arXiv subjects

Steffen Dereich

Publications and source records attributed to Steffen Dereich.

At least 37 records · Page 2Linked to original sources

arXiv2023

Quasi-processes for branching Markov chains

Potential theory is a central tool to understand and analyse Markov processes. In this article, we develop its probabilistic counterpart for branching Markov chains. Specifically, we examine versions of quasi-processes or interlacements that incorporate branching, referred to as branching quasi-processes. These processes are characterized by their occupation measures. If a certain decorability condition is fulfilled, there's an isomorphism between the set of branching quasi-processes and the set of excessive measures, where the excessive measures correspond to the occupation measures of the branching quasi-processes. Utilizing a branching quasi-process as an intensity measure for a Poisson point process leads to the formulation of random interlacements with branching. In cases where individuals reproduce with an average rate of one or less, we detail a construction that draws on classical interlacements. Our approach significantly employs the additive structure of branching Markov chains, with the spinal representation of the branching processes serving as a crucial technical tool.

math.PR↗

arXiv2023

On the existence of optimal shallow feedforward networks with ReLU activation

We prove existence of global minima in the loss landscape for the approximation of continuous target functions using shallow feedforward artificial neural networks with ReLU activation. This property is one of the fundamental artifacts separating ReLU from other commonly used activation functions. We propose a kind of closure of the search space so that in the extended space minimizers exist. In a second step, we show under mild assumptions that the newly added functions in the extension perform worse than appropriate representable ReLU networks. This then implies that the optimal response in the extended target space is indeed the response of a ReLU network.

cs.LG↗

arXiv2019

Central limit theorems for stochastic gradient descent with averaging for stable manifolds

In this article we establish new central limit theorems for Ruppert-Polyak averaged stochastic gradient descent schemes. Compared to previous work we do not assume that convergence occurs to an isolated attractor but instead allow convergence to a stable manifold. On the stable manifold the target function is constant and the oscillations in the tangential direction may be significantly larger than the ones in the normal direction. As we show, one still recovers a central limit theorem with the same rates as in the case of isolated attractors. Here we consider step-sizes $γ_n=n^{-γ}$ with $γ\in(\frac34,1)$, typically.

math.PR↗

arXiv2019

General multilevel adaptations for stochastic approximation algorithms

In this article we establish central limit theorems for multilevel Polyak-Ruppert averaged stochastic approximation schemes. We work under very mild technical assumptions and consider the slow regime in wich typical errors decay like $N^{-δ}$ with $δ\in(0,\frac 12)$ and the critical regime in which errors decay of order $N^{-1/2}\sqrt{\log N}$ in the runtime $N$ of the algorithm.

math.PR↗

arXiv2017

The shape of the emerging condensate in effective models of condensation

We consider effective models of condensation where the condensation occurs as time t goes to infinity. We provide natural conditions under which the build-up of the condensate occurs on a spatial scale of 1/t and has the universal form of a Gamma density. The exponential parameter of this density is determined only by the equation and the total mass of the condensate, while the power law parameter may in addition depend on the decay properties of the initial condition near the condensation point. We apply our results to some examples, including simple models of Bose-Einstein condensation.

math-ph↗

arXiv2017

Local neighbourhoods for first passage percolation on the configuration model

We consider first passage percolation on the configuration model. Once the network has been generated each edge is assigned an i.i.d. weight modeling the passage time of a message along this edge. Then independently two vertices are chosen uniformly at random, a sender and a recipient, and all edges along the geodesic connecting the two vertices are coloured in red (in the case that both vertices are in the same component). In this article we prove local limit theorems for the coloured graph around the recipient in the spirit of Benjamini and Schramm. We consider the explosive regime, in which case the random distances are of finite order, and the Malthusian regime, in which case the random distances are of logarithmic order.

math.PR↗

arXiv2017

Distances in scale free networks at criticality

Scale-free networks with moderate edge dependence experience a phase transition between ultrasmall and small world behaviour when the power law exponent passes the critical value of three. Moreover, there are laws of large numbers for the graph distance of two randomly chosen vertices in the giant component. When the degree distribution follows a pure power law these show the same asymptotic distances of $\frac{\log N}{\log\log N}$ at the critical value three, but in the ultrasmall regime reveal a difference of a factor two between the most-studied rank-one and preferential attachment model classes. In this paper we identify the critical window where this factor emerges. We look at models from both classes when the asymptotic proportion of vertices with degree at least~$k$ scales like $k^{-2} (\log k)^{2α+ o(1)}$ and show that for preferential attachment networks the typical distance is $\big(\frac{1}{1+α}+o(1)\big)\frac{\log N}{\log\log N}$ in probability as the number~$N$ of vertices goes to infinity. By contrast the typical distance in a rank one model with the same asymptotic degree sequence is $\big(\frac{1}{1+2α}+o(1)\big)\frac{\log N}{\log\log N}.$ As $α\to\infty$ we see the emergence of a factor two between the length of shortest paths as we approach the ultrasmall regime.

math.PR↗

arXiv2017

General multilevel adaptations for stochastic approximation algorithms

In this article we present and analyse new multilevel adaptations of stochastic approximation algorithms for the computation of a zero of a function $f\colon D \to \mathbb R^d$ defined on a convex domain $D\subset \mathbb R^d$, which is given as a parameterised family of expectations. Our approach is universal in the sense that having multilevel implementations for a particular application at hand it is straightforward to implement the corresponding stochastic approximation algorithm. Moreover, previous research on multilevel Monte Carlo can be incorporated in a natural way. This is due to the fact that the analysis of the error and the computational cost of our method is based on similar assumptions as used in Giles (2008) for the computation of a single expectation. Additionally, we essentially only require that $f$ satisfies a classical contraction property from stochastic approximation theory. Under these assumptions we establish error bounds in $p$-th mean for our multilevel Robbins-Monro and Polyak-Ruppert schemes that decay in the computational time as fast as the classical error bounds for multilevel Monte Carlo approximations of single expectations known from Giles (2008).

math.PR↗

arXiv2017

Non-extensive condensation in reinforced branching processes

We study a class of branching processes in which a population consists of immortal individuals equipped with a fitness value. Individuals produce offspring with a rate given by their fitness, and offspring may either belong to the same family, sharing the fitness of their parent, or be founders of new families, with a fitness sampled from a fitness distribution. Examples that can be embedded in this class are stochastic house-of-cards models, urn models with reinforcement, and the preferential attachment tree of Bianconi and Barabasi. Our focus is on the case when the fitness distribution has bounded support and regularly varying tail at the essential supremum. In this case there exists a condensation phase, in which asymptotically a proportion of mass in the empirical fitness distribution of the overall population condenses in the maximal fitness value. Our main results describe the asymptotic behaviour of the size and fitness of the largest family at a given time. In particular, we show that as time goes to infinity the size of the largest family is always negligible compared to the overall population size. This implies that condensation, when it arises, is non-extensive and emerges as a collective effort of several families none of which can create a condensate on its own. Our result disproves claims made in the physics literature in the context of preferential attachment trees.

math.PR↗

arXiv2016

Multilevel Monte Carlo for Lévy-driven SDEs: Central limit theorems for adaptive Euler schemes

In this article, we consider multilevel Monte Carlo for the numerical computation of expectations for stochastic differential equations driven by Lévy processes. The underlying numerical schemes are based on jump-adapted Euler schemes. We prove stable convergence of an idealised scheme. Further, we deduce limit theorems for certain classes of functionals depending on the whole trajectory of the process. In particular, we allow dependence on marginals, integral averages and the supremum of the process. The idealised scheme is related to two practically implementable schemes and corresponding central limit theorems are given. In all cases, we obtain errors of order $N^{-1/2}(\log N)^{1/2}$ in the computational time $N$ which is the same order as obtained in the classical set-up analysed by Giles [Oper. Res. 56 (2008) 607-617]. Finally, we use the central limit theorems to optimise the parameters of the multilevel scheme.

math.PR↗

arXiv2015

Real Self-Similar Processes Started from the Origin

Since the seminal work of Lamperti there is a lot of interest in the understanding of the general structure of self-similar Markov processes. Lamperti gave a representation of positive self-similar Markov processes with initial condition strictly larger than 0 which subsequently was extended to zero initial condition. For real self-similar Markov processes (rssMps) there is a generalization of Lamperti's representation giving a one-to-one correspondence between Markov additive processes and rssMps with initial condition different from the origin. We develop fluctuation theory for Markov additive processes and use Kuznetsov measures to construct the law of transient real self-similar Markov processes issued from the origin. The construction gives a pathwise representation through two-sided Markov additive processes extending the Lamperti-Kiu representation to the origin.

math.PR↗

arXiv2015

Random Interlacements via Kuznetsov Measures

The aim of this note is to give an alternative construction of interlacements - as introduced by Sznitman - which makes use of classical probabilistic potential theory. In particular, we outline that the intensity measure of an interlacement is known in probabilistic potential theory under the name "approximate Markov chain" or "quasi-process". We provide a simple construction of random interlacements through (unconditioned) two-sided Brownian motions (resp. two-sided random walks) involving Mitro's general construction of Kuznetsov measures and a Palm measures relation due to Fitzsimmons. In particular, we show that random interlacement is a Poisson cloud (`soup') of two-sided random walks (or Brownian motions) started in Lebesgue measure and restricted on being closest to the origin at time between 0 and 1 - modulus time-shift.

math.PR↗

arXiv2013

Robust analysis of preferential attachment models with fitness

The preferential attachment network with fitness is a dynamic random graph model. New vertices are introduced consecutively and a new vertex is attached to an old vertex with probability proportional to the degree of the old one multiplied by a random fitness. We concentrate on the typical behaviour of the graph by calculating the fitness distribution of a vertex chosen proportional to its degree. For a particular variant of the model, this analysis was first carried out by Borgs, Chayes, Daskalakis and Roch. However, we present a new method, which is robust in the sense that it does not depend on the exact specification of the attachment law. In particular, we show that a peculiar phenomenon, referred to as Bose-Einstein condensation, can be observed in a wide variety of models. Finally, we also compute the joint degree and fitness distribution of a uniformly chosen vertex.

math.PR↗

arXiv2013

Random networks with sublinear preferential attachment: The giant component

We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function f of its current degree. We give a criterion for the existence of a giant component, which is both necessary and sufficient, and which becomes explicit when f is linear. Otherwise it allows the derivation of explicit necessary and sufficient conditions, which are often fairly close. We give an explicit criterion to decide whether the giant component is robust under random removal of edges. We also determine asymptotically the size of the giant component and the empirical distribution of component sizes in terms of the survival probability and size distribution of a multitype branching random walk associated with f.

math.PR↗

arXiv2012

Emergence of condensation in Kingman's model of selection and mutation

We describe the onset of condensation in the simple model for the balance between selection and mutation given by Kingman in terms of a scaling limit theorem. Loosely speaking, this shows that the wave moving towards genes of maximal fitness has the shape of a gamma distribution. We conjecture that this wave shape is a universal phenomenon that can also be found in a variety of more complex models, well beyond the genetics context, and provide some further evidence for this.

math.PR↗

arXiv2012

Persistence probabilities for an integrated random walk bridge

We prove that an integrated simple random walk, where random walk and integrated random walk are conditioned to return to zero, has asymptotic probability $n^{-1/2}$ to stay positive. This question is motivated by so-called random polymer models and proves a conjecture by Caravenna and Deuschel.

math.PR↗

arXiv2011

Constructive quantization: approximation by empirical measures

In this article, we study the approximation of a probability measure $μ$ on $\mathbb{R}^{d}$ by its empirical measure $\hatμ_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$-th moment Wasserstein metric. In the case where $2p<d$, we establish refined upper and lower bounds for the error, a high-resolution formula. Moreover, we provide a universal estimate based on moments, a so-called Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

math.PR↗

arXiv2011

Typical distances in ultrasmall random networks

We show that in preferential attachment models with power-law exponent $τ\in(2,3)$ the distance between randomly chosen vertices in the giant component is asymptotically equal to $(4+o(1))\, \frac{\log\log N}{-\log (τ-2)}$, where $N$ denotes the number of nodes. This is twice the value obtained for several types of configuration models with the same power-law exponent. The extra factor reveals the different structure of typical shortest paths in preferential attachment graphs.

math.PR↗