Related papers: Arithmetic functions and learning theory

We introduce the notion of a reproducible algorithm in the context of learning. A reproducible learning algorithm is resilient to variations in its samples -- with high probability, it returns the exact same output when run on two samples…

We show that if $T$ is a dependent theory then so is its Keisler randomisation $T^R$. In order to do this we generalise the notion of a Vapnik-Chervonenkis class to families of $[0,1]$-valued functions (a \emph{continuous}…

This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises…

Conditional inference on arbitrary subsets of variables is a core problem in probabilistic inference with important applications such as masked language modeling and image inpainting. In recent years, the family of Any-Order Autoregressive…

The auto-correlations of arithmetic functions, such as the von Mangoldt function, the M\"obius function and the divisor function, are the subject of classical problems in analytic number theory. The function field analogues of these…

An earlier introduced characterization of nonuniform learnability that allows the sample size to depend on the hypothesis to which the learner is compared has been redefined using the measure theoretic approach. Where nonuniform…

We study distribution-free nonparametric regression following a notion of average smoothness initiated by Ashlagi et al. (2021), which measures the "effective" smoothness of a function with respect to an arbitrary unknown underlying…

The main objective of this paper is to look from the unique point of view at some phenomena arising in different areas of probability theory and mathematical statistics. We will try to understand what is common between classical…

This work studies the statistical limits of uniform convergence for offline policy evaluation (OPE) problems with model-based methods (for episodic MDP) and provides a unified framework towards optimal learning for several well-motivated…

Let $(X, T)$ be a topological dynamical system. We show that if each invariant measure of $(X, T)$ gives rise to a measure-theoretic dynamical system that is either: a. rigid along a sequence of "bounded prime volume" or b. admits a…

We show that the sum function of the M\"{o}bius function of a Beurling number system must satisfy the asymptotic bound $M(x)=o(x)$ if it satisfies the prime number theorem and its prime distribution function arises from a monotone…

Motivated by results on generic-case complexity in group theory, we apply the ideas of effective Baire category and effective measure theory to study complexity classes of functions which are "fractionally computable" by a partial…

In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to…

Uniform deviation bounds limit the difference between a model's expected loss and its loss on an empirical sample uniformly for all models in a learning problem. As such, they are a critical component to empirical risk minimization. In this…

A common challenge across all areas of machine learning is that training data is not distributed like test data, due to natural shifts, "blind spots," or adversarial examples; such test examples are referred to as out-of-distribution (OOD)…

We revisit the machine-learning (ML) approach to the universal density functional $F[\mathbf{n}]$ of the one-dimensional Hubbard model with a site-dependent random potential $\mathbf{v}=\{v_{i}\}$. We generate exact ground-state data via…

Machine learning algorithms use error function minimization to fit a large set of parameters in a preexisting model. However, error minimization eventually leads to a memorization of the training dataset, losing the ability to generalize to…

It is becoming increasingly important to understand the vulnerability of machine learning models to adversarial attacks. In this paper we study the feasibility of robust learning from the perspective of computational learning theory,…

The paper studies entire functions of finite order of growth for which a representation of the form $\psi(z) = 1+ O(|z|^{-\mu}), \mu >0,$ as $z\to \infty$, is valid on a fixed ray of the complex plane. The main result is the following.…

Inference in expressive probabilistic models is generally intractable, which makes them difficult to learn and limits their applicability. Sum-product networks are a class of deep models where, surprisingly, inference remains tractable even…