On Busemann subgradient methods for stochastic minimization in Hadamard spaces

We study the recently introduced Busemann subgradient method due to Goodwin, Lewis, López-Acedo and Nicolae, extending it to minimize the mean of a stochastic function over general Hadamard spaces. We prove a strong convergence theorem under a local compactness assumption and further prove weak convergence of the method over separable Hadamard spaces. The proof is based on a general (weak) convergence theorem for stochastic processes in Hadamard spaces which confine to a stochastic variant of quasi-Fejér monotonicity, together with a nonlinear variant of Pettis' theorem, which are of independent interest. Lastly, we provide best function value estimates as well as a strong convergence result under a strong convexity assumption, and in that latter case in particular derive explicit rates of convergence. We illustrate the practical use of the general mean minimization problem and the associated Busemann subgradient method through statistical models for phylogenetic inference via the Billera-Holmes-Vogtmann tree space.