Parallelization does not Accelerate Convex Optimization: Adaptivity Lower Bounds for Non-smooth Convex Minimization

In this paper we study the limitations of parallelization in convex optimization. A convenient approach to study parallelization is through the prism of \emph{adaptivity} which is an information theoretic measure of the parallel runtime of an algorithm. Informally, adaptivity is the number of sequential rounds an algorithm needs to make when it can execute polynomially-many queries in parallel at every round. For combinatorial optimization with black-box oracle access, the study of adaptivity has recently led to exponential accelerations in parallel runtime and the natural question is whether dramatic accelerations are achievable for convex optimization. Our main result is a spoiler. We show that, in general, parallelization does not accelerate convex optimization. In particular, for the problem of minimizing a non-smooth Lipschitz and strongly convex function with black-box oracle access we give information theoretic lower bounds that indicate that the number of adaptive rounds of any randomized algorithm exactly match the upper bounds of single-query-per-round (i.e. non-parallel) algorithms.