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Nicholas Pischke

Publications and source records attributed to Nicholas Pischke.

At least 19 recordsLinked to original sources
arXiv2026

A relaxed-inertial proximal point algorithm for strongly quasiconvex equilibrium problems on Hadamard manifolds

We study a proximal point type method for approximating solutions to equilibrium problems generated by pseudomonotone and strongly quasiconvex bifunctions over Hadamard manifolds, that is complete simply connected Riemannian manifolds of nonpositive sectional curvature. Next to the usual proximal step, the method we consider also incorporates an inertia step together with a subsequent over-relaxation, the latter of which is treated in the context of Hadamard manifolds, to our knowledge, for the first time. Making use of a quantitative approach towards such proximal methods for strongly quasiconvex optimization developed by the authors in previous work, we in particular provide effective arguments for the convergence of the method, yielding explicit, fast and very uniform rates of convergence for the distance of the iterates towards the solution. These results extend previous work by Grad, Lara and Marcavillaca on such a method over finite-dimensional Euclidean spaces for the first time to a nonlinear setting, with the quantitative estimates already being novel in the Euclidean case. In particular, our effective approach allows for a fine-grained view on the assumptions on the surrounding objects, so that we are able to either weaken or even fully discharge some previous assumptions.

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arXiv2026

On strongly quasiconvex pseudomonotone equilibrium problems in Hadamard spaces

We study two proximal point type methods for finding equilibrium points of pseudomonotone and strongly quasiconvex bifunctions. Extending results by A. Iusem and F. Lara, we prove the strong convergence of these methods over general complete geodesic metric spaces of nonpositive curvature, so-called Hadamard spaces. Our arguments are quite elementary and in particular effective, yielding sublinear non-asymptotic guarantees for the distance of the iterates towards the solution. These quantitative results are novel even in the context of Euclidean spaces, the original setting of the work by Iusem and Lara, and the simplicity of our arguments allows us to either weaken or even fully discharge various assumptions featuring in this previous work. We also provide an existence result for solutions of equilibrium problems generated by suitably semicontinuous, pseudomonotone and strongly quasiconvex bifunctions over general Hadamard spaces, and derive from this that every lower-semicontinuous strongly quasiconvex function over a Hadamard space has a minimizer, answering a question of the second author.

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arXiv2026

Weak convergence of the stochastic proximal point method in metric spaces

We prove the almost sure weak convergence of a stochastic proximal point method for minimizing a convex integral function in the general nonlinear context of complete geodesic metric spaces of nonpositive curvature (so-called Hadamard spaces), solving a problem of M. Ba\v{c}\'ak. This method, formulated in the context of a mild growth condition on the function which generalizes Lipschitz continuity, was previously only considered in the context of strong metric regularity conditions or in the context of locally compact spaces (all of which immediately induce strong convergence). In particular, this result is novel already over Hilbert spaces. The proof is a combination of a weak almost sure convergence theorem for stochastic processes in Hadamard spaces which confine to a stochastic variant of quasi-Fej\'er monotonicity, due to previous work of the author, together with a new argument for proving the almost sure convergence of the mean function values of the process towards the minimal value.

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arXiv2026

Avoiding logical strength in real analysis

In reverse mathematics, real numbers are traditionally represented by Cauchy sequences with a given rate of convergence. We work without rates and speak of slow Cauchy sequences. It turns out that almost all one-dimensional real analysis from the reverse mathematics book by Simpson can then be developed in theories that are conservative over $\mathsf{RCA}_0$. Specifically, we obtain clusters of equivalences with the infinite pigeonhole principle and the strong cohesive principle. The second cluster includes results like the Bolzano-Weierstrass and Arzel\`a-Ascoli theorems, which are traditionally associated with the stronger axiom of arithmetical comprehension, but also the Heine-Borel theorem, which is normally separated from these principles. This suggests two things: In elementary analysis, one can avoid logical strength to an extent that the traditional picture seems to forbid. And the division of the so-called reverse mathematics zoo into analytical and combinatorial principles may be less rigid than previously assumed.

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arXiv2026

An abstract effective convergence theorem for stochastic processes, with applications to stochastic approximation

We provide a general theorem on the asymptotic behavior of stochastic processes that conform to a relaxed supermartingale condition. The distinguishing feature of our result is that it provides quantitative convergence guarantees at a much higher level of abstraction and generality than is typically seen in the stochastic approximation literature, formulated in particular in terms of a general modulus $τ$ that, on an intuitive level, captures an effective variant of the uniqueness in expectation of associated solutions. Our convergence rate is highly uniform, depending on very few data beyond $τ$. We then demonstrate the utility of our result as a unifying framework by deriving new quantitative versions of several key concepts and theorems from stochastic approximation, including the Robbins-Siegmund theorem, Dvoretzky's convergence theorem, and the convergence of stochastic quasi-Fejér monotone sequences, the latter formulated in a novel and highly general metric context. Throughout, we isolate and discuss special cases of our results which allow for the construction of fast, and in particular linear, rates. Various applications of our results and our general methodology to stochastic approximation are discussed, and in particular explicitly derived in related work of the authors.

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arXiv2026

Convergence guarantees for stochastic algorithms solving non-unique problems in metric spaces

We prove a general quantitative theorem on the asymptotic behavior of stochastic quasi-Fejér monotone sequences in a broad metric context. Concretely, our result explicitly constructs a rate of convergence for such process, both in mean and almost surely, under an abstract stochastic regularity assumption, derived from previous work of Kohlenbach, López-Acedo and Nicolae [Isr. J. Math. 232(1), pp. 261-297, 2019] on such notions in a deterministic context. Our notion of regularity extends and unifies many common conditions from the literature, such as generalized contractivity for self maps, weak sharp minima and error bounds for real-valued functions, uniform monotonicity and global metric subregularity for set-valued operators, related Polyak-Łojasiewicz or Kurdyka-Łojasiewicz conditions, as well as expected sharp growth as e.g. studied by Asi and Duchi [SIAM J. Optim. 29(3), pp. 2257-2290, 2019]. The rate is moreover highly uniform, depending only on very few data of the surrounding objects. We also discuss special cases which allow for the construction of fast rates in the form of linear non-asymptotic guarantees. We conclude by presenting three concrete methods from stochastic approximation where our results yield new rates of convergence, including the classical example of the stochastic proximal point method, a randomized variant of the Krasnoselskii-Mann scheme for solving stochastic fixed-point equations, and a Busemann subgradient method recently introduced by Goodwin, Lewis, López-Acedo and Nicolae [Math. Program., to appear], all of which make use of our metric generality by being formulated over complete geodesic metric spaces of nonpositive curvature.

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arXiv2026

A systematic way of analysing proofs in probability theory

Over extended systems of finite type arithmetic, we utilize a formal representation of the outer measure to define a translation which allows for the systematic formalization of probabilistic statements. As a main result, this translation gives rise to novel probabilistic logical metatheorems in the style of proof mining, guaranteeing the extractability of computable bounds from (non-effective) proofs of probabilistic existence statements. We further show how the set-theoretically false principle of uniform boundedness due to Kohlenbach can be used to replicate logically strong continuity properties of probability measures in the context of these bound extraction theorems in a tame way, i.e. without affecting the computational complexity of the resulting bounds in question, all the while guaranteeing the validity of those bounds even over finitely additive probability spaces. This in particular provides a formal perspective on the elimination of the principle of $σ$-additivity during bound extraction, as previously only observed ad hoc in the practice of proof mining. In that context, we for the first time provide a proof-theoretic treatment of higher-type uniform boundedness principles and related contra-collection principles via Kohlenbach's monotone variant of Gödel's functional interpretation, which is of independent interest. All together, these new metatheorems provide a systematic proof-theoretic approach towards extracting various types of quantitative information for probabilistic theorems considered in the literature, justifying a range of recent applications to probability theory and stochastic optimization. This paper represents a major logical contribution to a recent advance of bringing the methods of proof mining to bear on probability theory, significantly extending previous work by the first and third author [Forum Math. Sigma, 13, e187 (2025)] in that direction.

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arXiv2026

Effective rates for continuous-time quasi-Fejér monotone dynamical systems

We provide quantitative convergence results for continuous-time dynamical systems in metric spaces that satisfy a continuous-time analog of quasi-Fejér monotonicity. More precisely, we provide a (strong) convergence result for such dynamical systems over compact metric spaces which is quantitatively outfitted with a continuous-time rate of metastability, which moreover can be explicitly and effectively constructed in a very uniform way, only depending on a few moduli representing quantitative witnesses to key properties of the dynamical system and a measure for the compactness of the space. We further show how this convergence result can be extended to non-compact spaces under a regularity assumption of the associated problem, where moreover rates of convergence can then be explicitly constructed which are similarly uniform. In both cases, already the associated ``infinitary'' convergence result is qualitatively novel in its present generality. Beyond this abstract quantitative theory for such dynamical systems, we motivate how the presently studied continuous-time variant of quasi-Fejér monotonicity naturally occurs as a unifying property of many dynamical systems and differential equations and inclusions, and in that way can be used to provide a comprehensive quantitative theory for many such dynamical systems. We illustrate this with three case studies for both classical first- and second-order dynamical systems in Hilbert spaces as well as (generalized) gradient flows and associated semigroups in nonlinear Hadamard spaces.

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arXiv2026

Generalized fluctuation bounds for stochastic algorithms in the presence of compactness

We provide a convergence result for sequences of random variables taking values in a metric space that satisfy a stochastic quasi-Fejér monotonicity condition, in the context of a (local) compactness assumption. Our result is quantitative in that we derive an explicit and effective construction which, in terms of only a few moduli representing quantitative witnesses to key properties of the sequence of random variables and the underlying metric space involved, provides a metastable rate of pointwise convergence, a type of generalized fluctuation bound. That quantitative result in particular relies on the development of a finitary theory of martingales, culminating in a fully finitary Robbins-Siegmund theorem. We outline how this result particularises to the circumstances of the seminal work of Combettes and Pesquet on stochastic quasi-Fejér monotone sequences in separable Hilbert spaces, and we provide an initial application by illustrating how these results can be used to provide a metastable rate of pointwise convergence for a stochastic Krasnoselskii-Mann scheme solving a stochastic common fixed point problem for nonexpansive maps over proper Hadamard spaces. This work is set in the context of recent applications of the logic-based methodology of proof mining to probability theory, and represents its most sophisticated case study to date.

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arXiv2026

On Busemann subgradient methods for stochastic minimization in Hadamard spaces

We study the recently introduced Busemann subgradient method due to Goodwin, Lewis, Nicolae and López-Acedo, 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 ergodic convergence of the method over Hadamard spaces satisfying condition $(\overline{Q}_4)$, a slight extension of the $(Q_4)$ condition of Kirk and Payanak, which in particular includes Hilbert spaces, $\mathbb{R}$-trees and spaces of constant curvature. 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 a strong convergence result under a strong convexity assumption, and in that case in particular derive explicit rates of convergence.

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arXiv2025

Asymptotic regularity of a generalised stochastic Halpern scheme

We provide abstract, general and highly uniform rates of asymptotic regularity for a generalized stochastic Halpern-style iteration, which incorporates a second mapping in the style of a Krasnoselskii-Mann iteration. This iteration is general in two ways: First, it incorporates stochasticity completely abstractly, rather than fixing a sampling method; second, it includes as special cases stochastic versions of various schemes from the optimization literature, including Halpern's iteration as well as a Krasnoselskii-Mann iteration with Tikhonov regularization terms in the sense of Boţ, Csetnek and Meier (where this stochastic variant of the latter is considered for the first time in this paper). For these specific cases, we obtain linear rates of asymptotic regularity, matching (or improving) the currently best known rates for these iterations in stochastic optimization, and quadratic rates of asymptotic regularity are obtained in the context of inner product spaces for the general iteration. We conclude by discussing how variance can be managed in practice through sampling methods in the style of minibatching, how our convergence rates can be adapted to provide oracle complexity bounds, and by sketching how the schemes presented here can be instantiated in the context of reinforcement learning to yield novel methods for Q-learning.

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arXiv2025

Mean-square and sublinear convergence of a stochastic proximal point algorithm in metric spaces of nonpositive curvature

We define a stochastic variant of the proximal point algorithm in the general setting of nonlinear Hadamard spaces for approximating zeros of the mean of a stochastically perturbed monotone vector field. Generalizing previous work by P. Bianchi, we prove the convergence of this method under a suitable strong monotonicity assumption in (separable) Hilbert-Hadamard spaces, that is assuming that all tangent spaces isometrically embed into Hilbert spaces (covering, but not being limited to, the setting of Hadamard manifolds). Moreover, our convergence proof is fully effective and allows for the construction of explicit rates of convergence for the iteration towards the (unique) solution both in mean and almost surely. These rates are moreover highly uniform, being independent of most data surrounding the iteration, space or distribution. In that generality, these rates are novel already in the context of Hilbert spaces. Sublinear nonasymptotic guarantees under additional second-moment conditions on the Yosida approximates and special cases of stochastic convex minimization are discussed.

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arXiv2025

A proof-theoretic metatheorem for nonlinear semigroups generated by an accretive operator and applications

We further develop the theoretical framework of proof mining, a program in mathematical logic that seeks to quantify and extract computational information from prima facie `non-computational' proofs from the mainstream mathematical literature. To that end, we establish logical metatheorems that allow for the treatment of proofs involving nonlinear semigroups generated by an accretive operator, structures which in particular arise in the study of the solutions and asymptotic behavior of differential equations. In that way, the here established metatheorems facilitate a theoretical basis for the application of methods from the proof mining program to the wide variety of mathematical results established in the context of that theory since the 1970's. We in particular illustrate the applicability of the new systems and their metatheorems introduced here by providing two case studies on two central results due to Reich and Plant, respectively, on the asymptotic behavior of said semigroups and the resolvents of their generators where we derive rates of convergence for the limits involved which are, moreover, polynomial in all data.

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arXiv2025

Generalized Fejér monotone sequences and their finitary content

We provide quantitative and abstract strong convergence results for sequences from a compact metric space satisfying a certain form of \emph{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) full Fejér monotonicity is relaxed to a partial variant and (3) the distance functions are allowed to vary along the iteration. For such sequences, the paper provides explicit and effective rates of metastability and even rates of convergence, the latter under a regularity assumption that generalizes the notion of metric regularity introduced by Kohlenbach, López-Acedo and Nicolae, itself an abstract generalization of many regularity notions from the literature. In the second part of the paper, we apply the abstract quantitative results established in the first part to two algorithms: one algorithm for approximating zeros of maximally monotone and maximally $ρ$-comonotone operators in Hilbert spaces (in the sense of Combettes and Pennanen as well as Bauschke, Moursi and Wang) that incorporates inertia terms \emph{every other} term and another algorithm for approximating zeros of monotone operators in Banach spaces (in the sense of Browder) that is only Fejér monotone w.r.t.\ a certain Bregman distance.

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arXiv2024

On the proximal point algorithm for strongly quasiconvex functions in Hadamard spaces

We prove the convergence of the proximal point algorithm for finding the unique minimizer of a strongly quasiconvex function in general nonlinear Hadamard spaces, generalizing a recent result due to F. Lara. Our argument is rather elementary and brief and relies only on a few properties of strongly quasiconvex functions and their proximal operators which are established here for the first time over these nonlinear spaces. In particular, our convergence proof is fully effective and actually yields fast (ranging up to linear) rates of convergence for the iterates towards the solution and for the function values towards the minimum. These rates are novel even in the context of Euclidean spaces.

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arXiv2024

Proof mining and probability theory

We extend the theoretical framework of proof mining by establishing general logical metatheorems that allow for the extraction of the computational content of theorems with prima facie "non-computational" proofs from probability theory, thereby unlocking a major branch of mathematics as a new area of application for these methods. Concretely, we devise proof-theoretically tame logical systems that, for one, allow for the formalization of proofs involving algebras of sets together with probability contents as well as associated Lebesgue integrals on them and which, for another, are amenable to proof-theoretic metatheorems in the style of proof mining that guarantee the extractability of effective and tame bounds from large classes of ineffective existence proofs in probability theory. Moreover, these extractable bounds are guaranteed to be highly uniform in the sense that they will be independent of all parameters relating to the underlying probability space, particularly regarding events or measures of them. As such, these results, in particular, provide the first logical explanation for the success and the observed uniformities of the previous ad hoc case studies of proof mining in these areas and further illustrate their extent. Beyond these systems, we provide extensions for the proof-theoretically tame treatment of $σ$-algebras and associated probability measures using an intensional approach to infinite unions. Lastly, we establish a general proof-theoretic transfer principle that allows for the lift of quantitative information on a relationship between different modes of convergence for sequences of real numbers to sequences of random variables.

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arXiv2023

On computational properties of Cauchy problems generated by accretive operators

In this paper, we provide quantitative versions of results on the asymptotic behavior of nonlinear semigroups generated by an accretive operator due to O. Nevanlinna and S. Reich as well as H.-K. Xu. These results themselves rely on a particular assumption on the underlying operator introduced by A. Pazy under the name of `convergence condition'. Based on logical techniques from `proof mining', a subdiscipline of mathematical logic, we derive various notions of a `convergence condition with modulus' which provide quantitative information on this condition in different ways. These techniques then also facilitate the extraction of quantitative information on the convergence results of Nevanlinna and Reich as well as Xu, in particular also in the form of rates of convergence which depend on these moduli for the convergence condition.

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arXiv2023

Rates of convergence for the asymptotic behavior of second-order Cauchy problems

We provide a quantitative version of a result due to Poffald and Reich on the asymptotic behavior of solutions of a second-order Cauchy problem generated by an accretive operator in the form of a rate of convergence. This quantitative result is then used to generalize a result of Xu on the asymptotic behavior of almost-orbits of the solution semigroup of a first-order Cauchy problem to this second-order case.

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