We present Nesterov-type acceleration techniques for Alternating Least Squares (ALS) methods applied to canonical tensor decomposition. While Nesterov acceleration turns gradient descent into an optimal first-order method for convex problems by adding a momentum term with a specific weight sequence, a direct application of this method and weight sequence to ALS results in erratic convergence behaviour or divergence. This is so because the tensor decomposition problem is non-convex and ALS is accelerated instead of gradient descent. We investigate how line search or restart mechanisms can be used to obtain effective acceleration. We first consider a cubic line search (LS) strategy for determining the momentum weight, showing numerically that the combined Nesterov-ALS-LS approach is competitive with or superior to other recently developed nonlinear acceleration techniques for ALS, including acceleration by nonlinear conjugate gradients (NCG) and LBFGS. As an alternative, we consider various restarting techniques, some of which are inspired by previously proposed restarting mechanisms for Nesterov's accelerated gradient method. We study how two key parameters, the momentum weight and the restart condition, should be set. Our extensive empirical results show that the Nesterov-accelerated ALS methods with restart can be dramatically more efficient than the stand-alone ALS or Nesterov accelerated gradient method, when problems are ill-conditioned or accurate solutions are required. The resulting methods perform competitively with or superior to existing acceleration methods for ALS, and additionally enjoy the benefit of being much simpler and easier to implement. On a large and ill-conditioned 71 x 1000 x 900 tensor consisting of readings from chemical sensors used for tracking hazardous gases, the restarted Nesterov-ALS method outperforms any of the existing methods by a large factor.
Nesterov's accelerated gradient method is known to exhibit oscillatory behavior on convex problems. An interesting discussion on this is provided in @cite_5 which formulates an ODE as the continuous time analogue of Nesterov's method. Such oscillatory behavior happens when the method approaches convergence, and can be alleviated by restarting the algorithm using the current iterate as the initial solution, usually resetting the sequence of momentum weights to its initial state close to 0. In @cite_5 an explanation is provided of why resetting the momentum weight to a small value is effective using the ODE formulation of Nesterov's accelerated gradient descent. In @cite_17 the use of adaptive restarting was explored for convex problems, and @cite_9 explored the use of adaptive restarting and adaptive momentum weight for nonlinear systems of equations resulting from finite element approximation of PDEs. Our work is the first study of a general Nesterov-accelerated ALS scheme.
@cite_5 @cite_9 @cite_17
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We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show that the continuous time ODE allows for a better understanding of Nesterov's scheme. As a byproduct, we obtain a family of schemes with similar convergence rates. The ODE interpretation also suggests restarting Nesterov's scheme leading to an algorithm, which can be rigorously proven to converge at a linear rate whenever the objective is strongly convex. Nesterov's momentum trick is famously known for accelerating gradient descent, and has been proven useful in building fast iterative algorithms. However, in the stochastic setting, counterexamples exist and prevent Nesterov's momentum from providing similar acceleration, even if the underlying problem is convex. We introduce Katyusha, a direct, primal-only stochastic gradient method to fix this issue. It has a provably accelerated convergence rate in convex (off-line) stochastic optimization. The main ingredient is Katyusha momentum, a novel "negative momentum" on top of Nesterov's momentum that can be incorporated into a variance-reduction based algorithm and speed it up. Since variance reduction has been successfully applied to a growing list of practical problems, our paper suggests that in each of such cases, one could potentially give Katyusha a hug. Nesterov's momentum trick is famously known for accelerating gradient descent, and has been proven useful in building fast iterative algorithms. However, in the stochastic setting, counterexamples exist and prevent Nesterov's momentum from providing similar acceleration, even if the underlying problem is convex. We introduce @math , a direct, primal-only stochastic gradient method to fix this issue. It has a provably accelerated convergence rate in convex (off-line) stochastic optimization. The main ingredient is @math , a novel "negative momentum" on top of Nesterov's momentum. It can be incorporated into a variance-reduction based algorithm and speed it up, both in terms of @math performance. Since variance reduction has been successfully applied to a growing list of practical problems, our paper suggests that in each of such cases, one could potentially try to give Katyusha a hug. Recently, stochastic momentum methods have been widely adopted in training deep neural networks. However, their convergence analysis is still underexplored at the moment, in particular for non-convex optimization. This paper fills the gap between practice and theory by developing a basic convergence analysis of two stochastic momentum methods, namely stochastic heavy-ball method and the stochastic variant of Nesterov's accelerated gradient method. We hope that the basic convergence results developed in this paper can serve the reference to the convergence of stochastic momentum methods and also serve the baselines for comparison in future development of stochastic momentum methods. The novelty of convergence analysis presented in this paper is a unified framework, revealing more insights about the similarities and differences between different stochastic momentum methods and stochastic gradient method. The unified framework exhibits a continuous change from the gradient method to Nesterov's accelerated gradient method and finally the heavy-ball method incurred by a free parameter, which can help explain a similar change observed in the testing error convergence behavior for deep learning. Furthermore, our empirical results for optimizing deep neural networks demonstrate that the stochastic variant of Nesterov's accelerated gradient method achieves a good tradeoff (between speed of convergence in training error and robustness of convergence in testing error) among the three stochastic methods.