Parallel distributed block coordinate descent methods based on pairwise comparison oracle

This paper provides a block coordinate descent algorithm to solve unconstrained optimization problems. Our algorithm uses only pairwise comparison of function values, which tells us only the order of function values over two points, and does not require computation of a function value itself or a gradient. Our algorithm iterates two steps: the direction estimate step and the search step. In the direction estimate step, a Newton-type search direction is estimated through a block coordinate descent-based computation method with the pairwise comparison. In the search step, a numerical solution is updated along the estimated direction. The computation in the direction estimate step can be easily parallelized, and thus, the algorithm works efficiently to find the minimizer of the objective function. Also, we theoretically derive an upper bound of the convergence rate for our algorithm and show that our algorithm achieves the optimal query complexity for specific cases. In numerical experiments, we show that our method efficiently finds the optimal solution compared to some existing methods based on the pairwise comparison.