De-singularity Subgradient for the $q$-th-Powered $\ell_p$-Norm Weber Location Problem

• URL: http://arxiv.org/abs/2412.15546v2
• Date: Mon, 03 Feb 2025 10:20:28 GMT
• Title: De-singularity Subgradient for the $q$-th-Powered $\ell_p$-Norm Weber Location Problem
• Abstract summary: The gradient of the objective does not exist at a considerable set of singular points.<n>In this paper, we establish the de-singularity subgradient for the $q$-th-powered $ell_p$-norm case with $1leqslant p$ and $1leqslant p2$.<n>We develop a $q$-th-powered $ell_p$-norm Weiszfeld Algorithm without Singularity ($q$P$p$NWAWS) for this problem, which ensures convergence and the descent property of the objective function.
• Score: 14.506533458989853
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The Weber location problem is widely used in several artificial intelligence scenarios. However, the gradient of the objective does not exist at a considerable set of singular points. Recently, a de-singularity subgradient method has been proposed to fix this problem, but it can only handle the $q$-th-powered $\ell_2$-norm case ($1\leqslant q<2$), which has only finite singular points. In this paper, we further establish the de-singularity subgradient for the $q$-th-powered $\ell_p$-norm case with $1\leqslant q\leqslant p$ and $1\leqslant p<2$, which includes all the rest unsolved situations in this problem. This is a challenging task because the singular set is a continuum. The geometry of the objective function is also complicated so that the characterizations of the subgradients, minimum and descent direction are very difficult. We develop a $q$-th-powered $\ell_p$-norm Weiszfeld Algorithm without Singularity ($q$P$p$NWAWS) for this problem, which ensures convergence and the descent property of the objective function. Extensive experiments on six real-world data sets demonstrate that $q$P$p$NWAWS successfully solves the singularity problem and achieves a linear computational convergence rate in practical scenarios.

Related papers

• Faster quantum linear system solver beyond the condition number [50.84794327094274]
  We present two quantum algorithms that produce the normalized solution $|xrangle$ of linear system $Ax=| b rangle$ to accuracy $$ with complexity independent of the condition number $=lVert A-1rVert$.<n>Our filtering-based solver is extremely simple with a favorable runtime prefactor.
  arXiv  Detail & Related papers  (2026-07-08T17:49:40Z)
• Statistical and computational challenges in ranking [53.03724383992195]
  We consider the problem of ranking $n$ experts according to their abilities, based on the correctness of their answers to $d$ questions.<n>We investigate here the existence of statistically optimal and computationally efficient procedures for this problem.
  arXiv  Detail & Related papers  (2025-12-24T11:18:06Z)
• Entangled Mean Estimation in High-Dimensions [36.97113089188035]
  We study the task of high-dimensional entangled mean estimation in the subset-of-signals model. We show that the optimal error (up to polylogarithmic factors) is $f(alpha,N) + sqrtD/(alpha N)$, where the term $f(alpha,N)$ is the error of the one-dimensional problem and the second term is the sub-Gaussian error rate.
  arXiv  Detail & Related papers  (2025-01-09T18:31:35Z)
• Private Algorithms for Stochastic Saddle Points and Variational Inequalities: Beyond Euclidean Geometry [20.068722738606287]
  We study saddle point problems (SSP) and variational inequalities (SVI) under the constraint of $(epsilon,delta)$-differential privacy (DP) We obtain a bound of $tildeObig(frac1sqrtn + fracsqrtdnepsilonbig)$ on the strong SP-gap, where $n$ is the number of samples and $d$ is the dimension. We provide the first analysis which obtains a bound on the strong VI-gap of $
  arXiv  Detail & Related papers  (2024-11-07T21:45:05Z)
• Monge-Kantorovich Fitting With Sobolev Budgets [6.748324975906262]
  We show that when $rho$ is concentrated near an $mtext-d$ set we may interpret this as a manifold learning problem with noisy data.<n>We quantify $nu$'s performance in approximating $rho$ via the Monge-Kantorovich $p$-cost $mathbbW_pp(rho, nu)$, and constrain the complexity by requiring $mathrmsupp nu$ to be coverable by an $f : mathbbRm
  arXiv  Detail & Related papers  (2024-09-25T01:30:16Z)
• A De-singularity Subgradient Approach for the Extended Weber Location Problem [12.263117303066831]
  We establish a de-singularity subgradient approach for the extended Weber location problem. We show that the proposed approach solves the problem, produces the same results as in the non-singularity cases, and shows a reasonable rate of linear convergence.
  arXiv  Detail & Related papers  (2024-05-11T09:22:44Z)
• Kernelized Normalizing Constant Estimation: Bridging Bayesian Quadrature and Bayesian Optimization [51.533164528799084]
  We show that to estimate the normalizing constant within a small relative error, the level of difficulty depends on the value of $lambda$. We find that this pattern holds true even when the function evaluations are noisy.
  arXiv  Detail & Related papers  (2024-01-11T07:45:09Z)
• The Computational Complexity of Finding Stationary Points in Non-Convex Optimization [53.86485757442486]
  Finding approximate stationary points, i.e., points where the gradient is approximately zero, of non-order but smooth objective functions is a computational problem. We show that finding approximate KKT points in constrained optimization is reducible to finding approximate stationary points in unconstrained optimization but the converse is impossible.
  arXiv  Detail & Related papers  (2023-10-13T14:52:46Z)
• Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end [114.3957763744719]
  We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems. We show that the algorithm finds the optimal solution in time $O*(2(0.5-c)n)$ for an $n$-independent constant $c$. We also show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case.
  arXiv  Detail & Related papers  (2022-12-03T02:45:23Z)
• Optimal Gradient Sliding and its Application to Distributed Optimization Under Similarity [121.83085611327654]
  We structured convex optimization problems with additive objective $r:=p + q$, where $r$ is $mu$-strong convex similarity. We proposed a method to solve problems master to agents' communication and local calls. The proposed method is much sharper than the $mathcalO(sqrtL_q/mu)$ method.
  arXiv  Detail & Related papers  (2022-05-30T14:28:02Z)
• A first-order primal-dual method with adaptivity to local smoothness [64.62056765216386]
  We consider the problem of finding a saddle point for the convex-concave objective $min_x max_y f(x) + langle Ax, yrangle - g*(y)$, where $f$ is a convex function with locally Lipschitz gradient and $g$ is convex and possibly non-smooth. We propose an adaptive version of the Condat-Vu algorithm, which alternates between primal gradient steps and dual steps.
  arXiv  Detail & Related papers  (2021-10-28T14:19:30Z)
• Projection-Free Algorithm for Stochastic Bi-level Optimization [17.759493152879013]
  This work presents the first projection-free algorithm to solve bi-level optimization problems, where the objective function depends on another optimization problem. The proposed $textbfStochastic $textbfF$rank-$textbfW$olfe ($textbfSCFW$) is shown to achieve a sample complexity of $mathcalO(epsilon-2)$ for convex objectives.
  arXiv  Detail & Related papers  (2021-10-22T11:49:15Z)
• An $L^2$ Analysis of Reinforcement Learning in High Dimensions with Kernel and Neural Network Approximation [9.088303226909277]
  This paper considers the situation where the function approximation is made using the kernel method or the two-layer neural network model. We establish an $tildeO(H3|mathcal A|frac14n-frac14)$ bound for the optimal policy with $Hn$ samples. Even though this result still requires a finite-sized action space, the error bound is independent of the dimensionality of the state space.
  arXiv  Detail & Related papers  (2021-04-15T21:59:03Z)
• Towards Tight Communication Lower Bounds for Distributed Optimisation [30.134447658488057]
  We consider a standard distributed optimisation setting where $N$ machines aim to jointly minimise the sum of the functions $sum_i = 1N f_i (x)$. Our main result provides the first fully unconditional bounds on total number of bits which need to be sent and received by the $N$ machines. We show that $Omega( Nd log d / Nvarepsilon)$ total bits need to be communicated between the machines to find an additive $epsilon$-approximation to the minimum of $sum_i = 1N
  arXiv  Detail & Related papers  (2020-10-16T08:10:02Z)
• Streaming Complexity of SVMs [110.63976030971106]
  We study the space complexity of solving the bias-regularized SVM problem in the streaming model. We show that for both problems, for dimensions of $frac1lambdaepsilon$, one can obtain streaming algorithms with spacely smaller than $frac1lambdaepsilon$.
  arXiv  Detail & Related papers  (2020-07-07T17:10:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.