SGD Generalizes Better Than GD (And Regularization Doesn't Help)
This provides a theoretical separation for optimization algorithms in machine learning, highlighting a fundamental advantage of SGD over GD in generalization, which is incremental but clarifies long-standing empirical observations.
The paper shows that stochastic gradient descent (SGD) generalizes better than full-batch gradient descent (GD) in stochastic convex optimization, with SGD achieving O(1/ε²) iterations for ε excess risk while GD may have Ω(1) error and requires Ω(1/ε⁴) iterations to match SGD.
We give a new separation result between the generalization performance of stochastic gradient descent (SGD) and of full-batch gradient descent (GD) in the fundamental stochastic convex optimization model. While for SGD it is well-known that $O(1/ε^2)$ iterations suffice for obtaining a solution with $ε$ excess expected risk, we show that with the same number of steps GD may overfit and emit a solution with $Ω(1)$ generalization error. Moreover, we show that in fact $Ω(1/ε^4)$ iterations are necessary for GD to match the generalization performance of SGD, which is also tight due to recent work by Bassily et al. (2020). We further discuss how regularizing the empirical risk minimized by GD essentially does not change the above result, and revisit the concepts of stability, implicit bias and the role of the learning algorithm in generalization.