New spherical codes by tuning an annealing solver

Abstract

The spherical code problem asks how to place $N$ points on a unit sphere in $D$ dimensions to maximize the smallest distance between two points. It is a non-smooth and non-convex max-min problem, where few problem sizes have known optimal solutions, some have constructions, but most still only have numerically obtained solutions. We use a simple solver, with seven parameters, that anneals a LogSumExp surrogate, minimizes with L-BFGS, applies a random perturbation between anneals, and repeats with no acceptance test until a fixed limit is reached. We search the solver parameters for every instance with $N \le 1024$, $3 \le D \le 32$, and $N > 2D$, a total of 29,670 instances, using 32 parameter trials. We then run the solver for 60 seconds with the best parameters for each instance, without utilizing any symmetries or existing solutions. After enforcing monotonicity of the results and numerically polishing the solutions, we improve the best known code in 4,577 instances. When a single set of solver parameters is used for all instances, with one 60-second episode, the number of improvements is still 3,649. From our records, we find 24 codes with an exact algebraic cosine. Among our ablations, we find that 1) the repeated anneal contributes most to improvements, 2) at our $32 \times 60$-second budget only fitting one parameter per-instance is better than fitting all seven, and 3) good parameters can be predicted from the instance size. We provide the solver code, the fitted parameters, and the codes.