This work addresses the problem of variance in stochastic gradient estimation for machine learning optimization. Deep learning relies on mini-batch methods such as stochastic gradient descent, which approximate full gradients but introduce noise, creating trade-offs between convergence stability, speed, and generalization. Existing methods, including variance reduction techniques (e.g., SVRG and SAG) and adaptive optimizers, aim to mitigate gradient noise but may introduce additional computational overhead. We propose a model-assisted sampling framework that interprets mini-batch gradients through survey sampling theory, treating the dataset as a fixed finite population and gradients as sample-based estimates. Our aim is to bridge machine learning optimization and survey sampling theory by combining their perspectives on sample-based estimation and variance reduction. By incorporating auxiliary gradient-prediction models, we construct more efficient gradient estimators, with uniform sampling arising as a special case when no auxiliary information is used. Our approach integrates easily with existing optimizers, improving efficiency without altering their dynamics. Empirical results on synthetic and six benchmark datasets show performance gains in 71-86% of the experiments, particularly for medium-sized input spaces in our benchmarks. Notably, with momentum-based optimizers such as AdamW, the proposed estimator achieves clearly better generalization in roughly half the training epochs compared to baseline estimator.
Generalization remains a pivotal challenge in deep learning, where traditional optimizers like Stochastic Gradient Descent (SGD) often converge to sharp minima, leading to overfitting and reduced performance on unseen data. Building on Sharpness-Aware Minimization (SAM), for seeking flat minima associated with improved generalization, we propose the Extragradient-Inspired Sharpness-Aware Minimization (EISAM), a novel optimizer that enhances generalization via the extragradient technique. EISAM uses a two-step update process: a prediction step investigating the geometry of the loss landscape and a perturbation step that refines updates with a base optimizer. This approach achieves better generalization performance than SAM. Crucially, EISAM reduces sensitivity to the perturbation radius, enhancing robustness, and simplifying the tuning across diverse settings. Extensive experiments on benchmark datasets demonstrate that EISAM consistently outperforms SGD, Adaptive Moment Estimation (Adam), and SAM in test accuracy and training efficiency across various architectures. Theoretical analysis further confirms that EISAM tightens the generalization bound by steering parameters toward flatter minima with reduced curvature. Accompanied by a thorough hyperparameter analysis, EISAM offers practical tuning guidance, establishing it as a robust, scalable, and broadly applicable optimization solution that advances both the theory and practice in deep learning.
Variance reduction (VR) methods employ stochastic gradients with decreasing variance, and they have been widely applied to solve large-scale optimization problems in machine learning because of their efficiency. Existing theoretical studies of VR methods are mainly focused on the convergence analysis, leaving the generalization behavior largely unexplored. In this paper, we bridge this gap by developing the first non-vacuous generalization analysis of the representative VR method: Stochastic Variance Reduced Gradient (SVRG), through the lens of algorithmic stability. In particular, we establish sharp stability bounds of the SVRG in both convex and strongly convex settings by exploiting its algorithmic structure. The obtained bounds are data-dependent, because the training errors are incorporated along the trajectory. Our analysis clarifies the interplay between optimization and generalization, leading to optimal excess population risk bounds in both settings. Our approach differs substantially from existing analyses of stochastic algorithms in the sense that we decompose the SVRG update as an SGD-like step plus a zero-mean correction term and then introduce novel Lyapunov functions to absorb the additional gradient terms induced by the reference points. Our analytical framework can be generalized to other VR methods, and we demonstrate the generalization by the well-known Stochastic Average Gradient Accelerated (SAGA) method.
Accelerating stochastic gradient methods with classical momentum schemes, such as Polyak's heavy ball, has proven highly successful in training large-scale machine learning models, particularly when combined with the hardware acceleration of large mini-batch computations. Yet, the effect of classical momentum on stochastic mini-batch optimization has been poorly understood theoretically, with prior works requiring strong noise assumptions and extremely large mini-batches. In this work, we develop a general theory of stochastic momentum acceleration for optimizing over quadratics in the interpolation regime, a popular abstraction for studying deep learning dynamics which also includes classical methods such as randomized Kaczmarz and coordinate descent. Our framework encompasses both heavy ball and Nesterov-style momentum, allows for arbitrary mini-batch sizes, and makes minimal assumptions on the stochastic noise. In particular, we show that acceleration from classical momentum is directly proportional to the gradient mini-batch size (up to a natural saturation point), thereby enabling perfect parallelization of mini-batch computations. Our theory also provides a simple choice for the momentum parameter, which is shown to be effective empirically.