Below you can see a list of past talks at the Proof Mining Seminar in this season. For forthcoming talks, see the main page.
Wednesday, July 31, 2024
Morenikeji Neri (University of Bath)
Proof mining and probability
Abstract:
Due to previous ad hoc case studies, the insight was made that a significant proportion of results in probability theory (and finite measure theory, in general) made use of infinite unions and sigma additivity in a very sparing way, allowing for a formalisation amenable to bound extraction, in the style of the classical proof mining metatheorems. This talk will discuss aspects of recently developed formal systems stemming from this perspective. While these systems entail many subtle details and features, we shall focus on those that allow us to discuss their key achievements. This includes a novel extension of Bezem’s majorizability that explains the uniformities in the extracted bounds of previous case studies and the formalisation of a strategy that transfers quantitative deterministic results to their corresponding probabilistic analogues. This is joint work with Nicholas Pischke.
Wednesday, March 20, 2024
Nicholas Pischke (TU Darmstadt)
Duality, Fréchet Differentiability and Bregman Distances in Hyperbolic Space
Abstract:
In the context of general hyperbolic metric spaces, I discuss a new notion of a dual system (akin to the influential notion from the context of topological vector spaces) that allows for a uniform study of different notions of duality for these nonlinear spaces. Over this abstract notion of duality, we lift various notions from convex analysis into this nonlinear setting, including Fréchet differentiability and Bregman distances. Further, we introduce a notion of a monotone operator relative to a given dual system and, using the new Fréchet derivatives, we study corresponding resolvents relative to a given gradient, generalizing the seminal notion of Eckstein from the linear setting. These resolvents are then related to corresponding notions of Bregman quasi-nonexpansive mappings which are introduced relative to this generalization of the classical Bregman distance and we prove a convergence result of an analogue of the proximal point algorithm (together with quantitative results on its convergence in form of a rate of metastability) using the abstract convergence results for generalized Fejér monotone sequences in metric spaces introduced in the preceding talk.
Wednesday, March 13, 2024
Nicholas Pischke (TU Darmstadt)
Generalized Fejér Monotone Sequences
Abstract:
I discuss recent results which provide quantitative and abstract strong convergence results for sequences from a compact metric space satisfying a certain form of generalized Fejér monotonicity where (1) the metric can be replaced by a much more general type of function measuring distances (including, in particular, certain Bregman distances), (2) these distance functions are allowed to vary along the iteration and (3) full Fejér monotonicity is relaxed to a certain partial variant. These novel convergence results are established using a preceding finitary and quantitative theorem which constructs a rate of metastability for the Cauchy property of such sequences. In the context of quantitative information, I also discuss the construction of rates of convergence for such sequences in the context of an additional metric regularity assumption as introduced by Kohlenbach, López-Acedo and Nicolae. At the end of the talk, I will shortly mention two methods from the literature that can be quantitatively analyzed using these results but a major application of the general theorems presented here will be given in a second talk in this seminar.
Wednesday, March 6, 2024
Pedro Pinto (TU Darmstadt)
On generalizations to Nonlinear Smooth Spaces
Abstract:
We shall discuss the notion of a nonlinear smooth space recently introduced in [3], which generalizes both CAT(0) spaces as well as smooth Banach spaces. Leaning on the proof-theoretical study carried out in [2], we established in [3] a nonlinear generalization of Reich’s theorem. This in turn allowed us to prove the convergence of several other methods. In particular, we obtained a unifying treatment of the previous proof mining studies [1, 4]. Moreover it also allowed for a nonlinear discussion of Chang’s reduction argument.
This talk will be a specialized version of the one delivered at the Oberwolfach Workshop last November. With the aim of identifying further application cases, we discuss the types of proofs that could potentially be generalized within this nonlinear framework. Furthermore, we present open questions associated with emerging concepts.
References:
[1] U. Kohlenbach, Quantitative analysis of a Halpern-type proximal point algorithm for accretive operators in Banach spaces. Journal of Nonlinear and Convex Analysis, 21(9):2125–2138, 2020.
[2] U. Kohlenbach and A. Sipoș, The finitary content of sunny nonexpansive retractions, Communications in Contemporary Mathematics 23.1: 1950093, 63pp, 2021.
[3] P. Pinto, Nonexpansive maps in nonlinear smooth spaces, submitted, 47pp, 2023.
[4] A. Sipoș, Abstract strongly convergent variants of the proximal point algorithm. Computational Optimization and Applications, 83(1):349–380, 2022.