Randomized Algorithms for Monotone Submodular Function Maximization on the Integer Lattice

• URL: http://arxiv.org/abs/2111.10175v1
• Date: Fri, 19 Nov 2021 12:07:16 GMT
• Title: Randomized Algorithms for Monotone Submodular Function Maximization on the Integer Lattice
• Authors: Alberto Schiabel and Vyacheslav Kungurtsev and Jakub Marecek
• Abstract summary: We consider the problem of maximizing a monotone submodular function on a bounded integer lattice subject to a cardinality constraint. In particular, we focus on maximizing DR-submodular functions, i.e., functions defined on the integer lattice that exhibit the diminishing returns property. Applying our proposed algorithm on the integer lattice is faster than the alternatives, including reducing a target problem to the set domain and then applying the fastest known set submodular algorithm.
• Score: 5.543220407902113
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Optimization problems with set submodular objective functions have many real-world applications. In discrete scenarios, where the same item can be selected more than once, the domain is generalized from a 2-element set to a bounded integer lattice. In this work, we consider the problem of maximizing a monotone submodular function on the bounded integer lattice subject to a cardinality constraint. In particular, we focus on maximizing DR-submodular functions, i.e., functions defined on the integer lattice that exhibit the diminishing returns property. Given any epsilon > 0, we present a randomized algorithm with probabilistic guarantees of O(1 - 1/e - epsilon) approximation, using a framework inspired by a Stochastic Greedy algorithm developed for set submodular functions by Mirzasoleiman et al. We then show that, on synthetic DR-submodular functions, applying our proposed algorithm on the integer lattice is faster than the alternatives, including reducing a target problem to the set domain and then applying the fastest known set submodular maximization algorithm.

Related papers

• Practical Parallel Algorithms for Non-Monotone Submodular Maximization [20.13836086815112]
  Submodular has found extensive applications in various domains within the field of artificial intelligence. One of the parallelizability of a submodular algorithm is its adaptive complexity, which indicates the number of rounds where a number of queries to the objective function can be executed in parallel. We propose the first algorithm with both provable approximation ratio and sub adaptive complexity for the problem of non-monotone submodepsular subject to a $k$-system.
  arXiv  Detail & Related papers  (2023-08-21T11:48:34Z)
• Supermodular Rank: Set Function Decomposition and Optimization [2.578242050187029]
  We define the supermodular rank of a function on a lattice. We analogously define the submodular rank. We use submodular decompositions to optimize set functions.
  arXiv  Detail & Related papers  (2023-05-24T02:09:28Z)
• Multi-block-Single-probe Variance Reduced Estimator for Coupled Compositional Optimization [49.58290066287418]
  We propose a novel method named Multi-block-probe Variance Reduced (MSVR) to alleviate the complexity of compositional problems. Our results improve upon prior ones in several aspects, including the order of sample complexities and dependence on strongity.
  arXiv  Detail & Related papers  (2022-07-18T12:03:26Z)
• Submodular + Concave [53.208470310734825]
  It has been well established that first order optimization methods can converge to the maximal objective value of concave functions. In this work, we initiate the determinant of the smooth functions convex body $$F(x) = G(x) +C(x)$. This class of functions is an extension of both concave and continuous DR-submodular functions for which no guarantee is known.
  arXiv  Detail & Related papers  (2021-06-09T01:59:55Z)
• Linear-Time Algorithms for Adaptive Submodular Maximization [17.19443570570189]
  First, we develop a well-studied adaptive submodular problem subject to a cardinality constraint. Second, we introduce the concept of fully adaptive submodularity. Our algorithm achieves a $frac1-1/e-epsilon4-2/e-2epsilon$ approximation ratio using only $O(nlogfrac1epsilon)$ number of function evaluations.
  arXiv  Detail & Related papers  (2020-07-08T15:54:28Z)
• Fast and Private Submodular and $k$-Submodular Functions Maximization with Matroid Constraints [27.070004659301162]
  We study the problem of maximizing monotone submodular functions subject to matroid constraints in the framework of differential privacy. We give the first $frac12$-approximation algorithm that preserves $k$submodular functions subject to matroid constraints.
  arXiv  Detail & Related papers  (2020-06-28T23:18:58Z)
• Continuous Submodular Function Maximization [91.17492610120324]
  Continuous submodularity is a class of functions with a wide spectrum of applications. We identify several applications of continuous submodular optimization, ranging from influence, MAP for inferences to inferences to field field.
  arXiv  Detail & Related papers  (2020-06-24T04:37:31Z)
• Submodular Maximization in Clean Linear Time [42.51873363778082]
  We provide the first deterministic algorithm that achieves the tight $1-1/e$ approximation guarantee for submodularimation under a cardinality (size) constraint. We show that when the cardinality allowed for a solution is constant, no algorithm making a sub-linear number of function evaluations can guarantee any constant ratio. We extensively evaluate our algorithms on multiple real-life machine learning applications, including movie recommendation, location summarization, twitter text summarization and video summarization.
  arXiv  Detail & Related papers  (2020-06-16T17:06:45Z)
• 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)
• From Sets to Multisets: Provable Variational Inference for Probabilistic Integer Submodular Models [82.95892656532696]
  Submodular functions have been studied extensively in machine learning and data mining. In this work, we propose a continuous DR-submodular extension for integer submodular functions. We formulate a new probabilistic model which is defined through integer submodular functions.
  arXiv  Detail & Related papers  (2020-06-01T22:20:45Z)

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.