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We analyze statistical features of the ``optimization landscape'' in a random version of one of the simplest constrained optimization problems of the least-square type: finding the best approximation for the solution of an overcomplete…
We introduce a new method for solving nonlinear continuous optimization problems with chance constraints. Our method is based on a reformulation of the probabilistic constraint as a quantile function. The quantile function is approximated…
Most commonly used \emph{adaptive} algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a…
In this paper, for $\mu$ and $\nu$ two probability measures on $\mathbb{R}^d$ with finite moments of order $\rho\ge 1$, we define the respective projections for the $W_\rho$-Wasserstein distance of $\mu$ and $\nu$ on the sets of probability…
$D$-optimal designs originate in statistics literature as an approach for optimal experimental designs. In numerical analysis points and weights resulting from maximal determinants turned out to be useful for quadrature and interpolation.…
Probability mass curves the data space with horizons. Let f be a multivariate probability density function with continuous second order partial derivatives. Consider the problem of estimating the true value of f(z) > 0 at a single point z,…
We consider the inverse elastic scattering of incident plane compressional and shear waves from the knowledge of the far field patterns. Specifically, three direct sampling methods for location and shape reconstruction are proposed using…
We study the problem of setting a price for a potential buyer with a valuation drawn from an unknown distribution $D$. The seller has "data"' about $D$ in the form of $m \ge 1$ i.i.d. samples, and the algorithmic challenge is to use these…
We propose an algorithmic framework, that employs active subspace techniques, for scalable global optimization of functions with low effective dimension (also referred to as low-rank functions). This proposal replaces the original…
We study regression of $1$-Lipschitz functions under a log-concave measure $\mu$ on $\mathbb{R}^d$. We focus on the high-dimensional regime where the sample size $n$ is subexponential in $d$, in which distribution-free estimators are…
In this paper, we present a novel derivative-free optimization framework for solving unconstrained stochastic optimization problems. Many problems in fields ranging from simulation optimization to reinforcement learning involve settings…
We consider the global minimization of smooth functions based solely on function evaluations. Algorithms that achieve the optimal number of function evaluations for a given precision level typically rely on explicitly constructing an…
This paper considers the recovery of a rank $r$ positive semidefinite matrix $X X^T\in\mathbb{R}^{n\times n}$ from $m$ scalar measurements of the form $y_i := a_i^T X X^T a_i$ (i.e., quadratic measurements of $X$). Such problems arise in a…
It is shown that over-parameterized neural networks can achieve minimax optimal rates of convergence (up to logarithmic factors) for learning functions from certain smooth function classes, if the weights are suitably constrained or…
We study the problem of estimating the value of a known smooth function $f$ at an unknown point $\boldsymbol{\mu} \in \mathbb{R}^n$, where each component $\mu_i$ can be sampled via a noisy oracle. Sampling more frequently components of…
Design of experiments, random search, initialization of population-based methods, or sampling inside an epoch of an evolutionary algorithm use a sample drawn according to some probability distribution for approximating the location of an…
Asymptotically-optimal motion planners such as RRT* have been shown to incrementally approximate the shortest path between start and goal states. Once an initial solution is found, their performance can be dramatically improved by…
Importance sampling is a popular variance reduction method for Monte Carlo estimation, where a notorious question is how to design good proposal distributions. While in most cases optimal (zero-variance) estimators are theoretically…
In this paper, we study a sampling problem where a source takes samples from a Wiener process and transmits them through a wireless channel to a remote estimator. Due to channel fading, interference, and potential collisions, the packet…
We propose a stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs. Our approach is based on a bi-objective viewpoint of chance-constrained programs that seeks solutions on the…