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Optimizing Monotone Chance-Constrained Submodular Functions Using Evolutionary Multi-Objective Algorithms

Aneta Neumann, Frank Neumann

TL;DR

This work presents a first runtime analysis of evolutionary multi-objective algorithms based on Pareto optimization for chance-constrained submodular functions and shows that the use of evolutionary multi-objective algorithms leads to significant performance improvements compared to state-of-the-art greedy algorithms for submodular optimization.

Abstract

Many real-world optimization problems can be stated in terms of submodular functions. Furthermore, these real-world problems often involve uncertainties which may lead to the violation of given constraints. A lot of evolutionary multi-objective algorithms following the Pareto optimization approach have recently been analyzed and applied to submodular problems with different types of constraints. We present a first runtime analysis of evolutionary multi-objective algorithms based on Pareto optimization for chance-constrained submodular functions. Here the constraint involves stochastic components and the constraint can only be violated with a small probability of alpha. We investigate the classical GSEMO algorithm for two different bi-objective formulations using tail bounds to determine the feasibility of solutions. We show that the algorithm GSEMO obtains the same worst case performance guarantees for monotone submodular functions as recently analyzed greedy algorithms for the case of uniform IID weights and uniformly distributed weights with the same dispersion when using the appropriate bi-objective formulation. As part of our investigations, we also point out situations where the use of tail bounds in the first bi-objective formulation can prevent GSEMO from obtaining good solutions in the case of uniformly distributed weights with the same dispersion if the objective function is submodular but non-monotone due to a single element impacting monotonicity. Furthermore, we investigate the behavior of the evolutionary multi-objective algorithms GSEMO, NSGA-II and SPEA2 on different submodular chance-constrained network problems. Our experimental results show that the use of evolutionary multi-objective algorithms leads to significant performance improvements compared to state-of-the-art greedy algorithms for submodular optimization.

Optimizing Monotone Chance-Constrained Submodular Functions Using Evolutionary Multi-Objective Algorithms

TL;DR

This work presents a first runtime analysis of evolutionary multi-objective algorithms based on Pareto optimization for chance-constrained submodular functions and shows that the use of evolutionary multi-objective algorithms leads to significant performance improvements compared to state-of-the-art greedy algorithms for submodular optimization.

Abstract

Many real-world optimization problems can be stated in terms of submodular functions. Furthermore, these real-world problems often involve uncertainties which may lead to the violation of given constraints. A lot of evolutionary multi-objective algorithms following the Pareto optimization approach have recently been analyzed and applied to submodular problems with different types of constraints. We present a first runtime analysis of evolutionary multi-objective algorithms based on Pareto optimization for chance-constrained submodular functions. Here the constraint involves stochastic components and the constraint can only be violated with a small probability of alpha. We investigate the classical GSEMO algorithm for two different bi-objective formulations using tail bounds to determine the feasibility of solutions. We show that the algorithm GSEMO obtains the same worst case performance guarantees for monotone submodular functions as recently analyzed greedy algorithms for the case of uniform IID weights and uniformly distributed weights with the same dispersion when using the appropriate bi-objective formulation. As part of our investigations, we also point out situations where the use of tail bounds in the first bi-objective formulation can prevent GSEMO from obtaining good solutions in the case of uniformly distributed weights with the same dispersion if the objective function is submodular but non-monotone due to a single element impacting monotonicity. Furthermore, we investigate the behavior of the evolutionary multi-objective algorithms GSEMO, NSGA-II and SPEA2 on different submodular chance-constrained network problems. Our experimental results show that the use of evolutionary multi-objective algorithms leads to significant performance improvements compared to state-of-the-art greedy algorithms for submodular optimization.

Paper Structure

This paper contains 24 sections, 7 theorems, 31 equations, 1 figure, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Chance-Constrained Optimization
  4. Submodular Optimization Problems
  5. Our Contribution
  6. Preliminaries
  7. Chance Constraint Evaluation Based on Tail Inequalities
  8. Multi-Objective Formulation
  9. Global SEMO
  10. Generalized Greedy Algorithm
  11. Runtime Analysis for Uniform IID Weights
  12. Runtime Analysis for Uniformly Distributed Weights with the Same Dispersion Using Tail Inequalities
  13. Analysis of computing feasible solutions using surrogate functions based on tail inequalities
  14. Example where Chebyshev's inequality may prevent progress
  15. Preventing optimal solutions
  16. ...and 9 more sections

Key Result

Theorem 1

Let $k=\min\{n+1, \lfloor C/a \rfloor +1\}$ and assume $\lfloor C/a \rfloor=\omega(1)$. Then the expected time until GSEMO using objective function $g$ has computed a $(1-o(1))(1-1/e)$-approximation for a given monotone submodular function under a chance constraint with uniform iid weights is $O(nk(

Figures (1)

  • Figure 1: Mean values for maximum coverage with degree-based chance constraints for NSGA-II on graph frb30-15-01 and frb35-17-01 (from left) for parent population size $\mu$ = $20, 50, 100, 200$ and offspring population size $\mu/2$.

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 4 more