Randomized Parallel Complexity of Weakly Convex Optimization

Kang Liu

We study the minimum number of query rounds needed to find approximate stationary points of nonsmooth weakly convex functions, measured by the gradient of the Moreau envelope. Each round permits multiple simultaneous queries. In the stated parameter regime, the standard subgradient rate is optimal in sufficiently high dimensions, even for randomized algorithms making polynomially many parallel queries per round. We then show how the order structure of the real line enables faster search: a deterministic algorithm using only function values achieves logarithmic round complexity, with a further reduction through parallel queries. Matching lower bounds hold even when each query returns the entire subdifferential. Together, these results clarify the role of dimension in determining the benefits of parallel queries for weakly convex optimization.