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SGD without Replacement: Sharper Rates for General Smooth Convex Functions
Dheeraj Nagaraj · Prateek Jain · Praneeth Netrapalli

Wed Jun 12 04:20 PM -- 04:25 PM (PDT) @ Room 103
We study stochastic gradient descent {\em without replacement} (\sgdwor) for smooth convex functions. \sgdwor is widely observed to converge faster than true \sgd where each sample is drawn independently {\em with replacement}~\cite{bottou2009curiously} and hence, is more popular in practice. But it's convergence properties are not well understood as sampling without replacement leads to coupling between iterates and gradients. By using method of exchangeable pairs to bound Wasserstein distance, we provide the first non-asymptotic results for \sgdwor when applied to {\em general smooth, strongly-convex} functions. In particular, we show that \sgdwor converges at a rate of $O(1/K^2)$ while \sgd~is known to converge at $O(1/K)$ rate, where $K$ denotes the number of passes over data and is required to be {\em large enough}. Existing results for \sgdwor in this setting require additional {\em Hessian Lipschitz assumption}~\cite{gurbuzbalaban2015random,haochen2018random}. For {\em small} $K$, we show \sgdwor can achieve same convergence rate as \sgd for {\em general smooth strongly-convex} functions. Existing results in this setting require $K=1$ and hold only for generalized linear models \cite{shamir2016without}. In addition, by careful analysis of the coupling, for both large and small $K$, we obtain better dependence on problem dependent parameters like condition number.

Author Information

Dheeraj Nagaraj (Massachusetts Institute of Technology)
Prateek Jain (Microsoft Research)
Praneeth Netrapalli (Microsoft Research)

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