Tropical Carathéodory with Matroids

Bárány’s colorful generalization of Carathéodory’s Theorem combines geometrical and combinatorial constraints. Kalai–Meshulam (2005) and Holmsen (2016) generalized Bárány’s theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert–Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth

Commuting and internet traffic congestion

We examine the fine microstructure of commuting in a game-theoretic setting with a continuum of commuters. Commuters' home and work locations can be heterogeneous. A commuter transport network is exogenous. Traffic speed is determined by link capacity and by local congestion at a time and place along a link, where local congestion at a time and place is endogenous. The model can be reinterpreted to apply to congestion on the internet. We find sufficient conditions for existence of equilibrium, that multiple equilibria are ubiquitous, and that the welfare properties of morning and evening commute equilibria differ on a generalization of a directed tree

Commuting and internet traffic congestion

We examine the fine microstructure of commuting in a game-theoretic setting with a continuum of commuters. Commuters' home and work locations can be heterogeneous. A commuter transport network is exogenous. Traffic speed is determined by link capacity and by local congestion at a time and place along a link, where local congestion at a time and place is endogenous. The model can be reinterpreted to apply to congestion on the internet. We find sufficient conditions for existence of equilibrium, that multiple equilibria are ubiquitous, and that the welfare properties of morning and evening commute equilibria differ on a tree

Optimization under Uncertainty with Applications to Multi-Agent Coordination

In this thesis several approaches for optimization and decision-making under uncertainty with a strong focus on applications in multi-agent systems are considered. These approaches are chance constrained optimization, random convex programs, and partially observable Markov decision processes

Comparing Machine Learning Algorithms by Union-Free Generic Depth

We propose a framework for descriptively analyzing sets of partial orders based on the concept of depth functions. Despite intensive studies in linear and metric spaces, there is very little discussion on depth functions for non-standard data types such as partial orders. We introduce an adaptation of the well-known simplicial depth to the set of all partial orders, the union-free generic (ufg) depth. Moreover, we utilize our ufg depth for a comparison of machine learning algorithms based on multidimensional performance measures. Concretely, we provide two examples of classifier comparisons on samples of standard benchmark data sets. Our results demonstrate promisingly the wide variety of different analysis approaches based on ufg methods. Furthermore, the examples outline that our approach differs substantially from existing benchmarking approaches, and thus adds a new perspective to the vivid debate on classifier comparison.Comment: arXiv admin note: substantial text overlap with arXiv:2304.0987

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Mathematical source references

This list of references is intended to be a convenient reference source for those interested in the historical origin of common mathematical ideas, The topics mentioned are mostly those met in a degree course in mathematics. For each entry the list attempts to give an exact source reference with comments about priority. There are now available other historical reference sources for mathematics on the internet but with a different style of presentation.<br/

The early historical roots of Lee-Yang theorem

A deep and detailed historiographical analysis of a particular case study concerning the so-called Lee-Yang theorem of theoretical statistical mechanics of phase transitions, has emphasized what real historical roots underlie such a case study. To be precise, it turned out that some well-determined aspects of entire function theory have been at the primeval origins of this important formal result of statistical physics.Comment: History of Physics case study. arXiv admin note: substantial text overlap with arXiv:1106.4348, arXiv:math/0601653, arXiv:0809.3087, arXiv:1311.0596 by other author

Black-Box Parallelization for Machine Learning

The landscape of machine learning applications is changing rapidly: large centralized datasets are replaced by high volume, high velocity data streams generated by a vast number of geographically distributed, loosely connected devices, such as mobile phones, smart sensors, autonomous vehicles or industrial machines. Current learning approaches centralize the data and process it in parallel in a cluster or computing center. This has three major disadvantages: (i) it does not scale well with the number of data-generating devices since their growth exceeds that of computing centers, (ii) the communication costs for centralizing the data are prohibitive in many applications, and (iii) it requires sharing potentially privacy-sensitive data. Pushing computation towards the data-generating devices alleviates these problems and allows to employ their otherwise unused computing power. However, current parallel learning approaches are designed for tightly integrated systems with low latency and high bandwidth, not for loosely connected distributed devices. Therefore, I propose a new paradigm for parallelization that treats the learning algorithm as a black box, training local models on distributed devices and aggregating them into a single strong one. Since this requires only exchanging models instead of actual data, the approach is highly scalable, communication-efficient, and privacy-preserving. Following this paradigm, this thesis develops black-box parallelizations for two broad classes of learning algorithms. One approach can be applied to incremental learning algorithms, i.e., those that improve a model in iterations. Based on the utility of aggregations it schedules communication dynamically, adapting it to the hardness of the learning problem. In practice, this leads to a reduction in communication by orders of magnitude. It is analyzed for (i) online learning, in particular in the context of in-stream learning, which allows to guarantee optimal regret and for (ii) batch learning based on empirical risk minimization where optimal convergence can be guaranteed. The other approach is applicable to non-incremental algorithms as well. It uses a novel aggregation method based on the Radon point that allows to achieve provably high model quality with only a single aggregation. This is achieved in polylogarithmic runtime on quasi-polynomially many processors. This relates parallel machine learning to Nick's class of parallel decision problems and is a step towards answering a fundamental open problem about the abilities and limitations of efficient parallel learning algorithms. An empirical study on real distributed systems confirms the potential of the approaches in realistic application scenarios