Strong laws, random monotone vector fields and gradient flows on metric spaces of nonpositive curvature

Abstract

Using a novel effective non-asymptotic concentration inequality, we establish a distribution-uniform generalization of Sturm's strong law of large numbers for inductive means of $L^1$-sequences of i.i.d. random variables on (separable) Hadamard spaces. Building on that, we establish a strong law of large numbers for integrable random monotone vector fields on (separable) Hilbert-Hadamard spaces, that is Hadamard spaces where all tangent cones isometrically embed into Hilbert spaces, extending a previous result of Salim set in Hilbert spaces. We use this latter strong law to establish a probabilistic Lie-Trotter-Kato formula for the gradient flow generated by an integral function over (separable) Hilbert-Hadamard spaces where all tangent cones are actually full Hilbert spaces. This application leverages the previous Lie-Trotter-Kato formula established for gradient flows of sums of convex functions by Stojković together with a result relating resolvent and gradient flow convergence established by Bačák, as well as a new result on the interchangeability of the subdifferential and the integral, which we establish for $L^2$-Lipschitz integrands. The last ingredient provides, to our knowledge, the first nonlinear version of (a particular case of) a previous result due variously to Ioffe and Tikhomirov, Levin, Hiriart-Urruty, Thibault as well as Rockafellar and Wets. Throughout the paper, we highlight various remaining open problems.