Closing the convergence gap of SGD without replacement
This work addresses a theoretical gap in optimization algorithms for machine learning practitioners, providing tighter convergence guarantees for a widely used but less analyzed variant of SGD.
The paper tackles the convergence gap of stochastic gradient descent without replacement (SGDo) by establishing an optimal convergence rate of O(1/T^2 + n^2/T^3) for quadratic functions and a new lower bound of Ω(n/T^2) for strongly convex sums of smooth functions, matching or improving prior bounds.
Stochastic gradient descent without replacement sampling is widely used in practice for model training. However, the vast majority of SGD analyses assumes data is sampled with replacement, and when the function minimized is strongly convex, an $\mathcal{O}\left(\frac{1}{T}\right)$ rate can be established when SGD is run for $T$ iterations. A recent line of breakthrough works on SGD without replacement (SGDo) established an $\mathcal{O}\left(\frac{n}{T^2}\right)$ convergence rate when the function minimized is strongly convex and is a sum of $n$ smooth functions, and an $\mathcal{O}\left(\frac{1}{T^2}+\frac{n^3}{T^3}\right)$ rate for sums of quadratics. On the other hand, the tightest known lower bound postulates an $Ω\left(\frac{1}{T^2}+\frac{n^2}{T^3}\right)$ rate, leaving open the possibility of better SGDo convergence rates in the general case. In this paper, we close this gap and show that SGD without replacement achieves a rate of $\mathcal{O}\left(\frac{1}{T^2}+\frac{n^2}{T^3}\right)$ when the sum of the functions is a quadratic, and offer a new lower bound of $Ω\left(\frac{n}{T^2}\right)$ for strongly convex functions that are sums of smooth functions.