On Parallel Complexity of Nonsmooth Convex Optimization

Abstract

AbstractWe consider the standard class of problems ƒ(x) → min, x ∈ Bn associated with convex continuous functions ƒ mapping the unit n-dimensional cube Bn into [0, 1]. It is known that the information complexity of the class with respect to the standard first-order oracle is, within an absolute constant factor, n ln (1/ϵ), ϵ < 12 being the required accuracy (measured in terms of ƒ). The question we are interested in is how the complexity can be reduced if one is allowed to use K copies of the oracle in parallel rather than a single oracle. We demonstrate that the "K-oracle complexity" is at least O(1)(n/ln(2Kn))1/3ln(1/ϵ)