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Rank-based zeroth-order (ZO) optimization -- which relies only on the ordering of function evaluations -- offers strong robustness to noise and monotone transformations, and underlies many successful algorithms such as CMA-ES, natural…
Suppose one has a collection of parameters indexed by a (possibly infinite dimensional) set. Given data generated from some distribution, the objective is to estimate the maximal parameter in this collection evaluated at this distribution.…
In this paper we establish new integral representations for the remainder term of the known asymptotic expansion of the logarithm of the Barnes $G$-function. Using these representations, we obtain explicit and numerically computable error…
The problem of determining the best achievable performance of arbitrary lossless compression algorithms is examined, when correlated side information is available at both the encoder and decoder. For arbitrary source-side information pairs,…
Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion…
Given a Dirichlet series $L(s) = \sum a_n n^{-s}$, the asymptotic growth rate of $\sum_{n\le X} a_n$ can be determined by a Tauberian theorem. Bounds on the error term are typically controlled by the size of $|L(\sigma+it)|$ for fixed real…
RRT* is one of the most widely used sampling-based algorithms for asymptotically-optimal motion planning. This algorithm laid the foundations for optimality in motion planning as a whole, and inspired the development of numerous new…
Hierarchical statistical models are widely employed in information science and data engineering. The models consist of two types of variables: observable variables that represent the given data and latent variables for the unobservable…
Recently, Anshu et al. introduced "partially" smoothed information measures and used them to derive tighter bounds for several information-processing tasks, including quantum state merging and privacy amplification against quantum…
We consider the asymptotic distribution of the IP sparsity function, which measures the minimal support of optimal IP solutions, and the IP to LP distance function, which measures the distance between optimal IP and LP solutions. We create…
Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in…
In this article, we provide a comprehensive analysis of the asymptotic behavior of Bell numbers, enhancing and unifying various results previously dispersed in the literature. We establish several explicit lower and upper bounds. The main…
We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree $n$ goes to $\infty$. These are defined on the interval $[-1,1]$ with weight function…
We consider the problem of placing n small balls of given radius in a certain domain subject to a force f in order to minimize the compliance of the configuration. Then we let n tend to infinity and look at the asymptotics of the…
In this paper we consider the initial value problem for a family of shallow water equations on the line $\R$ with various asymptotic conditions at infinity. In particular we construct solutions with prescribed asymptotic expansion as…
We propose an iterative estimating equations procedure for analysis of longitudinal data. We show that, under very mild conditions, the probability that the procedure converges at an exponential rate tends to one as the sample size…
In this paper, we consider asymptotics of the optimal value and the optimal solutions of parametric minimax estimation problems. Specifically, we consider estimators of the optimal value and the optimal solutions in a sample minimax problem…
We derive an asymptotic error formula for Gauss--Legendre quadrature applied to functions with limited regularity, using the contour-integral representation of the remainder term. To address the absence of uniformly valid approximations of…
Length generalization is the ability of a learning algorithm to learn a hypothesis which generalizes to longer inputs than the inputs in the training set. In this paper, we provide provable guarantees of length generalization for various…
We consider a general problem of finding a strategy that minimizes the exponential moment of a given cost function, with an emphasis on its relation to the more common criterion of minimization the expectation of the first moment of the…