We study the conditions under which one is able to efficiently apply variance-reduction and acceleration schemes on finite sum optimization problems. First, we show that, perhaps surprisingly, the finite sum structure by itself, is not sufficient for obtaining a complexity bound of $tilde{cO}((n+L/mu)ln(1/epsilon))$ for $L$-smooth and $mu$-strongly convex individual functions - one must also know which individual function is being referred to by the oracle at each iteration. Next, we show that for a broad class of first-order and coordinate-descent finite sum algorithms (including, e.g., SDCA, SVRG, SAG), it is not possible to get an `accelerated complexity bound of $tilde{cO}((n+sqrt{n L/mu})ln(1/epsilon))$, unless the strong convexity parameter is given explicitly. Lastly, we show that when this class of algorithms is used for minimizing $L$-smooth and convex finite sums, the optimal complexity bound is $tilde{cO}(n+L/epsilon)$, assuming that (on average) the same update rule is used in every iteration, and $tilde{cO}(n+sqrt{nL/epsilon})$, otherwise.
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