Yazhen Wang

Yuchen Zhou, Anru R. Zhang, Lili Zheng et al.

This paper studies a general framework for high-order tensor SVD. We propose a new computationally efficient algorithm, tensor-train orthogonal iteration (TTOI), that aims to estimate the low tensor-train rank structure from the noisy high-order tensor observation. The proposed TTOI consists of initialization via TT-SVD (Oseledets, 2011) and new iterative backward/forward updates. We develop the general upper bound on estimation error for TTOI with the support of several new representation lemmas on tensor matricizations. By developing a matching information-theoretic lower bound, we also prove that TTOI achieves the minimax optimality under the spiked tensor model. The merits of the proposed TTOI are illustrated through applications to estimation and dimension reduction of high-order Markov processes, numerical studies, and a real data example on New York City taxi travel records. The software of the proposed algorithm is available online$^6$.

Victor Luo, Yazhen Wang

Stochastic gradient descent (SGD) is often applied to train Deep Neural Networks (DNNs), and research efforts have been devoted to investigate the convergent dynamics of SGD and minima found by SGD. The influencing factors identified in the literature include learning rate, batch size, Hessian, and gradient covariance, and stochastic differential equations are used to model SGD and establish the relationships among these factors for characterizing minima found by SGD. It has been found that the ratio of batch size to learning rate is a main factor in highlighting the underlying SGD dynamics; however, the influence of other important factors such as the Hessian and gradient covariance is not entirely agreed upon. This paper describes the factors and relationships in the recent literature and presents numerical findings on the relationships. In particular, it confirms the four-factor and general relationship results obtained in Wang (2019), while the three-factor and associated relationship results found in Jastrzȩbski et al. (2018) may not hold beyond the considered special case.

Yazhen Wang

This paper investigates asymptotic behaviors of gradient descent algorithms (particularly accelerated gradient descent and stochastic gradient descent) in the context of stochastic optimization arising in statistics and machine learning where objective functions are estimated from available data. We show that these algorithms can be computationally modeled by continuous-time ordinary or stochastic differential equations. We establish gradient flow central limit theorems to describe the limiting dynamic behaviors of these computational algorithms and the large-sample performances of the related statistical procedures, as the number of algorithm iterations and data size both go to infinity, where the gradient flow central limit theorems are governed by some linear ordinary or stochastic differential equations like time-dependent Ornstein-Uhlenbeck processes. We illustrate that our study can provide a novel unified framework for a joint computational and statistical asymptotic analysis, where the computational asymptotic analysis studies dynamic behaviors of these algorithms with the time (or the number of iterations in the algorithms), the statistical asymptotic analysis investigates large sample behaviors of the statistical procedures (like estimators and classifiers) that the algorithms are applied to compute, and in fact the statistical procedures are equal to the limits of the random sequences generated from these iterative algorithms as the number of iterations goes to infinity. The joint analysis results based on the obtained gradient flow central limit theorems can identify four factors - learning rate, batch size, gradient covariance, and Hessian - to derive new theory regarding the local minima found by stochastic gradient descent for solving non-convex optimization problems.