The Proof Mining Seminar is a joint LOS/IMAR/ILDS seminar, featuring talks on recent results in proof mining.
Proof mining is a paradigm of research, concerned with the extraction of hidden finitary and combinatorial content from proofs that make use of highly infinitary principles. The new information is obtained after a logical analysis, using proof-theoretic tools, and can be both of a quantitative nature, such as algorithms and effective bounds, as well as of a qualitative nature, such as uniformities in the bounds or weakenings of the premises.
All seminars, except where otherwise indicated, will be on Wednesdays between 16:00 and 18:00, Bucharest time. The seminars are held remotely.
To receive announcements about the seminar, please send an email to proof-mining-seminar@ilds.ro.
Organizers: Ulrich Kohlenbach, Laurențiu Leuștean, Andrei Sipoș
Wednesday, April 8, 2026
Nicholas Pischke (University of Bath)
Rates of convergence for a stochastic proximal point algorithm in metric spaces of nonpositive curvature
Abstract:
I introduce a stochastic variant of the proximal point algorithm in the general setting of nonlinear (separable) Hadamard spaces for approximating zeros of the mean of a stochastically perturbed monotone vector field and prove its convergence under a suitable strong monotonicity assumption, together with a probabilistic independence assumption and a separability assumption on the tangent spaces. The convergence result is moreover furnished with an explicit rate of convergence for the iteration towards the (unique) solution both in mean and almost surely. While the rates themselves are, in that generality, already novel in the context of Hilbert spaces, the qualitative result itself is novel at least over Hadamard manifolds. At the end, I discuss how the quantitative result arises as a special case of a recent general quantitative study of rates of convergence for stochastic processes over metric spaces satisfying a stochastic variant of quasi-Fejer monotonicity under uniqueness assumptions, and I outline how such results can be extended to more general metric regularity assumptions.
The talk is partially based on joint work with Morenikeji Neri and Thomas Powell.