Related papers: Complexity of weakly null sequences

In this paper, we provide tight lower bounds for the oracle complexity of minimizing high-order H\"older smooth and uniformly convex functions. Specifically, for a function whose $p^{th}$-order derivatives are H\"older continuous with…

Building on Buchholz' assignment for ordinals below Bachmann-Howard ordinal, see Buchholz 2003, we introduce systems of fundamental sequences for two kinds of relativized $\vartheta$-function-based notation systems of strength…

This paper examines Automatic Complexity, a complexity notion introduced by Shallit and Wang in 2001. We demonstrate that there exists a normal sequence $T$ such that $I(T) = 0$ and $S(T) \leq 1/2$, where $I(T)$ and $S(T)$ are the lower and…

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\varphi:\ \Omega\times[0,\infty)\to [0,\infty)$ be a Musielak--Orlicz function. In this article, the authors establish the atomic characterizations of weak martingale…

This paper studies the lower bound complexity for the optimization problem whose objective function is the average of $n$ individual smooth convex functions. We consider the algorithm which gets access to gradient and proximal oracle for…

We explore the Weihrauch degree of the problems ``find a bad sequence in a non-well quasi order'' ($\mathsf{BS}$) and ``find a descending sequence in an ill-founded linear order'' ($\mathsf{DS}$). We prove that $\mathsf{DS}$ is strictly…

We provide improved convergence rates for constrained convex-concave min-max problems and monotone variational inequalities with higher-order smoothness. In min-max settings where the $p^{th}$-order derivatives are Lipschitz continuous, we…

We consider a non-convex constrained optimization problem, where the objective function is weakly convex and the constraint function is either convex or weakly convex. To solve this problem, we consider the classical switching subgradient…

We define a hierarchy of circuit complexity classes LD^i, whose depth are the inverse of a function in Ackermann hierarchy. Then we introduce extremely weak versions of length induction and construct a bounded arithmetic theory L^i_2 whose…

Let L(n-l+1/2,0) be the vertex operator algebra associated to an affine Lie algebra of type B_l^(1) at level n-l+1/2, for a positive integer n. We classify irreducible L(n-l+1/2,0)-modules and show that every L(n-l+1/2,0)-module is…

Zeroth-order optimization is a fundamental research topic that has been a focus of various learning tasks, such as black-box adversarial attacks, bandits, and reinforcement learning. However, in theory, most complexity results assert a…

For exponentially closed ordinals $\alpha$, we consider recognizability of constructible subsets of $\alpha$ for $\alpha$-(w)ITRMs and their distribution in the constructible hierarchy. In particular, for $\alpha$-ITRMs, we show that, there…

Zagier proved that the generating series for the traces of singular moduli is a \textit{weakly holomorphic} modular form of weight 3/2 on $\Gamma_0(4)$. Bruinier and Funke extended the results of Zagier to modular curves of arbitrary genus.…

A subset $A$ of an abelian group $G$ is sequenceable if there is an ordering $(a_1, \ldots, a_k)$ of its elements such that the partial sums $(s_0, s_1, \ldots, s_k)$, given by $s_0 = 0$ and $s_i = \sum_{j=1}^i a_i$ for $1 \leq i \leq k$,…

We relate the computational complexity of finite strings to universal representations of their underlying symmetries. First, Boolean functions are classified using the universal covering topologies of the circuits which enumerate them. A…

We study the subsymmetric basic sequence structure of variable exponent Lebesgue spaces $L_{P}$ built from index functions $P\colon\Omega\to(0,\infty]$ on $\sigma$-finite measure spaces $(\Omega,\Sigma,\mu)$. Specifically, we prove that if…

If a separable Banach space $X$ is such that for some nonquasireflexive Banach space $Y$ there exists a surjective strictly singular operator $T:X\to Y$ then for every countable ordinal $\alpha $ the dual of $X$ contains a subspace whose…

Given a sequence of real numbers, we consider its subsequences converging to possibly different limits and associate to each of them an index of convergence which depends on the density of the associated subsequences. This index turns out…

It is well known that besides oscillations, sequences bounded only in $L^1$ can also develop concentrations, and if the latter occurs, we can at most hope for weak$^*$ convergence in the sense of measures. Here we derive a new tool to…

The founding of the theory of cylindric algebras, by Alfred Tarski, was a conscious effort to create algebras out of first order predicate calculus. Let $n\in\omega$. The classes of non-commutative cylindric algebras ($NCA_n$) and weakened…