Jinlan Liu

Dongpo Xu, Jinlan Liu, Yinghua Lu et al.

The stochastic momentum method is a commonly used acceleration technique for solving large-scale stochastic optimization problems in artificial neural networks. Current convergence results of stochastic momentum methods under non-convex stochastic settings mostly discuss convergence in terms of the random output and minimum output. To this end, we address the convergence of the last iterate output (called last-iterate convergence) of the stochastic momentum methods for non-convex stochastic optimization problems, in a way conformal with traditional optimization theory. We prove the last-iterate convergence of the stochastic momentum methods under a unified framework, covering both stochastic heavy ball momentum and stochastic Nesterov accelerated gradient momentum. The momentum factors can be fixed to be constant, rather than time-varying coefficients in existing analyses. Finally, the last-iterate convergence of the stochastic momentum methods is verified on the benchmark MNIST and CIFAR-10 datasets.

Meixuan He, Yuqing Liang, Jinlan Liu et al.

Adam is a commonly used stochastic optimization algorithm in machine learning. However, its convergence is still not fully understood, especially in the non-convex setting. This paper focuses on exploring hyperparameter settings for the convergence of vanilla Adam and tackling the challenges of non-ergodic convergence related to practical application. The primary contributions are summarized as follows: firstly, we introduce precise definitions of ergodic and non-ergodic convergence, which cover nearly all forms of convergence for stochastic optimization algorithms. Meanwhile, we emphasize the superiority of non-ergodic convergence over ergodic convergence. Secondly, we establish a weaker sufficient condition for the ergodic convergence guarantee of Adam, allowing a more relaxed choice of hyperparameters. On this basis, we achieve the almost sure ergodic convergence rate of Adam, which is arbitrarily close to $o(1/\sqrt{K})$. More importantly, we prove, for the first time, that the last iterate of Adam converges to a stationary point for non-convex objectives. Finally, we obtain the non-ergodic convergence rate of $O(1/K)$ for function values under the Polyak-Lojasiewicz (PL) condition. These findings build a solid theoretical foundation for Adam to solve non-convex stochastic optimization problems.

Kun Zeng, Jinlan Liu, Zhixia Jiang et al.

The plain stochastic gradient descent and momentum stochastic gradient descent have extremely wide applications in deep learning due to their simple settings and low computational complexity. The momentum stochastic gradient descent uses the accumulated gradient as the updated direction of the current parameters, which has a faster training speed. Because the direction of the plain stochastic gradient descent has not been corrected by the accumulated gradient. For the parameters that currently need to be updated, it is the optimal direction, and its update is more accurate. We combine the advantages of the momentum stochastic gradient descent with fast training speed and the plain stochastic gradient descent with high accuracy, and propose a scaling transition from momentum stochastic gradient descent to plain stochastic gradient descent(TSGD) method. At the same time, a learning rate that decreases linearly with the iterations is used instead of a constant learning rate. The TSGD algorithm has a larger step size in the early stage to speed up the training, and training with a smaller step size in the later stage can steadily converge. Our experimental results show that the TSGD algorithm has faster training speed, higher accuracy and better stability. Our implementation is available at: https://github.com/kunzeng/TSGD.

Kun Zeng, Jinlan Liu, Zhixia Jiang et al.

Adaptive gradient algorithm (AdaGrad) and its variants, such as RMSProp, Adam, AMSGrad, etc, have been widely used in deep learning. Although these algorithms are faster in the early phase of training, their generalization performance is often not as good as stochastic gradient descent (SGD). Hence, a trade-off method of transforming Adam to SGD after a certain iteration to gain the merits of both algorithms is theoretically and practically significant. To that end, we propose a decreasing scaling transition scheme to achieve a smooth and stable transition from Adam to SGD, which is called DSTAdam. The convergence of the proposed DSTAdam is also proved in an online convex setting. Finally, the effectiveness of the DSTAdam is verified on the CIFAR-10/100 datasets. Our implementation is available at: https://github.com/kunzeng/DSTAdam.

Yiming Jiang, Jinlan Liu, Dongpo Xu et al.

Adam-type algorithms have become a preferred choice for optimisation in the deep learning setting, however, despite success, their convergence is still not well understood. To this end, we introduce a unified framework for Adam-type algorithms (called UAdam). This is equipped with a general form of the second-order moment, which makes it possible to include Adam and its variants as special cases, such as NAdam, AMSGrad, AdaBound, AdaFom, and Adan. This is supported by a rigorous convergence analysis of UAdam in the non-convex stochastic setting, showing that UAdam converges to the neighborhood of stationary points with the rate of $\mathcal{O}(1/T)$. Furthermore, the size of neighborhood decreases as $β$ increases. Importantly, our analysis only requires the first-order momentum factor to be close enough to 1, without any restrictions on the second-order momentum factor. Theoretical results also show that vanilla Adam can converge by selecting appropriate hyperparameters, which provides a theoretical guarantee for the analysis, applications, and further developments of the whole class of Adam-type algorithms.