On Tight Convergence Rates of Without-replacement SGD

For solving finite-sum optimization problems, SGD without replacement sampling is empirically shown to outperform SGD. Denoting by $n$ the number of components in the cost and $K$ the number of epochs of the algorithm , several recent works have shown convergence rates of without-replacement SGD that have better dependency on $n$ and $K$ than the baseline rate of $O(1/(nK))$ for SGD. However, there are two main limitations shared among those works: the rates have extra poly-logarithmic factors on $nK$, and denoting by $\kappa$ the condition number of the problem, the rates hold after $\kappa^c\log(nK)$ epochs for some $c>0$. In this work, we overcome these limitations by analyzing step sizes that vary across epochs.