Related papers: Nonlocal error bounds for piecewise affine functio…
We provide a framework for obtaining error bounds for linear conic problems without assuming constraint qualifications or regularity conditions. The key aspects of our approach are the notions of amenable cones and facial residual…
Error bounds, which refer to inequalities that bound the distance of vectors in a test set to a given set by a residual function, have proven to be extremely useful in analyzing the convergence rates of a host of iterative methods for…
We investigate the problem of fitting piecewise affine functions (PWA) to data. Our algorithm divides the input domain into finitely many polyhedral regions whose shapes are specified using a user-defined template such that the data points…
The complexity of continuous piecewise affine (CPA) functions can be measured by the number of pieces $p$ or the number of distinct affine functions $n$. For CPA functions on $\mathbb{R}^d$, this paper shows an upper bound of $p=O(n^{d+1})$…
A standard theorem in nonsmooth analysis states that a piecewise affine function $F:\mathbb R^n\rightarrow\mathbb R^n$ is surjective if it is coherently oriented in that the linear parts of its selection functions all have the same nonzero…
We consider the problem of designing piecewise affine policies for two-stage adjustable robust linear optimization problems under right-hand side uncertainty. It is well known that a piecewise affine policy is optimal although the number of…
We study the basic computational problem of detecting approximate stationary points for continuous piecewise affine (PA) functions. Our contributions span multiple aspects, including complexity, regularity, and algorithms. Specifically, we…
This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound. This bound encompasses various aspects, including general smoothness conditions, capacity…
In this paper we study local error bound moduli for a locally Lipschitz and regular function via its outer limiting subdifferential set. We show that the distance of 0 from the outer limiting subdifferential of the support function of the…
In this paper, we study the solution uniqueness of an individual feasible vector of a class of convex optimization problems involving convex piecewise affine functions and subject to general polyhedral constraints. This class of problems…
The aim of this paper is to study a wide class of non-convex sweeping processes with moving constraint whose translation and deformation are represented by regulated functions, i.e., functions of not necessarily bounded variation admitting…
For many large undirected models that arise in real-world applications, exact maximumlikelihood training is intractable, because it requires computing marginal distributions of the model. Conditional training is even more difficult, because…
This paper presents an asymptotically compatible error bound for the finite element method (FEM) applied to a nonlocal diffusion model. The analysis covers two scenarios: meshes with and without shape regularity. For shape-regular meshes,…
We develop efficient algorithms for optimizing piecewise smooth (PWS) functions where the underlying partition of the domain into smooth pieces is \emph{unknown}. For PWS functions satisfying a quadratic growth (QG) condition, we propose a…
We provide tools to help automate the error analysis of algorithms that evaluate simple functions over the floating-point numbers. The aim is to obtain tight relative error bounds for these algorithms, expressed as a function of the unit…
We present a brief survey of existing mistake bounds and introduce novel bounds for the Perceptron or the kernel Perceptron algorithm. Our novel bounds generalize beyond standard margin-loss type bounds, allow for any convex and Lipschitz…
In this paper, we establish sufficient conditions for the existence of error bounds at infinity for lower semicontinuous inequality systems. We also show that the existence of an error bound at infinity of constraint systems plays an…
The main result states that every convex set-valued function defined on a real interval with compact values in a locally convex space, admits an affine selection. In the case if the target space is a real line and the values are closed real…
We consider active, semi-supervised learning in an offline transductive setting. We show that a previously proposed error bound for active learning on undirected weighted graphs can be generalized by replacing graph cut with an arbitrary…
Upper semicontinuous (usc) functions arise in the analysis of maximization problems, distributionally robust optimization, and function identification, which includes many problems of nonparametric statistics. We establish that every usc…