Trang H. Tran

Trang H. Tran, Quoc Tran-Dinh, Lam M. Nguyen

The Stochastic Gradient Descent method (SGD) and its stochastic variants have become methods of choice for solving finite-sum optimization problems arising from machine learning and data science thanks to their ability to handle large-scale applications and big datasets. In the last decades, researchers have made substantial effort to study the theoretical performance of SGD and its shuffling variants. However, only limited work has investigated its shuffling momentum variants, including shuffling heavy-ball momentum schemes for non-convex problems and Nesterov's momentum for convex settings. In this work, we extend the analysis of the shuffling momentum gradient method developed in [Tran et al (2021)] to both finite-sum convex and strongly convex optimization problems. We provide the first analysis of shuffling momentum-based methods for the strongly convex setting, attaining a convergence rate of $O(1/nT^2)$, where $n$ is the number of samples and $T$ is the number of training epochs. Our analysis is a state-of-the-art, matching the best rates of existing shuffling stochastic gradient algorithms in the literature.

Duc Toan Nguyen, Trang H. Tran, Lam M. Nguyen

In this paper, we propose Adjusted Shuffling SARAH, a novel algorithm that integrates shuffling techniques with the well-known variance-reduced algorithm SARAH while dynamically adjusting the stochastic gradient weights in each update to enhance exploration. Our method achieves the best-known gradient complexity for shuffling variance reduction methods in a strongly convex setting. This result applies to any shuffling technique, which narrows the gap in the complexity analysis of variance reduction methods between uniform sampling and shuffling data. Furthermore, we introduce Inexact Adjusted Reshuffling SARAH, an inexact variant of Adjusted Shuffling SARAH that eliminates the need for full-batch gradient computations. This algorithm retains the same linear convergence rate as Adjusted Shuffling SARAH while showing an advantage in total complexity when the sample size is very large.

Trang H. Tran, Lam M. Nguyen, Quoc Tran-Dinh

We combine two advanced ideas widely used in optimization for machine learning: shuffling strategy and momentum technique to develop a novel shuffling gradient-based method with momentum, coined Shuffling Momentum Gradient (SMG), for non-convex finite-sum optimization problems. While our method is inspired by momentum techniques, its update is fundamentally different from existing momentum-based methods. We establish state-of-the-art convergence rates of SMG for any shuffling strategy using either constant or diminishing learning rate under standard assumptions (i.e.$L$-smoothness and bounded variance). When the shuffling strategy is fixed, we develop another new algorithm that is similar to existing momentum methods, and prove the same convergence rates for this algorithm under the $L$-smoothness and bounded gradient assumptions. We demonstrate our algorithms via numerical simulations on standard datasets and compare them with existing shuffling methods. Our tests have shown encouraging performance of the new algorithms.