  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.
