In this work, we propose a meta-learner based on ODE neural networks that learns gradients. This approach makes the optimizer is more flexible inducing an automatic inductive bias to the given task. Using the simplest Hamiltonian Neural Network we demonstrate that our method outperforms a meta-learner based on LSTM for an artificial task and the MNIST dataset with ReLU activations in the optimizee. Furthermore, it also surpasses the classic optimization methods for the artificial task and achieves comparable results for MNIST.
The Chinese University of Hong Kong（香港中文大学）；Rutgers University（罗格斯大学）；Florida State University（佛罗里达州立大学）
Stochastic gradient Hamiltonian Monte Carlo (SGHMC) is a variant of stochastic gradient with momentum where a controlled and properly scaled Gaussian noise is added to the stochastic gradients to steer the iterates towards a global minimum. Many works reported its empirical success in practice for solving stochastic non-convex optimization problems, in particular it has been observed to outperform overdamped Langevin Monte Carlo-based methods such as stochastic gradient Langevin dynamics (SGLD) in many applications. Although asymptotic global convergence properties of SGHMC are well known, its…
Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) is a momentum version of stochastic gradient descent with properly injected Gaussian noise to find a global minimum. In this paper, non-asymptotic convergence analysis of SGHMC is given in the context of non-convex optimization, where subsampling techniques are used over an i.i.d dataset for gradient updates. Our results complement those of [RRT17] and improve on those of [GGZ18].
We propose a new sampler that integrates the protocol of parallel tempering with the Nosé-Hoover (NH) dynamics. The proposed method can efficiently draw representative samples from complex posterior distributions with multiple isolated modes in the presence of noise arising from stochastic gradient. It potentially facilitates deep Bayesian learning on large datasets where complex multimodal posteriors and mini-batch gradient are encountered.
Gradient-based Monte Carlo sampling algorithms, like Langevin dynamics and Hamiltonian Monte Carlo, are important methods for Bayesian inference. In large-scale settings, full-gradients are not affordable and thus stochastic gradients evaluated on mini-batches are used as a replacement. In order to reduce the high variance of noisy stochastic gradients, Dubey et al. [2016] applied the standard variance reduction technique on stochastic gradient Langevin dynamics and obtained both theoretical and experimental improvements. In this paper, we apply the variance reduction tricks on Hamiltonian Mon…
Recurrent neural networks (RNNs) have gained a great deal of attention in solving sequential learning problems. The learning of long-term dependencies, however, remains challenging due to the problem of a vanishing or exploding hidden states gradient. By exploring further the recently established connections between RNNs and dynamical systems we propose a novel RNN architecture, which we call a Hamiltonian recurrent neural network (Hamiltonian RNN), based on a symplectic discretization of an appropriately chosen Hamiltonian system. The key benefit of this approach is that the corresponding RNN…
Rutgers Business School（罗格斯大学商学院）；The Chinese University of Hong Kong（香港中文大学）；Florida State University（佛罗里达州立大学）
Stochastic gradient Langevin dynamics (SGLD) and stochastic gradient Hamiltonian Monte Carlo (SGHMC) are two popular Markov Chain Monte Carlo (MCMC) algorithms for Bayesian inference that can scale to large datasets, allowing to sample from the posterior distribution of the parameters of a statistical model given the input data and the prior distribution over the model parameters. However, these algorithms do not apply to the decentralized learning setting, when a network of agents are working collaboratively to learn the parameters of a statistical model without sharing their individual data …
We study the convergence properties of gradient descent for training deep linear neural networks, i.e., deep matrix factorizations, by extending a previous analysis for the related gradient flow. We show that under suitable conditions on the step sizes gradient descent converges to a critical point of the loss function, i.e., the square loss in this article. Furthermore, we demonstrate that for almost all initializations gradient descent converges to a global minimum in the case of two layers. In the case of three or more layers we show that gradient descent converges to a global minimum on th…
Non-asymptotic convergence analysis of the stochastic gradient Hamiltonian Monte Carlo algorithm with discontinuous stochastic gradient with applications to training of ReLU neural networks
School of Mathematics, Renmin University of China, Beijing, China（中国人民大学数学学院）；Division of Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371 Singapore（南洋理工大学数学科学学院）；Financial Technology Thrust, Society Hub, The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China（香港科技大学（广州）金融科技中心）
Mila（米拉）；Université de Montréal（蒙特利尔大学）；Facebook AI Research（Facebook人工智能研究院）
The success of adversarial formulations in machine learning has brought renewed motivation for smooth games. In this work, we focus on the class of stochastic Hamiltonian methods and provide the first convergence guarantees for certain classes of stochastic smooth games. We propose a novel unbiased estimator for the stochastic Hamiltonian gradient descent (SHGD) and highlight its benefits. Using tools from the optimization literature we show that SHGD converges linearly to the neighbourhood of a stationary point. To guarantee convergence to the exact solution, we analyze SHGD with a decreasing…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Shanghai University（上海大学）
Deep Neural Networks (DNNs) training can be difficult due to vanishing and exploding gradients during weight optimization through backpropagation. To address this problem, we propose a general class of Hamiltonian DNNs (H-DNNs) that stem from the discretization of continuous-time Hamiltonian systems and include several existing DNN architectures based on ordinary differential equations. Our main result is that a broad set of H-DNNs ensures non-vanishing gradients by design for an arbitrary network depth. This is obtained by proving that, using a semi-implicit Euler discretization scheme, the b…
Hamiltonian Monte Carlo (HMC) sampling methods provide a mechanism for defining distant proposals with high acceptance probabilities in a Metropolis-Hastings framework, enabling more efficient exploration of the state space than standard random-walk proposals. The popularity of such methods has grown significantly in recent years. However, a limitation of HMC methods is the required gradient computation for simulation of the Hamiltonian dynamical system-such computation is infeasible in problems involving a large sample size or streaming data. Instead, we must rely on a noisy gradient estimate…
Shanghai Jiao Tong University（上海交通大学）；Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
The past few years have witnessed an increased interest in learning Hamiltonian dynamics in deep learning frameworks. As an inductive bias based on physical laws, Hamiltonian dynamics endow neural networks with accurate long-term prediction, interpretability, and data-efficient learning. However, Hamiltonian dynamics also bring energy conservation or dissipation assumptions on the input data and additional computational overhead. In this paper, we systematically survey recently proposed Hamiltonian neural network models, with a special emphasis on methodologies. In general, we discuss the majo…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Training deep neural networks (DNNs) can be difficult due to the occurrence of vanishing/exploding gradients during weight optimization. To avoid this problem, we propose a class of DNNs stemming from the time discretization of Hamiltonian systems. The time-invariant version of the corresponding Hamiltonian models enjoys marginal stability, a property that, as shown in previous works and for specific DNNs architectures, can mitigate convergence to zero or divergence of gradients. In the present paper, we formally study this feature by deriving and analysing the backward gradient dynamics in co…
The massive size of modern neural networks has motivated substantial recent interest in neural network quantization. We introduce Stochastic Markov Gradient Descent (SMGD), a discrete optimization method applicable to training quantized neural networks. The SMGD algorithm is designed for settings where memory is highly constrained during training. We provide theoretical guarantees of algorithm performance as well as encouraging numerical results.
University of Cambridge（剑桥大学）；Microsoft Research（微软研究院）；Alan Turing Institute（艾伦·图灵研究所）；University of Sevilla（塞维利亚大学）
Deep Gaussian Processes (DGPs) are hierarchical generalizations of Gaussian Processes that combine well calibrated uncertainty estimates with the high flexibility of multilayer models. One of the biggest challenges with these models is that exact inference is intractable. The current state-of-the-art inference method, Variational Inference (VI), employs a Gaussian approximation to the posterior distribution. This can be a potentially poor unimodal approximation of the generally multimodal posterior. In this work, we provide evidence for the non-Gaussian nature of the posterior and we apply the…
It has been shown that gradient descent can yield the zero training loss in the over-parametrized regime (the width of the neural networks is much larger than the number of data points). In this work, combining the ideas of some existing works, we investigate the gradient descent method for training two-layer neural networks for approximating some target continuous functions. By making use the generic chaining technique from probability theory, we show that gradient descent can yield an exponential convergence rate, while the width of the neural networks needed is independent of the size of th…
Policy gradient methods with actor-critic schemes demonstrate tremendous empirical successes, especially when the actors and critics are parameterized by neural networks. However, it remains less clear whether such "neural" policy gradient methods converge to globally optimal policies and whether they even converge at all. We answer both the questions affirmatively in the overparameterized regime. In detail, we prove that neural natural policy gradient converges to a globally optimal policy at a sublinear rate. Also, we show that neural vanilla policy gradient converges sublinearly to a statio…
We take a Hamiltonian-based perspective to generalize Nesterov's accelerated gradient descent and Polyak's heavy ball method to a broad class of momentum methods in the setting of (possibly) constrained minimization in Euclidean and non-Euclidean normed vector spaces. Our perspective leads to a generic and unifying nonasymptotic analysis of convergence of these methods in both the function value (in the setting of convex optimization) and in norm of the gradient (in the setting of unconstrained, possibly nonconvex, optimization). Our approach relies upon a time-varying Hamiltonian that produce…
We present a derivation of the gradients of feedforward neural networks using Fréchet calculus which is arguably more compact than the ones usually presented in the literature. We first derive the gradients for ordinary neural networks working on vectorial data and show how these derived formulas can be used to derive a simple and efficient algorithm for calculating a neural networks gradients. Subsequently we show how our analysis generalizes to more general neural network architectures including, but not limited to, convolutional networks.
Deep learning is vulnerable to adversarial examples. Many defenses based on randomized neural networks have been proposed to solve the problem, but fail to achieve robustness against attacks using proxy gradients such as the Expectation over Transformation (EOT) attack. We investigate the effect of the adversarial attacks using proxy gradients on randomized neural networks and demonstrate that it highly relies on the directional distribution of the loss gradients of the randomized neural network. We show in particular that proxy gradients are less effective when the gradients are more scattere…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Carnegie Mellon University（卡内基梅隆大学）
Adaptive gradient methods like AdaGrad are widely used in optimizing neural networks. Yet, existing convergence guarantees for adaptive gradient methods require either convexity or smoothness, and, in the smooth setting, only guarantee convergence to a stationary point. We propose an adaptive gradient method and show that for two-layer over-parameterized neural networks -- if the width is sufficiently large (polynomially) -- then the proposed method converges \emph{to the global minimum} in polynomial time, and convergence is robust, \emph{ without the need to fine-tune hyper-parameters such a…
We propose a new framework of variance-reduced Hamiltonian Monte Carlo (HMC) methods for sampling from an $L$-smooth and $m$-strongly log-concave distribution, based on a unified formulation of biased and unbiased variance reduction methods. We study the convergence properties for HMC with gradient estimators which satisfy the Mean-Squared-Error-Bias (MSEB) property. We show that the unbiased gradient estimators, including SAGA and SVRG, based HMC methods achieve highest gradient efficiency with small batch size under high precision regime, and require $\tilde{O}(N + κ^2 d^{\frac{1}{2}} \varep…
We introduce a new technique for gradient normalization during neural network training. The gradients are rescaled during the backward pass using normalization layers introduced at certain points within the network architecture. These normalization nodes do not affect forward activity propagation, but modify backpropagation equations to permit a well-scaled gradient flow that reaches the deepest network layers without experimenting vanishing or explosion. Results on tests with very deep neural networks show that the new technique can do an effective control of the gradient norm, allowing the u…
Informatics Institute（信息学研究所）；University of Amsterdam（阿姆斯特丹大学）
Approximate Bayesian computation (ABC) is a powerful and elegant framework for performing inference in simulation-based models. However, due to the difficulty in scaling likelihood estimates, ABC remains useful for relatively low-dimensional problems. We introduce Hamiltonian ABC (HABC), a set of likelihood-free algorithms that apply recent advances in scaling Bayesian learning using Hamiltonian Monte Carlo (HMC) and stochastic gradients. We find that a small number forward simulations can effectively approximate the ABC gradient, allowing Hamiltonian dynamics to efficiently traverse parameter…
Many physical phenomena are described by Hamiltonian mechanics using an energy function (the Hamiltonian). Recently, the Hamiltonian neural network, which approximates the Hamiltonian as a neural network, and its extensions have attracted much attention. This is a very powerful method, but its use in theoretical studies remains limited. In this study, by combining the statistical learning theory and Kolmogorov-Arnold-Moser (KAM) theory, we provide a theoretical analysis of the behavior of Hamiltonian neural networks when the learning error is not completely zero. A Hamiltonian neural network w…
Graph neural networks (GNNs) have achieved success in various inference tasks on graph-structured data. However, common challenges faced by many GNNs in the literature include the problem of graph node embedding under various geometries and the over-smoothing problem. To address these issues, we propose a novel graph information propagation strategy called Hamiltonian Dynamic GNN (HDG) that uses a Hamiltonian mechanics approach to learn node embeddings in a graph. The Hamiltonian energy function in HDG is learnable and can adapt to the underlying geometry of any given graph dataset. We demonst…
University of Washington（华盛顿大学）；Stanford University（斯坦福大学）
We show that the gradient norm $\|\nabla f(x)\|$ for $x \sim \exp(-f(x))$, where $f$ is strongly convex and smooth, concentrates tightly around its mean. This removes a barrier in the prior state-of-the-art analysis for the well-studied Metropolized Hamiltonian Monte Carlo (HMC) algorithm for sampling from a strongly logconcave distribution. We correspondingly demonstrate that Metropolized HMC mixes in $\tilde{O}(κd)$ iterations, improving upon the $\tilde{O}(κ^{1.5}\sqrt{d} + κd)$ runtime of (Dwivedi et. al. '18, Chen et. al. '19) by a factor $(κ/d)^{1/2}$ when the condition number $κ$ is lar…
Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
Deep neural networks can be fragile and sensitive to small input perturbations that might cause a significant change in the output. In this paper, we employ contraction theory to improve the robustness of neural ODEs (NODEs). A dynamical system is contractive if all solutions with different initial conditions converge to each other exponentially fast. As a consequence, perturbations in initial conditions become less and less relevant over time. Since in NODEs the input data corresponds to the initial condition of dynamical systems, we show contractivity can mitigate the effect of input perturb…
IHPC A*STAR（A*STAR高性能计算研究所）；Georgia State University（佐治亚州立大学）
Deep neural networks have achieved great success both in computer vision and natural language processing tasks. However, mostly state-of-art methods highly rely on external training or computing to improve the performance. To alleviate the external reliance, we proposed a gradient enhancement approach, conducted by the short circuit neural connections, to improve the gradient learning of deep neural networks. The proposed short circuit is a unidirectional connection that single back propagates the sensitive from the deep layer to the shallows. Moreover, the short circuit formulates to be a gra…
Symplectic Neural Networks for Learning Non-Separable Hamiltonians
Czech Technical University in Prague（捷克布拉格理工大学）；University of California at San Diego（加州大学圣地亚哥分校）；Athena Research and Innovation Center（雅典研究与创新中心）；HSBC Holdings（汇丰控股）
Understanding natural symmetries is key to making sense of our complex and ever-changing world. Recent work has shown that neural networks can learn such symmetries directly from data using Hamiltonian Neural Networks (HNNs). But HNNs struggle when trained on datasets where energy is not conserved. In this paper, we ask whether it is possible to identify and decompose conservative and dissipative dynamics simultaneously. We propose Dissipative Hamiltonian Neural Networks (D-HNNs), which parameterize both a Hamiltonian and a Rayleigh dissipation function. Taken together, they represent an impli…
Facebook（脸书）；University of Maryland, College Park（马里兰大学学院市分校）；DeepMind（深度思维）
This paper studies how neural network architecture affects the speed of training. We introduce a simple concept called gradient confusion to help formally analyze this. When gradient confusion is high, stochastic gradients produced by different data samples may be negatively correlated, slowing down convergence. But when gradient confusion is low, data samples interact harmoniously, and training proceeds quickly. Through theoretical and experimental results, we demonstrate how the neural network architecture affects gradient confusion, and thus the efficiency of training. Our results show that…
Learning partially observed systems with neural Hamiltonian ordinary differential equations
Department of Mathematics and Cybernetics, SINTEF Digital（数学与自动化系，SINTEF数字研究院）；Department of Physics, University of Oslo（物理系，奥斯陆大学）；Department of Data Science, Norwegian University of Life Science（数据科学系，挪威生命科学大学）
Hybrid machine learning based on Hamiltonian formulations has recently been successfully demonstrated for simple mechanical systems, both energy conserving and not energy conserving. We introduce a pseudo-Hamiltonian formulation that is a generalization of the Hamiltonian formulation via the port-Hamiltonian formulation, and show that pseudo-Hamiltonian neural network models can be used to learn external forces acting on a system. We argue that this property is particularly useful when the external forces are state dependent, in which case it is the pseudo-Hamiltonian structure that facilitate…
Well-designed molecular representations (fingerprints) are vital to combine medical chemistry and deep learning. Whereas incorporating 3D geometry of molecules (i.e. conformations) in their representations seems beneficial, current 3D algorithms are still in infancy. In this paper, we propose a novel molecular representation algorithm which preserves 3D conformations of molecules with a Molecular Hamiltonian Network (HamNet). In HamNet, implicit positions and momentums of atoms in a molecule interact in the Hamiltonian Engine following the discretized Hamiltonian equations. These implicit coor…
Hamiltonian Monte Carlo (HMC) is a state-of-the-art Markov chain Monte Carlo sampling algorithm for drawing samples from smooth probability densities over continuous spaces. We study the variant most widely used in practice, Metropolized HMC with the Störmer-Verlet or leapfrog integrator, and make two primary contributions. First, we provide a non-asymptotic upper bound on the mixing time of the Metropolized HMC with explicit choices of step-size and number of leapfrog steps. This bound gives a precise quantification of the faster convergence of Metropolized HMC relative to simpler MCMC algori…
Habana Labs – An Intel company（哈巴纳实验室——英特尔旗下公司）；Department of Electrical Engineering - Technion（以色列理工学院电气工程系）
While training can mostly be accelerated by reducing the time needed to propagate neural gradients back throughout the model, most previous works focus on the quantization/pruning of weights and activations. These methods are often not applicable to neural gradients, which have very different statistical properties. Distinguished from weights and activations, we find that the distribution of neural gradients is approximately lognormal. Considering this, we suggest two closed-form analytical methods to reduce the computational and memory burdens of neural gradients. The first method optimizes t…
Carnegie Mellon University（卡内基梅隆大学）；University of Southern California（南加州大学）；Peking University（北京大学）；Beijing Institute of Big Data Research（北京大数据研究院）；Massachusetts Institute of Technology（麻省理工学院）
Gradient descent finds a global minimum in training deep neural networks despite the objective function being non-convex. The current paper proves gradient descent achieves zero training loss in polynomial time for a deep over-parameterized neural network with residual connections (ResNet). Our analysis relies on the particular structure of the Gram matrix induced by the neural network architecture. This structure allows us to show the Gram matrix is stable throughout the training process and this stability implies the global optimality of the gradient descent algorithm. We further extend our …
Training recurrent neural networks (RNNs) with backpropagation through time (BPTT) has known drawbacks such as being difficult to capture longterm dependencies in sequences. Successful alternatives to BPTT have not yet been discovered. Recently, BP with synthetic gradients by a decoupled neural interface module has been proposed to replace BPTT for training RNNs. On the other hand, it has been shown that the representations learned with synthetic and real gradients are different though they are functionally identical. In this project, we explore ways of combining synthetic and real gradients w…
There has been a wave of interest in applying machine learning to study dynamical systems. We present a Hamiltonian neural network that solves the differential equations that govern dynamical systems. This is an equation-driven machine learning method where the optimization process of the network depends solely on the predicted functions without using any ground truth data. The model learns solutions that satisfy, up to an arbitrarily small error, Hamilton's equations and, therefore, conserve the Hamiltonian invariants. The choice of an appropriate activation function drastically improves the …
