. We then introduce momentum into these components and propose the \emph{momentum transformer}, which utilizes momentum to improve the accuracy of linear transformers while maintaining linear memory and computational complexities. Furthermore, we develop an adaptive strategy to compute the momentum value for our model based on the optimal momentum for quadratic optimization. This adaptive momentum eliminates the need to search for the optimal momentum value and further enhances the performance of the momentum transformer. A range of experiments on both autoregressive and non-autoregressive tasks, including image generation and machine translation, demonstrate that the momentum transformer outperforms popular linear transformers in training efficiency and accuracy.

Justin Baker, Hedi Xia, Yiwei Wang et al.

Learning neural ODEs often requires solving very stiff ODE systems, primarily using explicit adaptive step size ODE solvers. These solvers are computationally expensive, requiring the use of tiny step sizes for numerical stability and accuracy guarantees. This paper considers learning neural ODEs using implicit ODE solvers of different orders leveraging proximal operators. The proximal implicit solver consists of inner-outer iterations: the inner iterations approximate each implicit update step using a fast optimization algorithm, and the outer iterations solve the ODE system over time. The proximal implicit ODE solver guarantees superiority over explicit solvers in numerical stability and computational efficiency. We validate the advantages of proximal implicit solvers over existing popular neural ODE solvers on various challenging benchmark tasks, including learning continuous-depth graph neural networks and continuous normalizing flows.

Hedi Xia, Vai Suliafu, Hangjie Ji et al.

We propose heavy ball neural ordinary differential equations (HBNODEs), leveraging the continuous limit of the classical momentum accelerated gradient descent, to improve neural ODEs (NODEs) training and inference. HBNODEs have two properties that imply practical advantages over NODEs: (i) The adjoint state of an HBNODE also satisfies an HBNODE, accelerating both forward and backward ODE solvers, thus significantly reducing the number of function evaluations (NFEs) and improving the utility of the trained models. (ii) The spectrum of HBNODEs is well structured, enabling effective learning of long-term dependencies from complex sequential data. We verify the advantages of HBNODEs over NODEs on benchmark tasks, including image classification, learning complex dynamics, and sequential modeling. Our method requires remarkably fewer forward and backward NFEs, is more accurate, and learns long-term dependencies more effectively than the other ODE-based neural network models. Code is available at \url{https://github.com/hedixia/HeavyBallNODE}.

Thu Dinh, Bao Wang, Andrea L. Bertozzi et al.

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}.

Bao Wang, Difan Zou, Quanquan Gu et al.

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}.

Bao Wang, Stanley J. Osher

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}.

Bao Wang, Quanquan Gu, March Boedihardjo et al.

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}.

Stanley Osher, Bao Wang, Penghang Yin et al.

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}

Shih-Hsin Wang, Justin Baker, Cory Hauck et al.

The pioneering work of Oono and Suzuki [ICLR, 2020] and Cai and Wang [arXiv:2006.13318] initializes the analysis of the smoothness of graph convolutional network (GCN) features. Their results reveal an intricate empirical correlation between node classification accuracy and the ratio of smooth to non-smooth feature components. However, the optimal ratio that favors node classification is unknown, and the non-smooth features of deep GCN with ReLU or leaky ReLU activation function diminish. In this paper, we propose a new strategy to let GCN learn node features with a desired smoothness -- adapting to data and tasks -- to enhance node classification. Our approach has three key steps: (1) We establish a geometric relationship between the input and output of ReLU or leaky ReLU. (2) Building on our geometric insights, we augment the message-passing process of graph convolutional layers (GCLs) with a learnable term to modulate the smoothness of node features with computational efficiency. (3) We investigate the achievable ratio between smooth and non-smooth feature components for GCNs with the augmented message-passing scheme. Our extensive numerical results show that the augmented message-passing schemes significantly improve node classification for GCN and some related models.

Yunling Zheng, Carson Hu, Guang Lin et al.

We propose GLassoformer, a novel and efficient transformer architecture leveraging group Lasso regularization to reduce the number of queries of the standard self-attention mechanism. Due to the sparsified queries, GLassoformer is more computationally efficient than the standard transformers. On the power grid post-fault voltage prediction task, GLassoformer shows remarkably better prediction than many existing benchmark algorithms in terms of accuracy and stability.

Yifan Hua, Kevin Miller, Andrea L. Bertozzi et al.

We propose near-optimal overlay networks based on $d$-regular expander graphs to accelerate decentralized federated learning (DFL) and improve its generalization. In DFL a massive number of clients are connected by an overlay network, and they solve machine learning problems collaboratively without sharing raw data. Our overlay network design integrates spectral graph theory and the theoretical convergence and generalization bounds for DFL. As such, our proposed overlay networks accelerate convergence, improve generalization, and enhance robustness to clients failures in DFL with theoretical guarantees. Also, we present an efficient algorithm to convert a given graph to a practical overlay network and maintaining the network topology after potential client failures. We numerically verify the advantages of DFL with our proposed networks on various benchmark tasks, ranging from image classification to language modeling using hundreds of clients.

Tao Sun, Huaming Ling, Zuoqiang Shi et al.

Heavy ball momentum is crucial in accelerating (stochastic) gradient-based optimization algorithms for machine learning. Existing heavy ball momentum is usually weighted by a uniform hyperparameter, which relies on excessive tuning. Moreover, the calibrated fixed hyperparameter may not lead to optimal performance. In this paper, to eliminate the effort for tuning the momentum-related hyperparameter, we propose a new adaptive momentum inspired by the optimal choice of the heavy ball momentum for quadratic optimization. Our proposed adaptive heavy ball momentum can improve stochastic gradient descent (SGD) and Adam. SGD and Adam with the newly designed adaptive momentum are more robust to large learning rates, converge faster, and generalize better than the baselines. We verify the efficiency of SGD and Adam with the new adaptive momentum on extensive machine learning benchmarks, including image classification, language modeling, and machine translation. Finally, we provide convergence guarantees for SGD and Adam with the proposed adaptive momentum.

Bao Wang, Hedi Xia, Tan Nguyen et al.

We present and review an algorithmic and theoretical framework for improving neural network architecture design via momentum. As case studies, we consider how momentum can improve the architecture design for recurrent neural networks (RNNs), neural ordinary differential equations (ODEs), and transformers. We show that integrating momentum into neural network architectures has several remarkable theoretical and empirical benefits, including 1) integrating momentum into RNNs and neural ODEs can overcome the vanishing gradient issues in training RNNs and neural ODEs, resulting in effective learning long-term dependencies. 2) momentum in neural ODEs can reduce the stiffness of the ODE dynamics, which significantly enhances the computational efficiency in training and testing. 3) momentum can improve the efficiency and accuracy of transformers.

Tao Sun, Dongsheng Li, Bao Wang

Federated averaging (FedAvg) is a communication efficient algorithm for the distributed training with an enormous number of clients. In FedAvg, clients keep their data locally for privacy protection; a central parameter server is used to communicate between clients. This central server distributes the parameters to each client and collects the updated parameters from clients. FedAvg is mostly studied in centralized fashions, which requires massive communication between server and clients in each communication. Moreover, attacking the central server can break the whole system's privacy. In this paper, we study the decentralized FedAvg with momentum (DFedAvgM), which is implemented on clients that are connected by an undirected graph. In DFedAvgM, all clients perform stochastic gradient descent with momentum and communicate with their neighbors only. To further reduce the communication cost, we also consider the quantized DFedAvgM. We prove convergence of the (quantized) DFedAvgM under trivial assumptions; the convergence rate can be improved when the loss function satisfies the PŁ property. Finally, we numerically verify the efficacy of DFedAvgM.

Tao Sun, Dongsheng Li, Bao Wang

The stability and generalization of stochastic gradient-based methods provide valuable insights into understanding the algorithmic performance of machine learning models. As the main workhorse for deep learning, stochastic gradient descent has received a considerable amount of studies. Nevertheless, the community paid little attention to its decentralized variants. In this paper, we provide a novel formulation of the decentralized stochastic gradient descent. Leveraging this formulation together with (non)convex optimization theory, we establish the first stability and generalization guarantees for the decentralized stochastic gradient descent. Our theoretical results are built on top of a few common and mild assumptions and reveal that the decentralization deteriorates the stability of SGD for the first time. We verify our theoretical findings by using a variety of decentralized settings and benchmark machine learning models.

Bao Wang, Qiang Ye

Momentum plays a crucial role in stochastic gradient-based optimization algorithms for accelerating or improving training deep neural networks (DNNs). In deep learning practice, the momentum is usually weighted by a well-calibrated constant. However, tuning hyperparameters for momentum can be a significant computational burden. In this paper, we propose a novel \emph{adaptive momentum} for improving DNNs training; this adaptive momentum, with no momentum related hyperparameter required, is motivated by the nonlinear conjugate gradient (NCG) method. Stochastic gradient descent (SGD) with this new adaptive momentum eliminates the need for the momentum hyperparameter calibration, allows a significantly larger learning rate, accelerates DNN training, and improves final accuracy and robustness of the trained DNNs. For instance, SGD with this adaptive momentum reduces classification errors for training ResNet110 for CIFAR10 and CIFAR100 from $5.25\%$ to $4.64\%$ and $23.75\%$ to $20.03\%$, respectively. Furthermore, SGD with the new adaptive momentum also benefits adversarial training and improves adversarial robustness of the trained DNNs.

Zhijian Li, Bao Wang, Jack Xin

Deep Neural Networks (DNNs) needs to be both efficient and robust for practical uses. Quantization and structure simplification are promising ways to adapt DNNs to mobile devices, and adversarial training is the most popular method to make DNNs robust. In this work, we try to obtain both features by applying a convergent relaxation quantization algorithm, Binary-Relax (BR), to a robust adversarial-trained model, ResNets Ensemble via Feynman-Kac Formalism (EnResNet). We also discover that high precision, such as ternary (tnn) and 4-bit, quantization will produce sparse DNNs. However, this sparsity is unstructured under advarsarial training. To solve the problems that adversarial training jeopardizes DNNs' accuracy on clean images and the struture of sparsity, we design a trade-off loss function that helps DNNs preserve their natural accuracy and improve the channel sparsity. With our trade-off loss function, we achieve both goals with no reduction of resistance under weak attacks and very minor reduction of resistance under strong attcks. Together with quantized EnResNet with trade-off loss function, we provide robust models that have high efficiency.

Bao Wang, Tan M. Nguyen, Andrea L. Bertozzi et al.

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.

Lisa Maria Kreusser, Stanley J. Osher, Bao Wang

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.

Bao Wang, Alex T. Lin, Wei Zhu et al.

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.

Zuoqiang Shi, Bao Wang, Stanley J. Osher

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.

Wei Zhu, Qiang Qiu, Bao Wang et al.

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.

Bao Wang, Xiyang Luo, Fangbo Zhang et al.

We present a generic framework for spatio-temporal (ST) data modeling, analysis, and forecasting, with a special focus on data that is sparse in both space and time. Our multi-scaled framework is a seamless coupling of two major components: a self-exciting point process that models the macroscale statistical behaviors of the ST data and a graph structured recurrent neural network (GSRNN) to discover the microscale patterns of the ST data on the inferred graph. This novel deep neural network (DNN) incorporates the real time interactions of the graph nodes to enable more accurate real time forecasting. The effectiveness of our method is demonstrated on both crime and traffic forecasting.

Bao Wang, Xiyang Luo, Zhen Li et al.

We replace the output layer of deep neural nets, typically the softmax function, by a novel interpolating function. And we propose end-to-end training and testing algorithms for this new architecture. Compared to classical neural nets with softmax function as output activation, the surrogate with interpolating function as output activation combines advantages of both deep and manifold learning. The new framework demonstrates the following major advantages: First, it is better applicable to the case with insufficient training data. Second, it significantly improves the generalization accuracy on a wide variety of networks. The algorithm is implemented in PyTorch, and code will be made publicly available.

Bao Wang, Zhixiong Zhao, Duc D. Nguyen et al.

We present a feature functional theory - binding predictor (FFT-BP) for the protein-ligand binding affinity prediction. The underpinning assumptions of FFT-BP are as follows: i) representability: there exists a microscopic feature vector that can uniquely characterize and distinguish one protein-ligand complex from another; ii) feature-function relationship: the macroscopic features, including binding free energy, of a complex is a functional of microscopic feature vectors; and iii) similarity: molecules with similar microscopic features have similar macroscopic features, such as binding affinity. Physical models, such as implicit solvent models and quantum theory, are utilized to extract microscopic features, while machine learning algorithms are employed to rank the similarity among protein-ligand complexes. A large variety of numerical validations and tests confirms the accuracy and robustness of the proposed FFT-BP model. The root mean square errors (RMSEs) of FFT-BP blind predictions of a benchmark set of 100 complexes, the PDBBind v2007 core set of 195 complexes and the PDBBind v2015 core set of 195 complexes are 1.99, 2.02 and 1.92 kcal/mol, respectively. Their corresponding Pearson correlation coefficients are 0.75, 0.80, and 0.78, respectively.