Limitations on Variance-Reduction and Acceleration Schemes for Finite Sum Optimization

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 (O) over tilde((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 (O) over tilde((n + root nL/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 iteration complexity is bounded from below by Omega(n + L/epsilon), assuming that (on average) the same update rule is used in any iteration, and Omega(n + root nL/epsilon) otherwise.