Stochastic gradient descent with initial regularization

Abstract:We analyze a variant of stochastic gradient descent with initial regularization (SGDIR) and derive dimension-free upper bounds on its expected excess risk for the squared loss. In the noiseless case, we obtain new bounds for both averaged and non-averaged SGDIR under moment, source, and capacity assumptions. For a particular value of the source parameter, these bounds are of order $m^{-2}\log^{2}m$, where the number of training samples is of order $m$. For another value of the source parameter, we obtain, for any $\epsilon>0$, bounds of order $m^{-3+\epsilon}$, provided that the capacity parameter exceeds $\epsilon^{-1}$. We also establish a lower bound that matches our upper bounds in certain regimes up to a polylogarithmic factor. In the noisy case, we provide an instance-based comparison between SGDIR and ridge regression. Under general assumptions and a mild lower bound on the regularization parameter, we show that the expected excess risk of SGDIR is no larger than that of ridge regression, up to a polylogarithmic factor. Numerical experiments on synthetic and real data are consistent with our theoretical findings.

Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)

arXiv:2608.22953 [cs.LG]

(or arXiv:2608.22953v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.22953 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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