Factor Augmented High-Dimensional SGD

Stochastic gradient descent (SGD) has been a cornerstone of machine learning since the pioneering work of Robbins & Monro (1951). Beyond its algorithmic simplicity and scalability, SGD has also become a central object of theoretical study, with refined analyses linking its dynamics to implicit regularization, generalization performance, and algorithmic stability. For decades, theoretical analyses of SGD have largely resided within the realm of classical stochastic approximation (Polyak & Juditsky, 1992; Lai, 2003; Bottou et al., 2018), where the data dimension is considered fixed while the sample size tends to infinity. While this regime has yielded foundational insights, it no longer fully reflects the characteristics of modern learning systems. Contemporary applications often operate in regimes where data dimension, sample size, and model complexity grow together, calling for new theoretical tools and perspectives that go beyond traditional asymptotic analyses. In this study, we focus on the learning tasks involving high-dimensional predictors. When SGD is applied directly to such data, the dimensionality of the feature space propagates into the optimization process, resulting in a highdimensional (HD) parameter space. Algorithmically, one trending strategy is to approximate the gradient updates using a low-rank representation to reduce memory costs and accelerate computation (Wang et al., 2018; Vogels et al., 2019; Kozak et al., 2019; Kasiviswanathan, 2021; Zhao et al., 2024). Theoretically, despite the vast literature on SGD, convergence guarantees of HD-SGD remain limited (Garrigos & Gower, 2023; Li et al., 2025).