FBFL: A Field-Based Coordination Approach for Data Heterogeneity in Federated Learning

In the last years, Federated learning (FL) has become a popular solution to train machine learning models in domains with high privacy concerns. However, FL scalability and performance face significant challenges in real-world deployments where data across devices are non-independently and identically distributed (non-IID). The heterogeneity in data distribution frequently arises from spatial distribution of devices, leading to degraded model performance in the absence of proper handling. Additionally, FL typical reliance on centralized architectures introduces bottlenecks and single-point-of-failure risks, particularly problematic at scale or in dynamic environments. To close this gap, we propose Field-Based Federated Learning (FBFL), a novel approach leveraging macroprogramming and field coordination to address these limitations through: (i) distributed spatial-based leader election for personalization to mitigate non-IID data challenges; and (ii) construction of a self-organizing, hierarchical architecture using advanced macroprogramming patterns. Moreover, FBFL not only overcomes the aforementioned limitations, but also enables the development of more specialized models tailored to the specific data distribution in each subregion. This paper formalizes FBFL and evaluates it extensively using MNIST, FashionMNIST, and Extended MNIST datasets. We demonstrate that, when operating under IID data conditions, FBFL performs comparably to the widely-used FedAvg algorithm. Furthermore, in challenging non-IID scenarios, FBFL not only outperforms FedAvg but also surpasses other state-of-the-art methods, namely FedProx and Scaffold, which have been specifically designed to address non-IID data distributions. Additionally, we showcase the resilience of FBFL's self-organizing hierarchical architecture against server failures

The Expansion Problem for Infinite Trees

We study Ramsey like theorems for infinite trees and similar combinatorial tools. As an application we consider the expansion problem for tree algebras

A nonlinear analogue of additive commutators

We study a nonlinear analogue of additive commutators, known as \textit{polynomial commutators}, defined by $p(ab) - p(ba)$ for a polynomial $p \in F[x]$ and elements $a, b$ in an algebra $R$ over a field $F$. Originally introduced by Laffey and West for matrices over fields, this notion is here extended to broader algebraic settings. We first show that in division rings, polynomial commutators can generate maximal subfields and even the entire ring as an algebra. In the matrix setting, we prove that matrices similar to ones with zero diagonal are polynomial commutators, and under mild assumptions, every matrix can be written as a product of at most three such commutators. Furthermore, we demonstrate that the matrix algebra can be decomposed as the sum of its center and the linear span of all polynomial commutators. Using the theory of rational identities in division rings, we also exhibit that the trace of a polynomial commutator in the matrix ring can be nonzero in noncommutative cases. Lastly, we explore the size of polynomial commutators via matrix norms

Gromov-Witten and Welschinger invariants of del Pezzo varieties

In this paper, we establish formulas for computing genus-$0$ Gromov-Witten and Welschinger invariants of some del Pezzo varieties of dimension three by comparing to that of dimension two. These formulas are generalizations of that given in three-dimensional projective space by E. Brugallé and P. Georgieva in 2016.55 pages, 7 figure

Universal quantification makes automatic structures hard to decide

Automatic structures are first-order structures whose universe and relations can be represented as regular languages. It follows from the standard closure properties of regular languages that the first-order theory of an automatic structure is decidable. While existential quantifiers can be eliminated in linear time by application of a homomorphism, universal quantifiers are commonly eliminated via the identity $\forall{x}. Φ\equiv \neg (\exists{x}. \neg Φ)$. If $Φ$ is represented in the standard way as an NFA, a priori this approach results in a doubly exponential blow-up. However, the recent literature has shown that there are classes of automatic structures for which universal quantifiers can be eliminated by different means without this blow-up by treating them as first-class citizens and not resorting to double complementation. While existing lower bounds for some classes of automatic structures show that a singly exponential blow-up is unavoidable when eliminating a universal quantifier, it is not known whether there may be better approaches that avoid the naïve doubly exponential blow-up, perhaps at least in restricted settings. In this paper, we answer this question negatively and show that there is a family of NFA representing automatic relations for which the minimal NFA recognising the language after eliminating a single universal quantifier is doubly exponential, and deciding whether this language is empty is EXPSPACE-complete. The techniques underlying our EXPSPACE lower bound further enable us to establish new lower bounds for some fragments of Büchi arithmetic with a fixed number of quantifier alternations

Skew braces and Rota-Baxter operators on semi-direct products

This paper examines the connections between (relative) Rota--Baxter groups, skew left braces, and enlargements of these structures on naturally associated semi-direct products. Given a skew left brace, we define a new skew left brace, referred to as its square, on the natural semi-direct product of its additive and multiplicative groups. Further, the square construction is distinct from the previously known double construction arising as a special case of matched pairs of skew braces. This provides a method to construct a new bijective, non-degenerate solution to the Yang--Baxter equation from an existing solution arising from a skew left brace. We show that the square construction is functorial and integrates naturally into both the cohomological and extension-theoretic frameworks for (relative) Rota--Baxter groups and skew left braces. Furthermore, we provide a sufficient condition under which two isoclinic skew left braces yield isoclinic squares

Urn-driven random walks

The symmetric random walk is known to be recurrent in one and two dimensions, and becomes transient in three or higher dimensions. We compare the symmetric random walk to walks driven by certain \polya\ urns. We show that, in contrast, if the probabilities of the random walk are instead driven by a \polya-Eggenberger urn, the states are recurrent only in one dimension. Further consideration of exchangeability reveals that the walk is null recurrent. As soon as the underlying Markov chain of \polya\ walk gets in two dimensions or higher, there is a positive probability that the walker gets lost in the space, and the probability of her recurrence is less than 1. On the other hand, a walk driven by Friedman urn behaves like the symmetric random walk, being recurrent in one and two dimensions and transient in higher dimensions. As Friedman urn scheme is not exchangeable, it is considerably harder to determine the nature of the recurrence in one and two dimensions. Empirical evidence through simulation suggests that in one dimension Friedman walk is positive recurrent

Prime Factorization in Models of PV1_1

Assuming that no family of polynomial-size Boolean circuits can factorize a constant fraction of all products of two $n$-bit primes, we show that the bounded arithmetic theory $\text{PV}_1$, even when augmented by the sharply bounded choice scheme $BB(Σ^b_0)$, cannot prove that every number has some prime divisor. By the completeness theorem, it follows that under this assumption there is a model $M$ of $\text{PV}_1$ that contains a nonstandard number $m$ which has no prime factorization

Ramanujan’s congruences for partitions modulo 5,75, 7 and 1111

We obtain short and uniform proofs of Ramanujan's partition congruences modulo $5, 7$ and $11$