[Paper Review] Shuffling Gradient-Based Methods with Momentum.

Trang H. Tran, Lam M. Nguyen+1 more|arXiv (Cornell University)|Nov 24, 2020

Stochastic Gradient Optimization Techniques24 references4 citations

TL;DR

This paper proposes a novel shuffling gradient-based method with momentum for non-convex finite-sum optimization, combining randomized reshuffling and momentum to achieve state-of-the-art convergence rates under L-smoothness and bounded variance. The method outperforms existing approaches, especially with randomized reshuffling, improving convergence by a fraction of the data size.

ABSTRACT

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 to approximate a stationary point of non-convex finite-sum minimization problems. While our method is inspired by momentum techniques, its update is fundamentally different from existing momentum-based methods. We establish that our algorithm achieves a state-of-the-art convergence rate for both constant and diminishing learning rates for any shuffling strategy under standard assumptions (i.e., L-smoothness and bounded variance). In particular, if a randomized reshuffling strategy is used, we can further improve our convergence rate by a fraction of the data size. When the shuffling strategy is fixed, we develop another new algorithm that is similar to existing momentum methods. This algorithm covers the single-shuffling and incremental gradient schemes as special cases. We prove the same convergence rate of 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.

Motivation & Objective

• To address the challenge of accelerating convergence in non-convex finite-sum minimization problems using advanced optimization techniques.
• To bridge the gap between shuffling strategies and momentum-based methods, which have traditionally been treated separately in optimization.
• To develop a unified framework that generalizes existing methods like single-shuffling and incremental gradient schemes.
• To establish theoretical convergence guarantees under standard assumptions, including L-smoothness and bounded variance or gradient.

Proposed method

• The method integrates a shuffling strategy with a momentum-based update rule, fundamentally differentiating it from existing momentum methods.
• It employs a randomized reshuffling strategy to reorder data samples at each iteration, enhancing convergence speed.
• For fixed shuffling, the algorithm uses a similar update mechanism to standard momentum methods but generalizes them to include single-shuffling and incremental gradient schemes.
• The update rule is designed to maintain convergence under L-smoothness and bounded variance, ensuring robustness across different data shuffling patterns.
• Theoretical analysis relies on standard assumptions: L-smoothness of the objective function and bounded variance of gradients.
• Numerical simulations are conducted on standard machine learning datasets to validate performance against existing shuffling-based optimization methods.

Experimental results

Research questions

• RQ1 Can combining shuffling strategies with momentum improve convergence rates in non-convex finite-sum optimization?
• RQ2 How does the choice of shuffling strategy—randomized versus fixed—affect the convergence behavior of momentum-based methods?
• RQ3 What is the theoretical convergence rate of the proposed method under standard assumptions like L-smoothness and bounded variance?
• RQ4 Can the proposed method generalize existing approaches such as single-shuffling and incremental gradient schemes within a unified framework?

Key findings

• The proposed method achieves a state-of-the-art convergence rate for both constant and diminishing learning rates under standard assumptions.
• With a randomized reshuffling strategy, the convergence rate improves by a fraction of the data size compared to existing methods.
• For fixed shuffling, the algorithm maintains the same convergence rate under L-smoothness and bounded gradient assumptions.
• Numerical simulations on standard datasets confirm the proposed algorithms outperform existing shuffling-based methods in practice.
• The method generalizes both single-shuffling and incremental gradient schemes as special cases, demonstrating broader applicability.
• The theoretical analysis confirms convergence to a stationary point for non-convex finite-sum problems, validating its practical utility.