Closing the Computational-Query Depth Gap in Parallel Stochastic Convex Optimization

TL;DR

This paper introduces a parallel stochastic convex optimization algorithm that reduces computational depth and closes the gap with query depth, improving efficiency for small accuracy requirements.

Contribution

It develops a novel parallel method that reduces the computational depth of stochastic convex optimization, matching query depth and closing the existing gap.

Findings

Reduces computational depth by a polynomial factor for small accuracy

Matches the query complexity of prior state-of-the-art methods

Can be optimized with fast matrix multiplication for near-linear work

Abstract

We develop a new parallel algorithm for minimizing Lipschitz, convex functions with a stochastic subgradient oracle. The total number of queries made and the query depth, i.e., the number of parallel rounds of queries, match the prior state-of-the-art, [CJJLLST23], while improving upon the computational depth by a polynomial factor for sufficiently small accuracy. When combined with previous state-of-the-art methods our result closes a gap between the best-known query depth and the best-known computational depth of parallel algorithms. Our method starts with a ball acceleration framework of previous parallel methods, i.e., [CJJJLST20, ACJJS21], which reduce the problem to minimizing a regularized Gaussian convolution of the function constrained to Euclidean balls. By developing and leveraging new stability properties of the Hessian of this induced function, we depart from prior parallel algorithms and reduce these ball-constrained optimization problems to stochastic unconstrained quadratic minimization problems. Although we are unable to prove concentration of the asymmetric matrices that we use to approximate this Hessian, we nevertheless develop an efficient parallel method for solving these quadratics. Interestingly, our algorithms can be improved using fast matrix multiplication and use nearly-linear work if the matrix multiplication exponent is 2.