How Good is SGD with Random Shuffling?
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We study the performance of stochastic gradient descent (SGD) on smooth and strongly-convex finite-sum optimization problems. In contrast to the majority of existing theoretical works, which assume that individual functions are sampled with replacement, we focus here on popular but poorly-understood heuristics, which involve going over random permutations of the individual functions. This setting has been investigated in several recent works, but the optimal error rates remains unclear. In this paper, we provide lower bounds on the expected optimization error with these heuristics (using SGD with any constant step size), which elucidate their advantages and disadvantages. In particular, we prove that after k passes over n individual functions, if the functions are re-shuffled after every pass, the best possible optimization error for SGD is at least Ω(1/(nk)^2+1/nk^3), which partially corresponds to recently derived upper bounds, and we conjecture to be tight. Moreover, if the functions are only shuffled once, then the lower bound increases to Ω(1/nk^2). Since there are strictly smaller upper bounds for random reshuffling, this proves an inherent performance gap between SGD with single shuffling and repeated shuffling. As a more minor contribution, we also provide a non-asymptotic Ω(1/k^2) lower bound (independent of n) for cyclic gradient descent, where no random shuffling takes place.
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