Limitations on Variance-Reduction and Acceleration Schemes for Finite Sum Optimization
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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 ((n+L/μ)(1/ϵ)) for L-smooth and μ-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 ((n+√(n L/μ))(1/ϵ)), 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 (n+L/ϵ), assuming that (on average) the same update rule is used in every iteration, and (n+√(nL/ϵ)), otherwise.
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