Coordinate Descent Methods for DC Minimization

• URL: http://arxiv.org/abs/2109.04228v1
• Date: Thu, 9 Sep 2021 12:44:06 GMT
• Title: Coordinate Descent Methods for DC Minimization
• Authors: Ganzhao Yuan
• Abstract summary: Difference-of-Convex (DC) refers to the problem of minimizing the difference of two convex functions. This paper proposes a non-dimensional one-dimensional subproblem globally, and it is guaranteed to converge to a coordinatewise stationary point.
• Score: 12.284934135116515
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Difference-of-Convex (DC) minimization, referring to the problem of minimizing the difference of two convex functions, has been found rich applications in statistical learning and studied extensively for decades. However, existing methods are primarily based on multi-stage convex relaxation, only leading to weak optimality of critical points. This paper proposes a coordinate descent method for minimizing DC functions based on sequential nonconvex approximation. Our approach iteratively solves a nonconvex one-dimensional subproblem globally, and it is guaranteed to converge to a coordinate-wise stationary point. We prove that this new optimality condition is always stronger than the critical point condition and the directional point condition when the objective function is weakly convex. For comparisons, we also include a naive variant of coordinate descent methods based on sequential convex approximation in our study. When the objective function satisfies an additional regularity condition called \emph{sharpness}, coordinate descent methods with an appropriate initialization converge \emph{linearly} to the optimal solution set. Also, for many applications of interest, we show that the nonconvex one-dimensional subproblem can be computed exactly and efficiently using a breakpoint searching method. We present some discussions and extensions of our proposed method. Finally, we have conducted extensive experiments on several statistical learning tasks to show the superiority of our approach. Keywords: Coordinate Descent, DC Minimization, DC Programming, Difference-of-Convex Programs, Nonconvex Optimization, Sparse Optimization, Binary Optimization.

Related papers

• Trust-Region Sequential Quadratic Programming for Stochastic Optimization with Random Models [57.52124921268249]
  We propose a Trust Sequential Quadratic Programming method to find both first and second-order stationary points. To converge to first-order stationary points, our method computes a gradient step in each iteration defined by minimizing a approximation of the objective subject. To converge to second-order stationary points, our method additionally computes an eigen step to explore the negative curvature the reduced Hessian matrix.
  arXiv  Detail & Related papers  (2024-09-24T04:39:47Z)
• Stochastic Zeroth-Order Optimization under Strongly Convexity and Lipschitz Hessian: Minimax Sample Complexity [59.75300530380427]
  We consider the problem of optimizing second-order smooth and strongly convex functions where the algorithm is only accessible to noisy evaluations of the objective function it queries. We provide the first tight characterization for the rate of the minimax simple regret by developing matching upper and lower bounds.
  arXiv  Detail & Related papers  (2024-06-28T02:56:22Z)
• Variable Substitution and Bilinear Programming for Aligning Partially Overlapping Point Sets [48.1015832267945]
  This research presents a method to meet requirements through the minimization objective function of the RPM algorithm. A branch-and-bound (BnB) algorithm is devised, which solely branches over the parameters, thereby boosting convergence rate. Empirical evaluations demonstrate better robustness of the proposed methodology against non-rigid deformation, positional noise, and outliers, when compared with prevailing state-of-the-art transformations.
  arXiv  Detail & Related papers  (2024-05-14T13:28:57Z)
• An Accelerated Gradient Method for Convex Smooth Simple Bilevel Optimization [16.709026203727007]
  We present a novel bilevel optimization method that locally approximates the solution set of the lower-level problem. We measure the performance of our method in terms of suboptimality and infeasibility errors.
  arXiv  Detail & Related papers  (2024-02-12T22:34:53Z)
• Efficient Gradient Approximation Method for Constrained Bilevel Optimization [2.0305676256390934]
  Bilevel optimization has been developed with large-scale high-dimensional data. This paper considers a constrained bilevel problem with convex and non-differentiable approximations.
  arXiv  Detail & Related papers  (2023-02-03T19:34:56Z)
• Efficient Algorithms for High-Dimensional Convex Subspace Optimization via Strict Complementarity [19.24470467199451]
  We consider optimization problems in which the goal is find a $k$ subspace of $realsn$, $k$, which minimizes a convex smooth loss. While this problem is highly in high-dimensional regimes, it is difficult to find a global optimal solution. In this paper we present a natural. determinate optimal dimension relaxation for which convergence to the. global is straightforward.
  arXiv  Detail & Related papers  (2022-02-08T17:36:43Z)
• Coordinate Descent Methods for Fractional Minimization [7.716156977428555]
  We consider a class of structured fractional convex problems in which the numerator part objective is the sum of a differentiable convex non-linear function. This problem is difficult since it is non-smooth convergence. We propose two methods for solving this problem.
  arXiv  Detail & Related papers  (2022-01-30T00:47:04Z)
• Recent Theoretical Advances in Non-Convex Optimization [56.88981258425256]
  Motivated by recent increased interest in analysis of optimization algorithms for non- optimization in deep networks and other problems in data, we give an overview of recent results of theoretical optimization algorithms for non- optimization.
  arXiv  Detail & Related papers  (2020-12-11T08:28:51Z)
• A Feasible Level Proximal Point Method for Nonconvex Sparse Constrained Optimization [25.73397307080647]
  We present a new model of a general convex or non objective machine machine objectives. We propose an algorithm that solves a constraint with gradually relaxed point levels of each subproblem. We demonstrate the effectiveness of our new numerical scale problems.
  arXiv  Detail & Related papers  (2020-10-23T05:24:05Z)
• Aligning Partially Overlapping Point Sets: an Inner Approximation Algorithm [80.15123031136564]
  We propose a robust method to align point sets where there is no prior information about the value of the transformation. Our algorithm does not need regularization on transformation, and thus can handle the situation where there is no prior information about the values of the transformations. Experimental results demonstrate the better robustness of the proposed method over state-of-the-art algorithms.
  arXiv  Detail & Related papers  (2020-07-05T15:23:33Z)
• Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandits [99.70167985955352]
  We study the problem of zero-order optimization of a strongly convex function. We consider a randomized approximation of the projected gradient descent algorithm. Our results imply that the zero-order algorithm is nearly optimal in terms of sample complexity and the problem parameters.
  arXiv  Detail & Related papers  (2020-06-14T10:42:23Z)

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.