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Recently, there has been much interest in studying the convergence rates ofwithout-replacement SGD, and proving that it is faster than with-replacementSGD in the worst case. However, known lower bounds ignore the problem'sgeometry, including its condition number, whereas the upper bounds explicitlydepend on it. Perhaps surprisingly, we prove that when the condition number istaken into account, without-replacement SGD \emph{does not} significantlyimprove on with-replacement SGD in terms of worst-case bounds, unless thenumber of epochs (passes over the data) is larger than the condition number.Since many problems in machine learning and other areas are bothill-conditioned and involve large datasets, this indicates thatwithout-replacement does not necessarily improve over with-replacement samplingfor realistic iteration budgets. We show this by providing new lower and upperbounds which are tight (up to log factors), for quadratic problems withcommuting quadratic terms, precisely quantifying the dependence on the problemparameters.

Random Shuffling Beats SGD Only After Many Epochs on Ill-Conditioned Problems