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We study the approximation capacity of deep ReLU recurrent neural networks (RNNs) and explore the convergence properties of nonparametric least squares regression using RNNs. We derive upper bounds on the approximation error of RNNs for…
We prove bounds for the approximation and estimation of certain binary classification functions using ReLU neural networks. Our estimation bounds provide a priori performance guarantees for empirical risk minimization using networks of a…
This paper studies the approximation of generalized ridge functions, namely of functions which are constant along some submanifolds of $\mathbb{R}^N$. We introduce the notion of linear-sleeve functions, whose function values only depend on…
Derivative-free algorithms seek the minimum of a given function based only on function values queried at appropriate points. Although these methods are widely used in practice, their performance is known to worsen as the problem dimension…
Coherent lower previsions are general probabilistic models allowing incompletely specified probability distributions. However, for complete description of a coherent lower prevision -- even on finite underlying sample spaces -- an infinite…
We propose an alternative to $k$-nearest neighbors for functional data whereby the approximating neighboring curves are piecewise functions built from a functional sample. Using a locally defined distance function that satisfies…
We study the approximation of multivariate functions with tensor networks (TNs), providing some answers to the following two questions: ``what are the approximation capabilities of TNs for functions from classical smoothness classes?'' and…
In a nutshell, submodular functions encode an intuitive notion of diminishing returns. As a result, submodularity appears in many important machine learning tasks such as feature selection and data summarization. Although there has been a…
A regression problem with dependent data is considered. Regularity assumptions on the dependency of the data are introduced, and it is shown that under suitable structural assumptions on the regression function a deep recurrent neural…
Debiased machine learning (DML) offers an attractive way to estimate treatment effects in observational settings, where identification of causal parameters requires a conditional independence or unconfoundedness assumption, since it allows…
Universal Approximation Theorems establish the density of various classes of neural network function approximators in $C(K, \mathbb{R}^m)$, where $K \subset \mathbb{R}^n$ is compact. In this paper, we aim to extend these guarantees by…
Differential equations are a ubiquitous tool to study dynamics, ranging from physical systems to complex systems, where a large number of agents interact through a graph with non-trivial topological features. Data-driven approximations of…
This work studies approximation based on single-hidden-layer feedforward and recurrent neural networks with randomly generated internal weights. These methods, in which only the last layer of weights and a few hyperparameters are optimized,…
We derive fundamental lower bounds on the connectivity and the memory requirements of deep neural networks guaranteeing uniform approximation rates for arbitrary function classes in $L^2(\mathbb R^d)$. In other words, we establish a…
The machine learning literature contains several constructions for prediction intervals that are intuitively reasonable but ultimately ad-hoc in that they do not come with provable performance guarantees. We present methods from the…
Uncertainty quantification for neural operators remains an open problem in the infinite-dimensional setting due to the lack of finite-sample coverage guarantees over functional outputs. While conformal prediction offers finite-sample…
Learning the differential statistical dependency network between two contexts is essential for many real-life applications, mostly in the high dimensional low sample regime. In this paper, we propose a novel differential network estimator…
We investigate the modeling and the numerical solution of machine learning problems with prediction functions which are linear combinations of elements of a possibly infinite-dimensional dictionary. We propose a novel flexible composite…
In this paper, we study the finite-sample statistical rates of distributional temporal difference (TD) learning with linear function approximation. The aim of distributional TD learning is to estimate the return distribution of a discounted…
Deep neural networks have frequently been used to directly learn representations useful for a given task from raw input data. In terms of overall performance metrics, machine learning solutions employing deep representations frequently have…