Publications

Monographs

  • N. Tupitsa, P. Dvurechensky, D. Dvinskikh, A. Gasnikov, Section: Computational Optimal Transport, P.M. Pardalos, O.A. Prokopyev, eds., Encyclopedia of Optimization, Springer International Publishing, Cham, published online on 11.07.2023 pages, (Chapter Published), DOI 10.1007/978-3-030-54621-2_861-1 .

  • J. Polzehl, K. Tabelow, Magnetic Resonance Brain Imaging: Modeling and Data Analysis using R, 2nd Revised Edition, Series: Use R!, Springer International Publishing, Cham, 2023, 258 pages, (Monograph Published), DOI 10.1007/978-3-031-38949-8 .
    Abstract
    This book discusses the modeling and analysis of magnetic resonance imaging (MRI) data acquired from the human brain. The data processing pipelines described rely on R. The book is intended for readers from two communities: Statisticians who are interested in neuroimaging and looking for an introduction to the acquired data and typical scientific problems in the field; and neuroimaging students wanting to learn about the statistical modeling and analysis of MRI data. Offering a practical introduction to the field, the book focuses on those problems in data analysis for which implementations within R are available. It also includes fully worked examples and as such serves as a tutorial on MRI analysis with R, from which the readers can derive their own data processing scripts. The book starts with a short introduction to MRI and then examines the process of reading and writing common neuroimaging data formats to and from the R session. The main chapters cover three common MR imaging modalities and their data modeling and analysis problems: functional MRI, diffusion MRI, and Multi-Parameter Mapping. The book concludes with extended appendices providing details of the non-parametric statistics used and the resources for R and MRI data.The book also addresses the issues of reproducibility and topics like data organization and description, as well as open data and open science. It relies solely on a dynamic report generation with knitr and uses neuroimaging data publicly available in data repositories. The PDF was created executing the R code in the chunks and then running LaTeX, which means that almost all figures, numbers, and results were generated while producing the PDF from the sources.

  • CH. Bayer, P.K. Friz, M. Fukasawa, J. Gatheral, A. Jacquier, M. Rosenbaum, eds., Rough Volatility, Society for Industrial and Applied Mathematics, Philadelphia, 2023, xxviii + 263 pages, (Monograph Published), DOI 10.1137/1.9781611977783 .

Articles in Refereed Journals

  • D. Belomestny, J.G.M. Schoenmakers, Primal-dual regression approach for Markov decision processes with general state and action space, SIAM Journal on Control and Optimization, 62, pp. 650--679, DOI 10.1137/22M1526010 .
    Abstract
    We develop a regression based primal-dual martingale approach for solving finite time horizon MDPs with general state and action space. As a result, our method allows for the construction of tight upper and lower biased approximations of the value functions, and, provides tight approximations to the optimal policy. In particular, we prove tight error bounds for the estimated duality gap featuring polynomial dependence on the time horizon, and sublinear dependence on the cardinality/dimension of the possibly infinite state and action space. From a computational point of view the proposed method is efficient since, in contrast to usual duality-based methods for optimal control problems in the literature, the Monte Carlo procedures here involved do not require nested simulations.

  • J.M. Oeschger, K. Tabelow, S. Mohammadi, Investigating apparent differences between standard DKI and axisymmetric DKI and its consequences for biophysical parameter estimates, Magnetic Resonance in Medicine, published online on 03.02.2024, DOI 10.1002/mrm.30034 .

  • A. Rogozin, A. Beznosikov, D. Dvinskikh, D. Kovalev, P. Dvurechensky, A. Gasnikov, Decentralized saddle point problems via non-Euclidean mirror prox, Optimization Methods & Software, published online in Jan. 2024, DOI 10.1080/10556788.2023.2280062 .

  • P. Dvurechensky, A. Gasnikov, P. Ostroukhov, C.A. Uribe, A. Ivanova, Near-optimal tensor methods for minimizing the gradient norm of convex function, Optimization Methods & Software, Published online on 05.02.2024, DOI 10.1080/10556788.2023.2296443 .

  • O. Butkovsky, K. Dareiotis, M. Gerencsér, Optimal rate of convergence for approximations of SPDEs with non-regular drift, SIAM Journal on Numerical Analysis, 61 (2023), pp. 1103--1137, DOI 10.1137/21M1454213 .

  • O. Butkovsky, V. Margarint, Y. Yuan, Law of the SLE tip, Electronic Journal of Probability, 28 (2023), pp. 126/1--126/25, DOI 10.1214/23-EJP1015 .

  • F. Galarce Marín, K. Tabelow, J. Polzehl, Ch.P. Papanikas, V. Vavourakis, L. Lilaj, I. Sack, A. Caiazzo, Displacement and pressure reconstruction from magnetic resonance elastography images: Application to an in silico brain model, SIAM Journal on Imaging Sciences, 16 (2023), pp. 996--1027, DOI 10.1137/22M149363X .
    Abstract
    This paper investigates a data assimilation approach for non-invasive quantification of intracranial pressure from partial displacement data, acquired through magnetic resonance elastography. Data assimilation is based on a parametrized-background data weak methodology, in which the state of the physical system tissue displacements and pressure fields is reconstructed from partially available data assuming an underlying poroelastic biomechanics model. For this purpose, a physics-informed manifold is built by sampling the space of parameters describing the tissue model close to their physiological ranges, to simulate the corresponding poroelastic problem, and compute a reduced basis. Displacements and pressure reconstruction is sought in a reduced space after solving a minimization problem that encompasses both the structure of the reduced-order model and the available measurements. The proposed pipeline is validated using synthetic data obtained after simulating the poroelastic mechanics on a physiological brain. The numerical experiments demonstrate that the framework can exhibit accurate joint reconstructions of both displacement and pressure fields. The methodology can be formulated for an arbitrary resolution of available displacement data from pertinent images. It can also inherently handle uncertainty on the physical parameters of the mechanical model by enlarging the physics-informed manifold accordingly. Moreover, the framework can be used to characterize, in silico, biomarkers for pathological conditions, by appropriately training the reduced-order model. A first application for the estimation of ventricular pressure as an indicator of abnormal intracranial pressure is shown in this contribution.

  • O. Yufereva, M. Persiianov, P. Dvurechensky, A. Gasnikov, D. Kovalev, Decentralized convex optimization on time-varying networks with application to Wasserstein barycenters, Computational Management Science, published online on 16.12.2023, DOI 10.1007/s10287-023-00493-9 .

  • A. Agafonov, D. Kamzolov, P. Dvurechensky, A. Gasnikov, Inexact tensor methods and their application to stochastic convex optimization, Optimization Methods & Software, published online in Nov. 2023, DOI 10.1080/10556788.2023.2261604 .

  • D. Belomestny, J.G.M. Schoenmakers, From optimal martingales to randomized dual optimal stopping, Quantitative Finance, 23 (2023), pp. 1099--1113, DOI 10.1080/14697688.2023.2223242 .
    Abstract
    In this article we study and classify optimal martingales in the dual formulation of optimal stopping problems. In this respect we distinguish between weakly optimal and surely optimal martingales. It is shown that the family of weakly optimal and surely optimal martingales may be quite large. On the other hand it is shown that the Doob-martingale, that is, the martingale part of the Snell envelope, is in a certain sense the most robust surely optimal martingale under random perturbations. This new insight leads to a novel randomized dual martingale minimization algorithm that does`nt require nested simulation. As a main feature, in a possibly large family of optimal martingales the algorithm efficiently selects a martingale that is as close as possible to the Doob martingale. As a result, one obtains the dual upper bound for the optimal stopping problem with low variance.

  • F. Bourgey, S. De Marco, P.K. Friz, P. Pigato, Local volatility under rough volatility, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, 33 (2023), pp. 1119--1145, DOI 10.1111/mafi.12392 .

  • J. Diehl, K. Ebrahimi-Fard, N. Tapia, Generalized iterated-sums signatures, Journal of Algebra, 632 (2023), pp. 801--824, DOI 10.1016/j.jalgebra.2023.06.007 .
    Abstract
    We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F. Király and H. Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties.

  • K. Ebrahimi-Fard, F. Patras, N. Tapia, L. Zambotti, Shifted substitution in non-commutative multivariate power series with a view toward free probability, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 19 (2023), pp. 038/1--038/17, DOI 10.3842/SIGMA.2023.038 .

  • N. Kornilov, A. Gasnikov, P. Dvurechensky, D. Dvinskikh, Gradient free methods for non-smooth convex stochastic optimization with heavy-tailed noise on convex compact, Computational Management Science, 20 (2023), pp. 37/1--37/43, DOI 10.1007/s10287-023-00470-2 .

  • A. Vasin, A. Gasnikov, P. Dvurechensky, V. Spokoiny, Accelerated gradient methods with absolute and relative noise in the gradient, Optimization Methods & Software, published online in June 2023, DOI 10.1080/10556788.2023.2212503 .

  • S. Athreya, O. Butkovsky, K. , L. Mytnik, Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation, Communications on Pure and Applied Mathematics, published online on 29.11.2023, DOI 10.1002/cpa.22157 .

  • CH. Bayer, P. Hager, S. Riedel, J.G.M. Schoenmakers, Optimal stopping with signatures, The Annals of Applied Probability, 33 (2023), pp. 238--273, DOI 10.1214/22-AAP1814 .
    Abstract
    We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process. We consider classic and randomized stopping times represented by linear functionals of the associated rough path signature, and prove that maximizing over the class of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature. The only assumption on the process is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion which fail to be either semi-martingales or Markov processes.

  • CH. Bayer, M. Eigel, L. Sallandt, P. Trunschke, Pricing high-dimensional Bermudan options with hierarchical tensor formats, SIAM Journal on Financial Mathematics, ISSN 1945-497X, 14 (2023), pp. 383--406, DOI 10.1137/21M1402170 .

  • CH. Bayer, P. Friz, N. Tapia, Stability of deep neural networks via discrete rough paths, SIAM Journal on Mathematics of Data Science, 5 (2023), pp. 50--76, DOI 10.1137/22M1472358 .
    Abstract
    Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks. In particular we derive stability bounds in terms of the total p-variation of trained weights for any p ≥ 1.

  • CH. Bayer, Ch. Ben Hammouda, R.F. Tempone, Numerical smoothing with hierarchical adaptive sparse grids and quasi-Monte Carlo methods for efficient option pricing, Quantitative Finance, 23 (2023), pp. 209--227, DOI 10.1080/14697688.2022.2135455 .
    Abstract
    When approximating the expectation of a functional of a stochastic process, the efficiency and performance of deterministic quadrature methods, such as sparse grid quadrature and quasi-Monte Carlo (QMC) methods, may critically depend on the regularity of the integrand. To overcome this issue and reveal the available regularity, we consider cases in which analytic smoothing cannot be performed, and introduce a novel numerical smoothing approach by combining a root finding algorithm with one-dimensional integration with respect to a single well-selected variable. We prove that under appropriate conditions, the resulting function of the remaining variables is a highly smooth function, potentially affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (i.e., Brownian bridge and Richardson extrapolation on the weak error). This approach facilitates the effective treatment of high dimensionality. Our study is motivated by option pricing problems, and our focus is on dynamics where the discretization of the asset price is necessary. Based on our analysis and numerical experiments, we show the advantages of combining numerical smoothing with the ASGQ and QMC methods over ASGQ and QMC methods without smoothing and the Monte Carlo approach.

  • P.K. Friz, Th. Wagenhofer, Reconstructing volatility: Pricing of index options under rough volatility, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, 33 (2023), pp. 19--40, DOI 10.1111/mafi.12374 .

  • P.K. Friz, P. Zorin-Kranich , Rough semimartingales and $p$-variation estimates for martingale transforms, The Annals of Applied Probability, 51 (2023), pp. 397--441, DOI 10.1214/22-AOP1598 .

  • V. Spokoiny, Dimension free non-asymptotic bounds on the accuracy of high dimensional Laplace approximation, SIAM/ASA Journal on Uncertainty Quantification, 11 (2023), pp. 1044--1068, DOI 10.1137/22M1495688 .
    Abstract
    This note attempts to revisit the classical results on Laplace approximation in a modern non-asymptotic and dimension free form. Such an extension is motivated by applications to high dimensional statistical and optimization problems. The established results provide explicit non-asymptotic bounds on the quality of a Gaussian approximation of the posterior distribution in total variation distance in terms of the so called empheffective dimension ( dimL ). This value is defined as interplay between information contained in the data and in the prior distribution. In the contrary to prominent Bernstein - von Mises results, the impact of the prior is not negligible and it allows to keep the effective dimension small or moderate even if the true parameter dimension is huge or infinite. We also address the issue of using a Gaussian approximation with inexact parameters with the focus on replacing the Maximum a Posteriori (MAP) value by the posterior mean and design the algorithm of Bayesian optimization based on Laplace iterations. The results are specified to the case of nonlinear regression.

Contributions to Collected Editions

  • R. Danabalan, M. Hintermüller, Th. Koprucki, K. Tabelow, MaRDI: Building research data infrastructures for mathematics and the mathematical sciences, in: 1st Conference on Research Data Infrastructure (CoRDI) -- Connecting Communities, Y. Sure-Vetter, C. Goble, eds., 1 of Proceedings of the Conference on Research Data Infrastructure, TIB Open Publishing, Hannover, 2023, pp. 69/1--69/4, DOI 10.52825/cordi.v1i.397 .
    Abstract
    MaRDI is building a research data infrastructure for mathematics and beyond based on semantic technologies (metadata, ontologies, knowledge graphs) and data repositories. Focusing on the algorithms, models and workflows, the MaRDI infrastructure will connect with other disciplines and NFDI consortia on data processing methods, solving real world problems and support mathematicians on research datamanagement

  • S. Abdurakhmon, M. Danilova, E. Gorbunov, S. Horvath, G. Gauthier, P. Dvurechensky, P. Richtarik, High-probability bounds for stochastic optimization and variational inequalities: The case of unbounded variance, in: Proceedings of the 40th International Conference on Machine Learning, A. Krause, E. Brunskill, K. Cho, B. Engelhardt, S. Sabato, J. Scarlett, eds., 202 of Proceedings of Machine Learning Research, 2023, pp. 29563--29648.

  • T. Boege, R. Fritze, Ch. Görgen, J. Hanselman, D. Iglezakis, L. Kastner, Th. Koprucki, T. Krause, Ch. Lehrenfeld, S. Polla, M. Reidelbach, Ch. Riedel, J. Saak, B. Schembera, K. Tabelow, M. Weber, Research-data management planning in the German mathematical community, 130 of EMS Magazine, European Mathematical Society, 2023, pp. 40--47, DOI 10.4171/MAG/152 .

  • P. Dvurechensky, A. Gasnikov, A. Tiurin, V. Zholobov, Unifying framework for accelerated randomized methods in convex optimization, in: Foundations of Modern Statistics. FMS 2019, D. Belomestny, C. Butucea, E. Mammen, E. Moulines , M. Reiss, V.V. Ulyanov, eds., 425 of Springer Proceedings in Mathematics & Statistics, Springer, Cham, 2023, pp. 511--561, DOI 10.1007/978-3-031-30114-8_15 .

Preprints, Reports, Technical Reports

  • D. Belomestny, J.G.M. Schoenmakers, V. Zorina, Weighted mesh algorithms for general Markov decision processes: Convergence and tractability, Preprint no. 3088, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3088 .
    Abstract, PDF (401 kByte)
    We introduce a mesh-type approach for tackling discrete-time, finite-horizon Markov Decision Processes (MDPs) characterized by state and action spaces that are general, encompassing both finite and infinite (yet suitably regular) subsets of Euclidean space. In particular, for bounded state and action spaces, our algorithm achieves a computational complexity that is tractable in the sense of Novak & Wozniakowski, and is polynomial in the time horizon. For unbounded state space the algorithm is “semi-tractable” in the sense that the complexity is proportional to ε -c with some dimension independent c ≥ 2, for achieving an accuracy ε and polynomial in the time horizon with degree linear in the underlying dimension. As such the proposed approach has some flavor of the randomization method by Rust which deals with infinite horizon MDPs and uniform sampling in compact state space. However, the present approach is essentially different due to the finite horizon and a simulation procedure due to general transition distributions, and more general in the sense that it encompasses unbounded state space. To demonstrate the effectiveness of our algorithm, we provide illustrations based on Linear-Quadratic Gaussian (LQG) control problems.

  • L. Schmitz, N. Tapia, Free generators and Hoffman's isomorphism for the two-parameter shuffle algebra, Preprint no. 3087, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3087 .
    Abstract, PDF (239 kByte)
    Signature transforms have recently been extended to data indexed by two and more parameters. With free Lyndon generators, ideas from B-algebras and a novel two-parameter Hoffman exponential, we provide three classes of isomorphisms between the underlying two-parameter shuffle and quasi-shuffle algebras. In particular, we provide a Hopf algebraic connection to the (classical, one-parameter) shuffle algebra over the extended alphabet of connected matrix compositions.

  • C. Bellingeri, E. Ferrucci, N. Tapia, Branched Itô formula and natural Itô--Stratonovich isomorphism, Preprint no. 3083, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3083 .
    Abstract, PDF (510 kByte)
    Branched rough paths define integration theories that may fail to satisfy the usual integration by parts identity. The intrinsically-defined projection of the Connes-Kreimer Hopf algebra onto its primitive elements defined by Broadhurst and Kreimer, and further studied by Foissy, allows us to view it as a commutative B?-algebra and thus to write an explicit change- of-variable formula for solutions to rough differential equations. This formula, which is realised by means of an explicit morphism from the Grossman-Larson Hopf algebra to the Hopf algebra of differential operators, restricts to the well-known Itô formula for semimartingales. We establish an isomorphism with the shuffle algebra over primitives, extending Hoffman?s exponential for the quasi shuffle algebra, and in particular the usual Itô-Stratonovich correction formula for semimartingales. We place special emphasis on the one-dimensional case, in which certain rough path terms can be expressed as polynomials in the extended trace path, which for semimartingales restrict to the well-known Kailath-Segall polynomials. We end by describing an algebraic framework for generating examples of branched rough paths, and, motivated by the recent literature on stochastic processes, exhibit a few such examples above scalar 1/4-fractional Brownian motion, two of which are ?truly branched?, i.e. not quasi- geometric.

  • CH. Bayer, L. Pelizzari, J.G.M. Schoenmakers, Primal and dual optimal stopping with signatures, Preprint no. 3068, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3068 .
    Abstract, PDF (458 kByte)
    We propose two signature-based methods to solve the optimal stopping problem - that is, to price American options - in non-Markovian frameworks. Both methods rely on a global approximation result for Lp-functionals on rough path-spaces, using linear functionals of robust, rough path signatures. In the primal formulation, we present a non-Markovian generalization of the fa- mous Longstaff--Schwartz algorithm, using linear functionals of the signature as regression basis. For the dual formulation, we parametrize the space of square-integrable martingales using linear functionals of the signature, and apply a sample average approximation. We prove convergence for both methods and present first numerical examples in non-Markovian and non-semimartingale regimes.

  • J.A. Dekker, R.J.A. Laeven, J.G.M. Schoenmakers, M.H. Vellekoop, Optimal stopping with randomly arriving opportunities to stop, Preprint no. 3056, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3056 .
    Abstract, PDF (701 kByte)
    We develop methods to solve general optimal stopping problems with opportunities to stop that arrive randomly. Such problems occur naturally in applications with market frictions. Pivotal to our approach is that our methods operate on random rather than deterministic time scales. This enables us to convert the original problem into an equivalent discrete-time optimal stopping problem with natural number valued stopping times and a possibly infinite horizon. To numerically solve this problem, we design a random times least squares Monte Carlo method. We also analyze an iterative policy improvement procedure in this setting. We illustrate the efficiency of our methods and the relevance of randomly arriving opportunities in a few examples.

  • CH. Bayer, S. Breneis, Efficient option pricing in the rough Heston model using weak simulation schemes, Preprint no. 3045, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3045 .
    Abstract, PDF (569 kByte)
    We provide an efficient and accurate simulation scheme for the rough Heston model in the standard ($H>0$) as well as the hyper-rough regime ($H > -1/2$). The scheme is based on low-dimensional Markovian approximations of the rough Heston process derived in [Bayer and Breneis, arXiv:2309.07023], and provides weak approximation to the rough Heston process. Numerical experiments show that the new scheme exhibits second order weak convergence, while the computational cost increases linear with respect to the number of time steps. In comparison, existing schemes based on discretization of the underlying stochastic Volterra integrals such as Gatheral's HQE scheme show a quadratic dependence of the computational cost. Extensive numerical tests for standard and path-dependent European options and Bermudan options show the method's accuracy and efficiency.

  • CH. Bayer, S. Breneis, Weak Markovian approximations of rough Heston, Preprint no. 3044, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3044 .
    Abstract, PDF (834 kByte)
    The rough Heston model is a very popular recent model in mathematical finance; however, the lack of Markov and semimartingale properties poses significant challenges in both theory and practice. A way to resolve this problem is to use Markovian approximations of the model. Several previous works have shown that these approximations can be very accurate even when the number of additional factors is very low. Existing error analysis is largely based on the strong error, corresponding to the L2 distance between the kernels. Extending earlier results by [Abi Jaber and El Euch, SIAM Journal on Financial Mathematics 10(2):309?349, 2019], we show that the weak error of the Markovian approximations can be bounded using the L1-error in the kernel approximation for general classes of payoff functions for European style options. Moreover, we give specific Markovian approximations which converge super-polynomially in the number of dimensions, and illustrate their numerical superiority in option pricing compared to previously existing approximations. The new approximations also work for the hyper-rough case H > -1/2.

  • P. Bank, Ch. Bayer, P. Friz, L. Pelizzari, Rough PDEs for local stochastic volatility models, Preprint no. 3034, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3034 .
    Abstract, PDF (575 kByte)
    In this work, we introduce a novel pricing methodology in general, possibly non-Markovian local stochastic volatility (LSV) models. We observe that by conditioning the LSV dynamics on the Brownian motion that drives the volatility, one obtains a time-inhomogeneous Markov process. Using tools from rough path theory, we describe how to precisely understand the conditional LSV dynamics and reveal their Markovian nature. The latter allows us to connect the conditional dynamics to so-called rough partial differential equations (RPDEs), through a Feynman-Kac type of formula. In terms of European pricing, conditional on realizations of one Brownian motion, we can compute conditional option prices by solving the corresponding linear RPDEs, and then average over all samples to find unconditional prices. Our approach depends only minimally on the specification of the volatility, making it applicable for a wide range of classical and rough LSV models, and it establishes a PDE pricing method for non-Markovian models. Finally, we present a first glimpse at numerical methods for RPDEs and apply them to price European options in several rough LSV models.

  • P. Dvurechensky, J.-J. Zhu, Kernel mirror prox and RKHS gradient flow for mixed functional Nash equilibrium, Preprint no. 3032, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3032 .
    Abstract, PDF (436 kByte)
    Kernel mirror prox and RKHS gradient flow for mixed functional Nash equilibrium Pavel Dvurechensky , Jia-Jie Zhu Abstract The theoretical analysis of machine learning algorithms, such as deep generative modeling, motivates multiple recent works on the Mixed Nash Equilibrium (MNE) problem. Different from MNE, this paper formulates the Mixed Functional Nash Equilibrium (MFNE), which replaces one of the measure optimization problems with optimization over a class of dual functions, e.g., the reproducing kernel Hilbert space (RKHS) in the case of Mixed Kernel Nash Equilibrium (MKNE). We show that our MFNE and MKNE framework form the backbones that govern several existing machine learning algorithms, such as implicit generative models, distributionally robust optimization (DRO), and Wasserstein barycenters. To model the infinite-dimensional continuous- limit optimization dynamics, we propose the Interacting Wasserstein-Kernel Gradient Flow, which includes the RKHS flow that is much less common than the Wasserstein gradient flow but enjoys a much simpler convexity structure. Time-discretizing this gradient flow, we propose a primal-dual kernel mirror prox algorithm, which alternates between a dual step in the RKHS, and a primal step in the space of probability measures. We then provide the first unified convergence analysis of our algorithm for this class of MKNE problems, which establishes a convergence rate of O(1/N ) in the deterministic case and O(1/√N) in the stochastic case. As a case study, we apply our analysis to DRO, providing the first primal-dual convergence analysis for DRO with probability-metric constraints.

  • CH. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations, Preprint no. 3013, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3013 .
    Abstract, PDF (838 kByte)
    We present an adaptive algorithm for effectively solving rough differential equations (RDEs) using the log- ODE method. The algorithm is based on an error representation formula that accurately describes the contribution of local errors to the global error. By incorporating a cost model, our algorithm efficiently deter- mines whether to refine the time grid or increase the order of the log-ODE method. In addition, we provide several examples that demonstrate the effectiveness of our adapctive algorithm in solving RDEs.

Talks, Poster

  • O. Butkovsky, New developments in regularization by noise for SDEs, Finnish Mathematical Days 2024, January 4 - 5, 2024, Aalto University, Finnish Mathematical Society, Helsinki, Finland, January 4, 2024.

  • O. Butkovsky, Weak and strong well-posedness and local times for SDEs driven by fractional Brownian motion with integrable drift (online talk), 18th Oxford-Berlin Young Researcher's Meeting on Applied Stochastic Analysis, University of Oxford, Mathematical Institute, UK, January 6, 2024.

  • A. Kroshnin, Robust k-means in metric spaces, Seminar Modern Methods in Applied Stochastics and Nonparametric Statistics, WIAS Berlin, February 6, 2024.

  • L. Pelizzari, Optimal control in energy markets using rough analysis & deep networks (online talk), MATH+ Spotlight talk (online event), January 24, 2024.

  • L. Pelizzari, Primal and dual optimal stopping with signatures, 18th Oxford-Berlin Young Researcher's Meeting on Applied Stochastic Analysis, January 4 - 6, 2024, University of Oxford, Mathematical Institute, UK, January 5, 2024.

  • P.K. Friz, Analyzing classes of SPDEs via RSDEs, Stochastic Partial Differential Equations, February 12 - 16, 2024, Universtität Wien, Erwin Schrödinger International Institute for Mathematics and Physics (ESI), Austria, February 16, 2024.

  • V. Spokoiny, Inference for nonlinear inverse problems, The Mathematics of Data, January 21 - 26, 2024, National University of Singapore, Institute for Mathematical Sciences, Singapore, January 23, 2024.

  • A. Shehu, MaRDI: The mathematical research data initiative, 2023 NFDI4DS Conference and Consortium Meeting, November 10, 2023, Fraunhofer FOKUS, November 10, 2023.

  • S. Breneis, American options under rough Heston, 11th General AMaMeF Conference, June 26 - 30, 2023, Universität Bielefeld, Center for Mathematical Economics, June 30, 2023.

  • S. Breneis, Path-dependent options under rough Heston, 4th Workshop on Stochastic Methods in Finance and Physics, Heraklion, Kreta, Greece, July 17 - 21, 2023.

  • S. Breneis, Pricing American options under rough Heston, Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2023, Thuwal, Saudi Arabia, May 21 - June 1, 2023.

  • S. Breneis, Pricing-path dependent options under rough Heston, CDT-IRTG Summer School 2023, September 3 - 8, 2023, Templin, September 3, 2023.

  • S. Breneis, Pricing-path dependent options under rough Heston, 18. Doktorand:innentreffen der Stochastik 2023, August 21 - 23, 2023, Universität Heidelberg, Fachbereich Mathematik, August 23, 2023.

  • S. Breneis, Weak Markovian approximations of rough Heston, 17th Oxford-Berlin Young Researcher's Meeting on Applied Stochastic Analysis, April 27 - 29, 2023, WIAS & TU Berlin, April 27, 2023.

  • O. Butkovsky, Regularization by noise for SDEs and SPDEs beyond the Brownian case, Probability Seminar, Université Paris-Saclay, CentraleSupélec, France, May 25, 2023.

  • O. Butkovsky, Stochastic equations with singular drift driven by fractional Brownian motion, 17th Oxford-Berlin Young Researcher's Meeting on Applied Stochastic Analysis, April 27 - 29, 2023, WIAS & TU Berlin, April 28, 2023.

  • O. Butkovsky, Stochastic equations with singular drift driven by fractional Brownian motion, 43rd Conference on Stochastic Processes and their Applications, August 23 - July 28, 2023, Bernoulli Society, Portugal, July 25, 2023.

  • O. Butkovsky, Stochastic equations with singular drift driven by fractional Brownian motion (online talk), Non-local Operators, Probability and Singularities (online event), researchseminars.org, April 4, 2023.

  • O. Butkovsky, Stochastic sewing, John--Nirenberg inequality, and taming singularities for regularization by noise, Mean Field, Interactions with Singular Kernels and their Approximations 2023, December 18, 2023, Institut Henri Poincaré, Paris, France, December 18, 2023.

  • O. Butkovsky, Stochastic sewing, John--Nirenberg inequality, and taming singularities for regularization by noise: A very practical guide, SDEs with Low Regularity Coefficients: Theory and Numerics, September 20 - 22, 2023, University of Torino, Department of Mathematics, Italy, September 22, 2023.

  • O. Butkovsky, Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise, 14th Conference on Monte Carlo Methods and Applications, June 26 - 30, 2023, Sorbonne University, Paris, France, June 29, 2023.

  • L. Pelizzari, Primal-dual optimal stopping with signatures, Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2023, Thuwal, Saudi Arabia, May 26 - June 1, 2023.

  • L. Pelizzari, Rough PDEs and local stochastic volatility, Volatility is rough, Isle of Skye Workshop, May 21 - 25, 2023, Sabhal Mòr Ostaig, Sleat, Isle of Skye, UK, May 25, 2023.

  • L. Pelizzari, Rough PDEs for local stochastic volatility models, 17th Oxford-Berlin Young Researcher's Meeting on Applied Stochastic Analysis, April 27 - 29, 2023, WIAS & TU Berlin, April 27, 2023.

  • N. Tapia, Branched Itô formula, SFI: Structural Aspects of Signatures and Rough Paths, August 28 - September 1, 2023, The Norwegian Academy of Science and Letters, Centre for Advanced Study (CAS), Oslo, Norway, August 31, 2023.

  • N. Tapia, Branched Itô formula, Mini-Workshop ``Combinatorial and Algebraic Structures in Rough Analysis and Related Fields'', November 26 - December 2, 2023, Mathematisches Forschungsinstitut Oberwolfach, November 30, 2023.

  • N. Tapia, Branched Itô formula, Imperial College London, Mathematical Institute, UK, November 7, 2023.

  • N. Tapia, Stability of deep neural networks via discrete rough paths, Oxford Stochastic Analysis and Mathematical Finance Seminar, University of Oxford, Mathematical Institute, UK, February 13, 2023.

  • A. Kroshnin, Sobolev space of measure-valued functions, Variational and Information Flows in Machine Learning and Optimal Transport, November 19 - 24, 2023, Mathematisches Forschungsinstitut Oberwolfach, November 20, 2023.

  • CH. Bayer, D. Kreher, M. Landstorfer, W. Kenmoe Nzali, Volatile electricity markets and battery storage: A model-based approach for optimal control, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

  • CH. Bayer, P. Friz, J.G.M. Schoenmakers, V. Spokoiny, N. Tapia, L. Pelizzari, Optimal control in energy markets using rough analysis and deep networks, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

  • CH. Bayer, Markovian approximations to rough volatility models, Volatility is rough, Isle of Skye Workshop, May 21 - 25, 2023, Sabhal Mòr Ostaig, Sleat, Isle of Skye, UK, May 25, 2023.

  • CH. Bayer, Markovian approximations to rough volatility models, Stochastics around Finance, August 28 - 30, 2023, Kanazawa University, Natural Science and Technology, Kanazawa, Japan, August 28, 2023.

  • CH. Bayer, Markovian approximations to rough volatility models, Heriot-Watt University, Mathematical Institute, Edinburgh, UK, November 15, 2023.

  • CH. Bayer, Optimal stopping with signatures, Probabilistic Methods, Signatures, Cubature and Geometry, January 9 - 11, 2023, University of York, Department of Mathematics, UK, January 9, 2023.

  • CH. Bayer, Optimal stopping with signatures, Quantitative Finance Conference, April 12 - 15, 2023, University of Cambridge, Centre for Financial Research, UK, April 13, 2023.

  • CH. Bayer, Optimal stopping with signatures, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00322 ``Methodological Advancement in Rough Paths and Data Science'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 24, 2023.

  • CH. Bayer, Optimal stopping with signatures, Workshop on Stochastic Control Theory, October 25 - 26, 2023, KTH Royal Institute of Technology, Department of Mathematics, Stockholm, Sweden, October 26, 2023.

  • CH. Bayer, Optimal stopping with signatures, University of Dundee, School of Science and Engineering, UK, November 13, 2023.

  • CH. Bayer, Optimal stopping with signatures (online talk), North British Probability Seminar, University of Edinburgh, UK, March 29, 2023.

  • CH. Bayer, Rough PDEs for local stochastic volatility models, Rough Volatility Workshop, November 21 - 22, 2023, Sorbonne Université, Institut Henri Poincaré, Paris, France.

  • CH. Bayer, Signatures and applications, 4th Workshop on Stochastic Methods in Finance and Physics, July 17 - 21, 2023, Institute of Applied and Computational Mathematics (IACM), Heraklion, Kreta, Greece.

  • CH. Bayer, Non-Markovian models in finance, Stochastic Numerics and Statistical Learning: Theory and Applications 2023 Workshop, May 26 - June 1, 2023, King Abdullah University of Science and Technology, Computer, Electrical and Mathematical Sciences, Thuwal, Saudi Arabia, May 27, 2023.

  • C. Cárcamo Sanchez, F. Galarce Marín, A. Caiazzo, I. Sack, K. Tabelow, Quantitative tissue pressure imaging via PDE-informed assimilation of MR-data, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

  • P. Dvurechensky, C. Geiersbach, M. Hintermüller, A. Kannan, S. Kater, Equilibria for distributed multi-modal energy systems under uncertainty, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

  • P. Dvurechensky, Decentralized local stochastic extra-gradient for variational inequalities, Thematic Einstein Semester Conference on Mathematical Optimization for Machine Learning, September 13 - 15, 2023, Mathematics Research Cluster MATH+, Berlin, September 14, 2023.

  • P. Dvurechensky, Decentralized local stochastic extra-gradient for variational inequalities, European Conference on Computational Optimization (EUCCO), Session ``Optimization under Uncertainty'', September 25 - 27, 2023, Universität Heidelberg, September 25, 2023.

  • P. Dvurechensky, Hessian barrier algorithms for non-convex conic optimization, 20th Workshop on Advances in Continuous Optimization, August 22 - 25, 2023, Corvinus University, Institute of Mathematical Statistics and Modelling, Budapest, August 25, 2023.

  • P.K. Friz, On rough stochastic differential equations, SPDEs, Optimal Control and Mean Field Games -- Analysis, Numerics and Applications, July 10 - 14, 2023, Universität Bielefeld, Center for Interdisciplinary Research (ZiF), July 11, 2023.

  • P.K. Friz, Rough paths for local (possibly rough) stochastic volatility, Lie-Størmer Colloquium Analytic and Probabilistic Aspects of Rough Paths, November 27 - 29, 2023, Norwegian Academy of Science and Letters, Oslo, Norway, November 27, 2023.

  • A. Kroshnin, Robust k-means clustering in metric spaces, Workshop on Statistics in Metric Spaces, October 11 - 13, 2023, Center for Research in Economics and Statistics (CREST), UMR 9194, Palaiseau, France, October 12, 2023.

  • A. Kroshnin, Robust k-means clustering in metric spaces, Rencontres de Statistique Mathématique, December 18 - 22, 2023, Centre International de Rencontres Mathématiques (CIRM), Marseille, France, December 20, 2023.

  • A. Kroshnin, Entropic Wasserstein barycenters, Interpolation of Measures, January 24 - 25, 2023, Lagrange Mathematics and Computation Research Center, Huawei, Paris, France, January 24, 2023.

  • J.G.M. Schoenmakers, Optimal stopping with randomly arriving opportunities, Stochastische Analysis und Stochastik der Finanzmärkte, Humboldt-Universität zu Berlin, Institut für Mathematik, November 23, 2023.

  • J.G.M. Schoenmakers, Primal-dual regression approach for Markov decision processes with general state and action spaces, SPDEs, Optimal Control and Mean Field Games -- Analysis, Numerics and Applications, July 11 - 14, 2023, Universität Bielefeld, Center for Interdisciplinary Research (ZiF), July 12, 2023.

  • V. Spokoiny, Bayesian inference for complex models, MIA 2023 -- Mathematics and Image Analysis, February 1 - 3, 2023, Berlin, February 3, 2023.

  • V. Spokoiny, Bayesian inference using mixed Laplace approximation with applications to error-in-operator models, New York University, Courant Institute of Mathematical Sciences and Center for Data Science, USA, October 3, 2023.

  • V. Spokoiny, Estimation and inference for error-in-operator model, Mathematics in Armenia: Advances and Perspectives, July 2 - 8, 2023, Yerevan State University and National Academy of Sciences, Institute of Mathematics, Yerevan, Armenia, July 3, 2023.

  • V. Spokoiny, Estimation and inference for error-in-operator model, Lecture Series Trends in Statistics, National University of Singapore, Department of Mathematics, Singapore, August 25, 2023.

  • V. Spokoiny, Estimation and inference for error-in-operator model, Massachusetts Institute of Technology, Department of Mathematics, Cambridge, USA, September 29, 2023.

  • V. Spokoiny, Inference in error-in-operator model, Tel Aviv University, Department of Statistics, Israel, March 30, 2023.

  • V. Spokoiny, Marginal Laplace approximation and Gaussian mixtures, Optimization and Statistical Learning, OSL2023, January 15 - 20, 2023, Les Houches School of Physics, France, January 17, 2023.

  • K. Tabelow, MaRDI: Building research data infrastructures for mathematics and the mathematical science, 1st Conference on Research Data Infrastructure (CoRDI), September 12 - 14, 2023, Karlsruhe Institute of Technology (KIT), September 12, 2023.

  • K. Tabelow, Mathematical research data management in interdisciplinary research, Workshop on Biophysics-based Modeling and Data Assimilation in Medical Imaging (Hybrid Event), WIAS Berlin, August 31, 2023.

  • J.-J. Zhu, From gradient flow force-balance to robust machine learning, Basque Center for Applied Mathematics, Bilbao, Spain, October 31, 2023.

External Preprints

  • O. Butkovsky, S. Gallay, Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime, Preprint no. arXiv:2311.12013, Cornell University, 2023, DOI 10.48550/arXiv.2311.12013 .

  • O. Butkovsky, K. , L. Mytnik, Stochastic equations with singular drift driven by fractional Brownian motion, Preprint no. arXiv:2302.11937, Cornell University, 2023, DOI 10.48550/arXiv.2302.11937 .

  • O. Yufereva, M. Persiianov, P. Dvurechensky, A. Gasnikov, D. Kovalev, Decentralized convex optimization on time-varying networks with application to Wasserstein barycenters, Preprint no. arXiv:2205.15669, Cornell University, 2023, DOI 10.48550/arXiv.2205.15669 .

  • K. Ebrahimi-Fard, F. Patras, N. Tapia, L. Zambotti, Shifted substitution in non-commutative multivariate power series with a view toward free probability, Preprint no. arXiv:2204.01445, Cornell University, 2023, DOI 10.48550/arXiv.2204.01445 .

  • D. Gergely, B. Fricke, J.M. Oeschger, L. Ruthotto, P. Freund, K. Tabelow, S. Mohammadi, ACID: A comprehensive toolbox for image processing and modeling of brain, spinal cord, and ex vivo diffusion MRI data, Preprint no. bioRxiv:2023.10.13.562027, Cold Spring Harbor Laboratory, 2023, DOI 10.1101/2023.10.13.562027 .

  • E. Gladin, A. Gasnikov, P. Dvurechensky, Accuracy certificates for convex minimization with inexact Oracle, Preprint no. arXiv:2310.00523, Cornell University, 2023, DOI 10.48550/arXiv.2310.00523 .
    Abstract
    Accuracy certificates for convex minimization problems allow for online verification of the accuracy of approximate solutions and provide a theoretically valid online stopping criterion. When solving the Lagrange dual problem, accuracy certificates produce a simple way to recover an approximate primal solution and estimate its accuracy. In this paper, we generalize accuracy certificates for the setting of inexact first-order oracle, including the setting of primal and Lagrange dual pair of problems. We further propose an explicit way to construct accuracy certificates for a large class of cutting plane methods based on polytopes. As a by-product, we show that the considered cutting plane methods can be efficiently used with a noisy oracle even thought they were originally designed to be equipped with an exact oracle. Finally, we illustrate the work of the proposed certificates in the numerical experiments highlighting that our certificates provide a tight upper bound on the objective residual.

  • E. Gorbunov, A. Sadiev, D. Dolinova, S. Horvát, G. Gidel, P. Dvurechensky, A. Gasnikov, P. Richtárik, High-probability convergence for composite and distributed stochastic minimization and variational inequalities with heavy-tailed noise, Preprint no. arXiv:2310.01860, Cornell University, 2023, DOI 10.48550/arXiv.2310.01860 .
    Abstract
    High-probability analysis of stochastic first-order optimization methods under mild assumptions on the noise has been gaining a lot of attention in recent years. Typically, gradient clipping is one of the key algorithmic ingredients to derive good high-probability guarantees when the noise is heavy-tailed. However, if implemented naïvely, clipping can spoil the convergence of the popular methods for composite and distributed optimization (Prox-SGD/Parallel SGD) even in the absence of any noise. Due to this reason, many works on high-probability analysis consider only unconstrained non-distributed problems, and the existing results for composite/distributed problems do not include some important special cases (like strongly convex problems) and are not optimal. To address this issue, we propose new stochastic methods for composite and distributed optimization based on the clipping of stochastic gradient differences and prove tight high-probability convergence results (including nearly optimal ones) for the new methods. Using similar ideas, we also develop new methods for composite and distributed variational inequalities and analyze the high-probability convergence of these methods.

  • N. Kornilov, A. Gasnikov, P. Dvurechensky, D. Dvinskikh, Gradient free methods for non-smooth convex stochastic optimization with heavy-tailed noise on convex compact, Preprint no. arXiv:2304.02442, Cornell University, 2023, DOI 10.48550/arXiv.2304.02442 .

  • N. Kornilov, E. Gorbunov, M. Alkousa, F. Stonyakin, P. Dvurechensky, A. Gasnikov, Intermediate gradient methods with relative inexactness, Preprint no. arXiv:2310.00506, Cornell University, 2023, DOI 10.48550/arXiv.2310.00506 .
    Abstract
    This paper is devoted to first-order algorithms for smooth convex optimization with inexact gradi- ents. Unlike the majority of the literature on this topic, we consider the setting of relative rather than absolute inexactness. More precisely, we assume that an additive error in the gradient is propor- tional to the gradient norm, rather than being globally bounded by some small quantity. We propose a novel analysis of the accelerated gradient method under relative inexactness and strong convex- ity and improve the bound on the maximum admissible error that preserves the linear convergence of the algorithm. In other words, we analyze how robust is the accelerated gradient method to the relative inexactness of the gradient information. Moreover, based on the Performance Estimation Problem (PEP) technique, we show that the obtained result is optimal for the family of accelerated algorithms we consider. Motivated by the existing intermediate methods with absolute error, i.e., the methods with convergence rates that interpolate between slower but more robust non-accelerated algorithms and faster, but less robust accelerated algorithms, we propose an adaptive variant of the intermediate gradient method with relative error in the gradient.

  • J.M. Oeschger, K. Tabelow, S. Mohammadi, Investigating apparent differences between standard DKI and axisymmetric DKI and its consequences for biophysical parameter estimates, Preprint no. bioRxiv:2023.06.21.545891, Cold Spring Harbor Laboratory, 2023, DOI 10.1101/2023.06.21.545891 .

  • D.A. Pasechnyuk, M. Persiianov, P. Dvurechensky, A. Gasnikov, Algorithms for Euclidean-regularised optimal transport, Preprint no. arXiv:2307.00321, Cornell University, 2023, DOI 10.48550/arXiv.2307.00321 .

  • A. Sadiev, E. Gorbunov, S. Horváth, G. Gidel, P. Dvurechensky, A. Gasnikov, P. Peter, High-probability bounds for stochastic optimization and variational inequalities: The case of unbounded variance, Preprint no. arXiv:2302.00999, Cornell University, 2023, DOI 10.48550/arXiv.2302.00999 .

  • B. Schembera, F. Wübbeling, H. Kleikamp, Ch. Bledinger, J. Fiedler, M. Reidelbach, A. Shehu, B. Schmidt, Th. Koprucki, D. Iglezakis, D. Göddeke, Ontologies for models and algorithms in applied mathematics and related disciplines, Preprint no. arXiv:2310.20443, Cornell University, 2023, DOI 10.48550/arXiv.2310.20443 .
    Abstract
    In applied mathematics and related disciplines, the modeling-simulationoptimization workflow is a prominent scheme, with mathematical models and numerical algorithms playing a crucial role. For these types of mathematical research data, the Mathematical Research Data Initiative has developed, merged and implemented ontologies and knowledge graphs. This contributes to making mathematical research data FAIR by introducing semantic technology and documenting the mathematical foundations accordingly. Using the concrete example of microfracture analysis of porous media, it is shown how the knowledge of the underlying mathematical model and the corresponding numerical algorithms for its solution can be represented by the ontologies.

  • V. Spokoiny, Concentration of a high dimensional sub-Gaussian vector, Preprint no. arXiv:2305.07885, Cornell University, 2023, DOI 10.48550/arXiv.2305.07885 .

  • V. Spokoiny, Mixed Laplace approximation for marginal posterior and Bayesian inference in error-in-operator model, Preprint no. arXiv:2305.08193, Cornell University, 2023, DOI 10.48550/arXiv.2305.09336 .

  • V. Spokoiny, Nonlinear regression: Finite sample guarantees, Preprint no. arXiv:2305.08193, Cornell University, 2023, DOI 10.48550/arXiv.2305.08193 .