arXiv subjects

Thomas Powell

Publications and source records attributed to Thomas Powell.

At least 19 recordsLinked to original sources

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-Fej\'er 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, L\'opez-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-{\L}ojasiewicz or Kurdyka-{\L}ojasiewicz 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, L\'opez-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.

math.OC↗

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-Fej\'er 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-Fej\'er 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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arXiv2025

An approximate zero-one law via the Dialectica interpretation

Zero-one laws state that probabilistic events of a certain type must occur with probability either $0$ or $1$, and nothing in between. We formulate a syntactic zero-one law, which enjoys good logical properties while being broadly applicable in probability theory. Then, inspired by G\"odel's Dialectica interpretation, we finitise it: The result is an approximate zero-one law which states that events with a particular finite structure occur with probability close to $0$ or $1$ up to an arbitrary degree of precision. This approximate zero-one law is equivalent - over classical logic - to the original zero-one law, but in contrast to the latter, is formulated entirely in terms of finite unions and intersections of events. Furthermore, in line with recent logical metatheorems for probability, it admits a computational interpretation, which in turn facilitates a quantitative analysis of theorems whose proof makes use of zero-one laws. Concrete applications in this spirit, over a variety of different settings, are discussed.

math.LO↗

arXiv2025

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 $\tau$ 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 $\tau$. 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-Fej\'er 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.

math.OC↗

arXiv2025

On the algorithmic structure of Dialectica realisers

G\"odel's Dialectica interpretation is a fundamental tool for the extraction of computational content from proofs, and plays a central role in today's proof mining program. In the past decades, it has also been studied from the perspective of programming languages, and our contribution is in that direction. Specifically, we present Dialectica as a collection of rules in the style of Hoare logic, where Dialectica is now viewed as a language for specifying procedural programs that come with a forward and backward direction. This viewpoint captures the interesting dynamics of realisers extracted by the Dialectica interpretation, and we illustrate this by defining a generalised backpropagation semantics for a fragment of this language. We envisage this work as providing a base for several future developments, both theoretical and practical, which we outline at the end.

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arXiv2024

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 Bo\c{t}, 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.

math.OC↗

arXiv2024

A quantitative Robbins-Siegmund theorem

The Robbins-Siegmund theorem is one of the most important results in stochastic optimization, where it is widely used to prove the convergence of stochastic algorithms. We provide a quantitative version of the theorem, establishing a bound on how far one needs to look in order to locate a region of \emph{metastability} in the sense of Tao. Our proof involves a metastable analogue of Doob's theorem for $L_1$-supermartingales along with a series of technical lemmas that make precise how quantitative information propagates through sums and products of stochastic processes. In this way, our paper establishes a general methodology for finding metastable bounds for stochastic processes that can be reduced to supermartingales, and therefore for obtaining quantitative convergence information across a broad class of stochastic algorithms whose convergence proof relies on some variation of the Robbins-Siegmund theorem. We conclude by discussing how our general quantitative result might be used in practice.

math.OC↗

arXiv2024

On quantitative convergence for stochastic processes: Crossings, fluctuations and martingales

We develop a general framework for extracting highly uniform bounds on local stability for stochastic processes in terms of information on fluctuations or crossings. This includes a large class of martingales: As a corollary of our main abstract result, we obtain a quantitative version of Doob's convergence theorem for $L_1$-sub- and supermartingales, but more importantly, demonstrate that our framework readily extends to more complex stochastic processes such as almost-supermartingales, thus paving the way for future applications in stochastic optimization. Fundamental to our approach is the use of ideas from logic, particularly a careful analysis of the quantifier structure of probabilistic statements and the introduction of a number of abstract notions that represent stochastic convergence in a quantitative manner. In this sense, our work falls under the 'proof mining' program, and indeed, our quantitative results provide new examples of the phenomenon, recently made precise by the first author and Pischke, that many proofs in probability theory are proof-theoretically tame, and amenable to the extraction of quantitative data that is both of low complexity and independent of the underlying probability space.

math.PR↗

arXiv2023

Proofs as stateful programs: A first-order logic with abstract Hoare triples, and an interpretation into an imperative language

We introduce an extension of first-order logic that comes equipped with additional predicates for reasoning about an abstract state. Sequents in the logic comprise a main formula together with pre- and postconditions in the style of Hoare logic, and the axioms and rules of the logic ensure that the assertions about the state compose in the correct way. The main result of the paper is a realizability interpretation of our logic that extracts programs into a mixed functional/imperative language. All programs expressible in this language act on the state in a sequential manner, and we make this intuition precise by interpreting them in a semantic metatheory using the state monad. Our basic framework is very general, and our intention is that it can be instantiated and extended in a variety of different ways. We outline in detail one such extension: A monadic version of Heyting arithmetic with a wellfounded while rule, and conclude by outlining several other directions for future work.

cs.LO↗

arXiv2022

A computational study of a class of recursive inequalities

We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results concerning rates of convergence, setting out conditions under which computable rates are possible, and when not, providing corresponding rates of metastability. We then demonstrate how the aforementioned quantitative results can be applied to extract computational information from a range of proofs in nonlinear analysis. Here we provide both a new case study on subgradient algorithms, and give overviews of a selection of recent results which each involve an instance of our main recursive inequality. This paper contains the definitions of all relevant concepts from both proof theory and mathematical analysis, and as such, we hope that it is accessible to a general audience.

math.LO↗

arXiv2021

Rates of convergence for asymptotically weakly contractive mappings in normed spaces

We study Krasnoselskii-Mann style iterative algorithms for approximating fixpoints of asymptotically weakly contractive mappings, with a focus on providing generalised convergence proofs along with explicit rates of convergence. More specifically, we define a new notion of being asymptotically $\psi$-weakly contractive with modulus, and present a series of abstract convergence theorems which both generalise and unify known results from the literature. Rates of convergence are formulated in terms of our modulus of contractivity, in conjunction with other moduli and functions which form quantitative analogues of additional assumptions that are required in each case. Our approach makes use of ideas from proof theory, in particular our emphasis on abstraction and on formulating our main results in a quantitative manner. As such, the paper can be seen as a contribution to the proof mining program.

math.FA↗

arXiv2020

On the computational content of Zorn's lemma

We give a computational interpretation to an abstract instance of Zorn's lemma formulated as a wellfoundedness principle in the language of arithmetic in all finite types. This is achieved through G\"odel's functional interpretation, and requires the introduction of a novel form of recursion over non-wellfounded partial orders whose existence in the model of total continuous functionals is proven using domain theoretic techniques. We show that a realizer for the functional interpretation of open induction over the lexicographic ordering on sequences follows as a simple application of our main results.

cs.LO↗

arXiv2019

A note on the finitization of Abelian and Tauberian theorems

We present finitary formulations of two well known results concerning infinite series, namely Abel's theorem, which establishes that if a series converges to some limit then its Abel sum converges to the same limit, and Tauber's theorem, which presents a simple condition under which the converse holds. Our approach is inspired by proof theory, and in particular G\"{o}del's functional interpretation, which we use to establish quantitative version of both of these results.

math.LO↗

arXiv2019

Rates of convergence for iterative solutions of equations involving set-valued accretive operators

This paper studies proofs of strong convergence of various iterative algorithms for computing the unique zeros of set-valued accretive operators that also satisfy some weak form of uniform accretivity at zero. More precisely, we extract explicit rates of convergence from these proofs which depend on a modulus of uniform accretivity at zero, a concept first introduced by A. Koutsoukou-Argyraki and the first author in 2015. Our highly modular approach, which is inspired by the logic-based proof mining paradigm, also establishes that a number of seemingly unrelated convergence proofs in the literature are actually instances of a common pattern.

math.OC↗

arXiv2019

A unifying framework for continuity and complexity in higher types

We set up a parametrised monadic translation for a class of call-by-value functional languages, and prove a corresponding soundness theorem. We then present a series of concrete instantiations of our translation, demonstrating that a number of fundamental notions concerning higher-order computation, including termination, continuity and complexity, can all be subsumed into our framework. Our main goal is to provide a unifying scheme which brings together several concepts which are often treated separately in the literature. However, as a by-product, we also obtain (i) a method for extracting moduli of continuity for closed functionals of type $(\mathbb{N}\to\mathbb{N})\to\mathbb{N}$ definable in (extensions of) System T, and (ii) a characterisation of the time complexity of bar recursion.

cs.LO↗

arXiv2019

An algorithmic approach to the existence of ideal objects in commutative algebra

The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn's lemma, and thus pose a challenge from a computational point of view. Giving a constructive meaning to ideal objects is a problem which dates back to Hilbert's program, and today is still a central theme in the area of dynamical algebra, which focuses on the elimination of ideal objects via syntactic methods. In this paper, we take an alternative approach based on Kreisel's no counterexample interpretation and sequential algorithms. We first give a computational interpretation to an abstract maximality principle in the countable setting via an intuitive, state based algorithm. We then carry out a concrete case study, in which we give an algorithmic account of the result that in any commutative ring, the intersection of all prime ideals is contained in its nilradical.

cs.LO↗

arXiv2019

Dependent choice as a termination principle

We introduce a new formulation of the axiom of dependent choice that can be viewed as an abstract termination principle, which generalises the recursive path orderings used to establish termination of rewrite systems. We consider several variants of our termination principle, and relate them to general termination theorems in the literature.

cs.LO↗

arXiv2019

A new metastable convergence criterion and an application in the theory of uniformly convex Banach spaces

We study a convergence criterion which generalises the notion of being monotonically decreasing, and introduce a quantitative version of this criterion, a so called metastable rate of asymptotic decreasingness. We then present a concrete application in the fixed point theory of uniformly convex Banach spaces, in which we carry out a quantitative analysis of a convergence proof of Kirk and Sims. More precisely, we produce a rate of metastability (in the sense of Tao) for the Picard iterates of mappings T which satisfy a variant of the convergence criterion, and whose fixed point set has nonempty interior.

math.FA↗