  We study to what extent may stochastic gradient descent (SGD) be understood
as a "conventional" learning rule that achieves generalization performance by
obtaining a good fit to training data. We consider the fundamental stochastic
convex optimization framework, where (one pass, without-replacement) SGD is
classically known to minimize the population risk at rate $O(1/\sqrt n)$, and
prove that, surprisingly, there exist problem instances where the SGD solution
exhibits both empirical risk and generalization gap of $\Omega(1)$.
Consequently, it turns out that SGD is not algorithmically stable in any sense,
and its generalization ability cannot be explained by uniform convergence or
any other currently known generalization bound technique for that matter (other
than that of its classical analysis). We then continue to analyze the closely
related with-replacement SGD, for which we show that an analogous phenomenon
does not occur and prove that its population risk does in fact converge at the
optimal rate. Finally, we interpret our main results in the context of
without-replacement SGD for finite-sum convex optimization problems, and derive
upper and lower bounds for the multi-epoch regime that significantly improve
upon previously known results.

Machine Learning;Optimization and Control;Machine Learning