Tensor networks have demonstrated significant value for machine learning in a myriad of different applications. However, optimizing tensor networks using standard gradient descent has proven to be difficult in practice. Tensor networks suffer from initialization problems resulting in exploding or vanishing gradients and require extensive hyperparameter tuning. Efforts to overcome these problems usually depend on specific network architectures, or ad hoc prescriptions. In this paper we address the problems of initialization and hyperparameter tuning, making it possible to train tensor networks using established machine learning techniques. We introduce a `copy node' method that successfully initializes arbitrary tensor networks, in addition to a gradient based regularization technique for bond dimensions. We present numerical results that show that the combination of techniques presented here produces quantum inspired tensor network models with far fewer parameters, while improving generalization performance.
Microsoft Research（微软研究院）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We present a primal only derivation of Mirror Descent as a "partial" discretization of gradient flow on a Riemannian manifold where the metric tensor is the Hessian of the Mirror Descent potential. We contrast this discretization to Natural Gradient Descent, which is obtained by a "full" forward Euler discretization. This view helps shed light on the relationship between the methods and allows generalizing Mirror Descent to general Riemannian geometries, even when the metric tensor is {\em not} a Hessian, and thus there is no "dual."
Salzburg University of Applied Sciences（萨尔茨堡应用科学大学）
In this work we present an "out-of-the-box" application of Machine Learning (ML) optimizers for an industrial optimization problem. We introduce a piecewise polynomial model (spline) for fitting of $\mathcal{C}^k$-continuos functions, which can be deployed in a cam approximation setting. We then use the gradient descent optimization context provided by the machine learning framework TensorFlow to optimize the model parameters with respect to approximation quality and $\mathcal{C}^k$-continuity and evaluate available optimizers. Our experiments show that the problem solution is feasible using T…
Sapienza Università di Roma（罗马大学）；INFN – Sezione di Roma1（意大利国家核物理研究所罗马第一分部）；CNR-Nanotec（意大利国家研究委员会纳米技术研究所）
In many high-dimensional estimation problems the main task consists in minimizing a cost function, which is often strongly non-convex when scanned in the space of parameters to be estimated. A standard solution to flatten the corresponding rough landscape consists in summing the losses associated to different data points and obtain a smoother empirical risk. Here we propose a complementary method that works for a single data point. The main idea is that a large amount of the roughness is uncorrelated in different parts of the landscape. One can then substantially reduce the noise by evaluating…
Structural Health Monitoring (SHM) provides an economic approach which aims to enhance understanding the behavior of structures by continuously collects data through multiple networked sensors attached to the structure. This data is then utilized to gain insight into the health of a structure and make timely and economic decisions about its maintenance. The generated SHM sensing data is non-stationary and exists in a correlated multi-way form which makes the batch/off-line learning and standard two-way matrix analysis unable to capture all of these correlations and relationships. In this sense…
Multi-way data analysis has become an essential tool for capturing underlying structures in higher-order datasets stored in tensor $\mathcal{X} \in \mathbb{R} ^{I_1 \times \dots \times I_N} $. $CANDECOMP/PARAFAC$ (CP) decomposition has been extensively studied and applied to approximate $\mathcal{X}$ by $N$ loading matrices $A^{(1)}, \dots, A^{(N)}$ where $N$ represents the order of the tensor. We propose a new efficient CP decomposition solver named NeCPD for non-convex problem in multi-way online data based on stochastic gradient descent (SGD) algorithm. SGD is very useful in online setting …
Georgia Institute of Technology（佐治亚理工学院）；IBM Research（IBM研究院）；Universidad San Pablo-CEU, CEU Universities（CEU大学圣巴勃罗校区）
This work focuses on designing low complexity hybrid tensor networks by considering trade-offs between the model complexity and practical performance. Firstly, we exploit a low-rank tensor-train deep neural network (TT-DNN) to build an end-to-end deep learning pipeline, namely LR-TT-DNN. Secondly, a hybrid model combining LR-TT-DNN with a convolutional neural network (CNN), which is denoted as CNN+(LR-TT-DNN), is set up to boost the performance. Instead of randomly assigning large TT-ranks for TT-DNN, we leverage Riemannian gradient descent to determine a TT-DNN associated with small TT-ranks.…
Duke University（杜克大学）；Princeton University（普林斯顿大学）；Stanford University（斯坦福大学）
Over-parametrization is an important technique in training neural networks. In both theory and practice, training a larger network allows the optimization algorithm to avoid bad local optimal solutions. In this paper we study a closely related tensor decomposition problem: given an $l$-th order tensor in $(R^d)^{\otimes l}$ of rank $r$ (where $r\ll d$), can variants of gradient descent find a rank $m$ decomposition where $m > r$? We show that in a lazy training regime (similar to the NTK regime for neural networks) one needs at least $m = Ω(d^{l-1})$, while a variant of gradient descent can fi…
A quantum generalization of Natural Gradient Descent is presented as part of a general-purpose optimization framework for variational quantum circuits. The optimization dynamics is interpreted as moving in the steepest descent direction with respect to the Quantum Information Geometry, corresponding to the real part of the Quantum Geometric Tensor (QGT), also known as the Fubini-Study metric tensor. An efficient algorithm is presented for computing a block-diagonal approximation to the Fubini-Study metric tensor for parametrized quantum circuits, which may be of independent interest.
Given recent deep learning results that demonstrate the ability to effectively optimize high-dimensional non-convex functions with gradient descent optimization on GPUs, we ask in this paper whether symbolic gradient optimization tools such as Tensorflow can be effective for planning in hybrid (mixed discrete and continuous) nonlinear domains with high dimensional state and action spaces? To this end, we demonstrate that hybrid planning with Tensorflow and RMSProp gradient descent is competitive with mixed integer linear program (MILP) based optimization on piecewise linear planning domains (w…
Microsoft Research New England（微软新英格兰研究院）；University of California Irvine（加州大学欧文分校）；University of California Berkeley（加州大学伯克利分校）；Cornell University（康奈尔大学）
We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points. In this paper we identify strict saddle property for non-convex problem that allows for efficient optimization. Using this property we show that stochastic gradient descent converges to a local minimum in a polynomial number of iterations. To the best of our knowledge this is the first work that gives global convergence guarantees for stochastic gradient desc…
We present a new method for online prediction and learning of tensors ($N$-way arrays, $N >2$) from sequential measurements. We focus on the specific case of 3-D tensors and exploit a recently developed framework of structured tensor decompositions proposed in [1]. In this framework it is possible to treat 3-D tensors as linear operators and appropriately generalize notions of rank and positive definiteness to tensors in a natural way. Using these notions we propose a generalization of the matrix exponentiated gradient descent algorithm [2] to a tensor exponentiated gradient descent algorithm …
Here we utilize a low-rank tensor model (LTM) as a function approximator, combined with the gradient descent method, to solve eigenvalue problems including the Laplacian operator and the harmonic oscillator. Experimental results show the superiority of the polynomial-based low-rank tensor model (PLTM) compared to the tensor neural network (TNN). We also test such low-rank architectures for the classification problem on the MNIST dataset.
National Bank of Canada（加拿大国家银行）；Université du Québec à Montréal（魁北克大学蒙特利尔分校）
Tensor decomposition, a collection of factorization techniques for multidimensional arrays, are among the most general and powerful tools for scientific analysis. However, because of their increasing size, today's data sets require more complex tensor decomposition involving factorization with multiple matrices and diagonal tensors such as DEDICOM or PARATUCK2. Traditional tensor resolution algorithms such as Stochastic Gradient Descent (SGD), Non-linear Conjugate Gradient descent (NCG) or Alternating Least Square (ALS), cannot be easily applied to complex tensor decomposition or often lead to…
We study the problem of learning one-hidden-layer neural networks with Rectified Linear Unit (ReLU) activation function, where the inputs are sampled from standard Gaussian distribution and the outputs are generated from a noisy teacher network. We analyze the performance of gradient descent for training such kind of neural networks based on empirical risk minimization, and provide algorithm-dependent guarantees. In particular, we prove that tensor initialization followed by gradient descent can converge to the ground-truth parameters at a linear rate up to some statistical error. To the best …
Training neural networks is a challenging non-convex optimization problem, and backpropagation or gradient descent can get stuck in spurious local optima. We propose a novel algorithm based on tensor decomposition for guaranteed training of two-layer neural networks. We provide risk bounds for our proposed method, with a polynomial sample complexity in the relevant parameters, such as input dimension and number of neurons. While learning arbitrary target functions is NP-hard, we provide transparent conditions on the function and the input for learnability. Our training method is based on tenso…
High-dimension Tensor Completion via Gradient-based Optimization Under Tensor-train Format
Graduate School of Engineering, Saitama Institute of Technology, Japan（日本埼玉科技大学工学研究科）；Tensor Learning Unit, RIKEN Center for Advanced Intelligence Project (AIP), Japan（日本RIKEN先进人工智能项目（AIP）张量学习单元）；School of Automation, Guangdong University of Technology, China（广东技术大学自动化学院）；School of Computer Science and Technology, Hangzhou Dianzi University, China（杭州电子科技大学计算机科学与技术学院）
Smooth Reparameterizations of Functions on Simplicial Product Spaces: Applications to Probabilistic Tensor Decomposition and Functional Data Registration
Johns Hopkins University（约翰斯·霍普金斯大学）；University of Virginia（弗吉尼亚大学）；UCLouvain（鲁汶大学（法语））
Institute of Science and Technology for Brain-inspired Intelligence（脑启发智能科学与技术研究院）；Fudan University（复旦大学）；Institute of Neuroscience（神经科学研究所）；University of Oregon（俄勒冈大学）
Off-policy algorithms, in which a behavior policy differs from the target policy and is used to gain experience for learning, have proven to be of great practical value in reinforcement learning. However, even for simple convex problems such as linear value function approximation, these algorithms are not guaranteed to be stable. To address this, alternative algorithms that are provably convergent in such cases have been introduced, the most well known being gradient descent temporal difference (GTD) learning. This algorithm and others like it, however, tend to converge much more slowly than c…
This work proposes a novel general-purpose estimator for supervised machine learning (ML) based on tensor trains (TT). The estimator uses TTs to parametrize discretized functions, which are then optimized using Riemannian gradient descent under the form of a tensor completion problem. Since this optimization is sensitive to initialization, it turns out that the use of other ML estimators for initialization is crucial. This results in a competitive, fast ML estimator with lower memory usage than many other ML estimators, like the ones used for the initialization.
University of Edinburgh（爱丁堡大学）；AndPlus LLC（AndPlus有限责任公司）
Neural networks have been able to achieve groundbreaking accuracy at tasks conventionally considered only doable by humans. Using stochastic gradient descent, optimization in many dimensions is made possible, albeit at a relatively high computational cost. By splitting training data into batches, networks can be distributed and trained vastly more efficiently and with minimal accuracy loss. We have explored the mathematics behind efficiently implementing tensor-based batch backpropagation algorithms. A common approach to batch training is iterating over batch items individually. Explicitly usi…
Unsupervised learning aims at the discovery of hidden structure that drives the observations in the real world. It is essential for success in modern machine learning. Latent variable models are versatile in unsupervised learning and have applications in almost every domain. Training latent variable models is challenging due to the non-convexity of the likelihood objective. An alternative method is based on the spectral decomposition of low order moment tensors. This versatile framework is guaranteed to estimate the correct model consistently. My thesis spans both theoretical analysis of tenso…
Particle based optimization algorithms have recently been developed as sampling methods that iteratively update a set of particles to approximate a target distribution. In particular Stein variational gradient descent has gained attention in the approximate inference literature for its flexibility and accuracy. We empirically explore the ability of this method to sample from multi-modal distributions and focus on two important issues: (i) the inability of the particles to escape from local modes and (ii) the inefficacy in reproducing the density of the different regions. We propose an annealin…
Moscow Institute of Physics and Technology（莫斯科物理技术学院）
We prove that if an activation function satisfies some mild conditions and number of neurons in a two-layered fully connected neural network with this activation function is beyond a certain threshold, then gradient descent on quadratic loss function finds the optimal weights of input layer for global minima in linear time. This threshold value is an improvement over previously obtained values. We hypothesise that this bound cannot be improved by the method we are using in this work.
A Scalable Factorization Approach for High-Order Structured Tensor Recovery
Department of Computer Science and Engineering, the Ohio State University（计算机科学与工程系，俄亥俄州立大学）；Department of Electrical Engineering, Colorado School of Mines（电气工程系，科罗拉多矿学院）
Natural Riemannian gradient for learning functional tensor networks
Institute of Mathematics, University of Augsburg（奥格斯堡大学数学研究所）；Department of Mathematics, Technical University of Munich（慕尼黑工业大学数学系）；Institute of Mathematics & Centre for Advanced Analytics and Predictive Sciences, University of Augsburg（奥格斯堡大学数学研究所与高级分析与预测科学中心）
We introduce a novel adaptive damping technique for an inertial gradient system which finds application as a gradient descent algorithm for unconstrained optimisation. In an example using the non-convex Rosenbrock's function, we show an improvement on existing momentum-based gradient optimisation methods. Also using Lyapunov stability analysis, we demonstrate the performance of the continuous-time version of the algorithm. Using numerical simulations, we consider the performance of its discrete-time counterpart obtained by using the symplectic Euler method of discretisation.
The Swiss AI Lab IDSIA / USI / SUPSI（瑞士人工智能实验室IDSIA / USI / SUPSI）
We combine Recurrent Neural Networks with Tensor Product Representations to learn combinatorial representations of sequential data. This improves symbolic interpretation and systematic generalisation. Our architecture is trained end-to-end through gradient descent on a variety of simple natural language reasoning tasks, significantly outperforming the latest state-of-the-art models in single-task and all-tasks settings. We also augment a subset of the data such that training and test data exhibit large systematic differences and show that our approach generalises better than the previous state…
University of Chicago（芝加哥大学）；University of Chicago, Booth School of Business（芝加哥大学布斯商学院）
In many applications, such as classification of images or videos, it is of interest to develop a framework for tensor data instead of an ad-hoc way of transforming data to vectors due to the computational and under-sampling issues. In this paper, we study convergence and statistical properties of two-dimensional canonical correlation analysis \citep{Lee2007Two} under an assumption that data come from a probabilistic model. We show that carefully initialized the power method converges to the optimum and provide a finite sample bound. Then we extend this framework to tensor-valued data and propo…
Stochastic gradient descent based algorithms are typically used as the general optimization tools for most deep learning models. A Restricted Boltzmann Machine (RBM) is a probabilistic generative model that can be stacked to construct deep architectures. For RBM with Bernoulli inputs, non-Euclidean algorithm such as stochastic spectral descent (SSD) has been specifically designed to speed up the convergence with improved use of the gradient estimation by sampling methods. However, the existing algorithm and corresponding theoretical justification depend on the assumption that the possible conf…
Tensor completion aimes at recovering missing data, and it is one of the popular concerns in deep learning and signal processing. Among the higher-order tensor decomposition algorithms, the recently proposed fully-connected tensor network decomposition (FCTN) algorithm is the most advanced. In this paper, by leveraging the superior expression of the fully-connected tensor network (FCTN) decomposition, we propose a new tensor completion method named the fully connected tensor network weighted optization(FCTN-WOPT). The algorithm performs a composition of the completed tensor by initialising the…
Faculty of computer and information science, University of Ljubljana（卢布尔雅那大学计算机与信息科学学院）；Faculty of mathematics and physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）
We discuss two solvable grokking (generalisation beyond overfitting) models in a rule learning scenario. We show that grokking is a phase transition and find exact analytic expressions for the critical exponents, grokking probability, and grokking time distribution. Further, we introduce a tensor-network map that connects the proposed grokking setup with the standard (perceptron) statistical learning theory and show that grokking is a consequence of the locality of the teacher model. As an example, we analyse the cellular automata learning task, numerically determine the critical exponent and …
Università degli Studi di Milano（米兰大学）；Ecole Normale Supérieure, Paris（巴黎高等师范学院）；ICREA（加泰罗尼亚研究与高级研究机构）；Universitat Pompeu Fabra, Barcelona（庞培法布拉大学（巴塞罗那））；HEC Paris（巴黎高等商学院）
Mirror descent with an entropic regularizer is known to achieve shifting regret bounds that are logarithmic in the dimension. This is done using either a carefully designed projection or by a weight sharing technique. Via a novel unified analysis, we show that these two approaches deliver essentially equivalent bounds on a notion of regret generalizing shifting, adaptive, discounted, and other related regrets. Our analysis also captures and extends the generalized weight sharing technique of Bousquet and Warmuth, and can be refined in several ways, including improvements for small losses and a…
This work analyzes the solution trajectory of gradient-based algorithms via a novel basis function decomposition. We show that, although solution trajectories of gradient-based algorithms may vary depending on the learning task, they behave almost monotonically when projected onto an appropriate orthonormal function basis. Such projection gives rise to a basis function decomposition of the solution trajectory. Theoretically, we use our proposed basis function decomposition to establish the convergence of gradient descent (GD) on several representative learning tasks. In particular, we improve …
Coordinate descent methods employ random partial updates of decision variables in order to solve huge-scale convex optimization problems. In this work, we introduce new adaptive rules for the random selection of their updates. By adaptive, we mean that our selection rules are based on the dual residual or the primal-dual gap estimates and can change at each iteration. We theoretically characterize the performance of our selection rules and demonstrate improvements over the state-of-the-art, and extend our theory and algorithms to general convex objectives. Numerical evidence with hinge-loss su…
National University of Defense Technology（国防科技大学）；Dongguan University of Technology（东莞理工学院）
We provide a new theoretical analysis framework to investigate online gradient descent in the dynamic environment. Comparing with the previous work, the new framework recovers the state-of-the-art dynamic regret, but does not require extra gradient queries for every iteration. Specifically, when functions are $α$ strongly convex and $β$ smooth, to achieve the state-of-the-art dynamic regret, the previous work requires $O(κ)$ with $κ= \fracβα$ queries of gradients at every iteration. But, our framework shows that the query complexity can be improved to be $O(1)$, which does not depend on $κ$. T…
Second-order optimizers hold intriguing potential for deep learning, but suffer from increased cost and sensitivity to the non-convexity of the loss surface as compared to gradient-based approaches. We introduce a coordinate descent method to train deep neural networks for classification tasks that exploits global convexity of the cross-entropy loss in the weights of the linear layer. Our hybrid Newton/Gradient Descent (NGD) method is consistent with the interpretation of hidden layers as providing an adaptive basis and the linear layer as providing an optimal fit of the basis to data. By alte…
INRIA Lille（法国国家信息与自动化研究所里尔中心）；Politecnico di Milano（米兰理工大学）
In this paper, we propose a novel approach to automatically determine the batch size in stochastic gradient descent methods. The choice of the batch size induces a trade-off between the accuracy of the gradient estimate and the cost in terms of samples of each update. We propose to determine the batch size by optimizing the ratio between a lower bound to a linear or quadratic Taylor approximation of the expected improvement and the number of samples used to estimate the gradient. The performance of the proposed approach is empirically compared with related methods on popular classification tas…
Many recent problems in signal processing and machine learning such as compressed sensing, image restoration, matrix/tensor recovery, and non-negative matrix factorization can be cast as constrained optimization. Projected gradient descent is a simple yet efficient method for solving such constrained optimization problems. Local convergence analysis furthers our understanding of its asymptotic behavior near the solution, offering sharper bounds on the convergence rate compared to global convergence analysis. However, local guarantees often appear scattered in problem-specific areas of machine …
Mila - Université de Montréal（米拉 - 蒙特利尔大学）；Facebook AI Research（脸书人工智能研究院）；CIFAR（加拿大高等研究院）
Optimization algorithms that leverage gradient covariance information, such as variants of natural gradient descent (Amari, 1998), offer the prospect of yielding more effective descent directions. For models with many parameters, the covariance matrix they are based on becomes gigantic, making them inapplicable in their original form. This has motivated research into both simple diagonal approximations and more sophisticated factored approximations such as KFAC (Heskes, 2000; Martens & Grosse, 2015; Grosse & Martens, 2016). In the present work we draw inspiration from both to propose a novel a…
Indian Institute of Information Technology, Allahabad（印度阿拉哈巴德信息技术学院）；PathPartner Technology Pvt. Ltd.（PathPartner科技私人有限公司）；Indian Statistical Institute（印度统计研究所）；Techno India University（Techno印度大学）
The convolutional neural networks (CNNs) are generally trained using stochastic gradient descent (SGD) based optimization techniques. The existing SGD optimizers generally suffer with the overshooting of the minimum and oscillation near minimum. In this paper, we propose a new approach, hereafter referred as AdaInject, for the gradient descent optimizers by injecting the second order moment into the first order moment. Specifically, the short-term change in parameter is used as a weight to inject the second order moment in the update rule. The AdaInject optimizer controls the parameter update,…
There is an increased interest in building data analytics frameworks with advanced algebraic capabilities both in industry and academia. Many of these frameworks, e.g., TensorFlow and BIDMach, implement their compute-intensive primitives in two flavors---as multi-thread routines for multi-core CPUs and as highly-parallel kernels executed on GPU. Stochastic gradient descent (SGD) is the most popular optimization method for model training implemented extensively on modern data analytics platforms. While the data-intensive properties of SGD are well-known, there is an intense debate on which of t…
