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Sum of Squares Submodularity

Anna Deza, Georgina Hall

TL;DR

This work builds a new bridge between discrete optimization and real algebraic geometry by connecting sum of squares-based algebraic certificates to a fundamental discrete structure, submodularity.

Abstract

We introduce the notion of $t$-sum of squares (sos) submodularity, which is a hierarchy, indexed by $t$, of sufficient algebraic conditions for certifying submodularity of set functions. We show that, for fixed $t$, each level of the hierarchy can be verified via a semidefinite program of size polynomial in $n$, the size of the ground set of the set function. This is particularly relevant given existing hardness results around testing whether a set function is submodular (Crama, 1989). We derive several equivalent algebraic characterizations of $t$-sos submodularity and identify submodularity-preserving operations that also preserve $t$-sos submodularity. We further present a complete classification of the cases for which submodularity and $t$-sos submodularity coincide, as well as examples of $t$-sos-submodular functions. We demonstrate the usefulness of $t$-sos submodularity through three applications: (i) a new convex approach to submodular regression, involving minimal manual tuning; (ii) a systematic procedure to derive lower bounds on the submodularity ratio in approximate submodular maximization, and (iii) improved difference-of-submodular decompositions for difference-of-submodular optimization. Overall, our work builds a new bridge between discrete optimization and real algebraic geometry by connecting sum of squares-based algebraic certificates to a fundamental discrete structure, submodularity.

Sum of Squares Submodularity

TL;DR

This work builds a new bridge between discrete optimization and real algebraic geometry by connecting sum of squares-based algebraic certificates to a fundamental discrete structure, submodularity.

Abstract

We introduce the notion of -sum of squares (sos) submodularity, which is a hierarchy, indexed by , of sufficient algebraic conditions for certifying submodularity of set functions. We show that, for fixed , each level of the hierarchy can be verified via a semidefinite program of size polynomial in , the size of the ground set of the set function. This is particularly relevant given existing hardness results around testing whether a set function is submodular (Crama, 1989). We derive several equivalent algebraic characterizations of -sos submodularity and identify submodularity-preserving operations that also preserve -sos submodularity. We further present a complete classification of the cases for which submodularity and -sos submodularity coincide, as well as examples of -sos-submodular functions. We demonstrate the usefulness of -sos submodularity through three applications: (i) a new convex approach to submodular regression, involving minimal manual tuning; (ii) a systematic procedure to derive lower bounds on the submodularity ratio in approximate submodular maximization, and (iii) improved difference-of-submodular decompositions for difference-of-submodular optimization. Overall, our work builds a new bridge between discrete optimization and real algebraic geometry by connecting sum of squares-based algebraic certificates to a fundamental discrete structure, submodularity.

Paper Structure

This paper contains 32 sections, 36 theorems, 145 equations, 4 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Submodular functions
  4. Multilinear extensions of set functions
  5. Hardness of testing submodularity
  6. Ideals, varieties, and sum of squares polynomials
  7. Theory of $t$-sos-submodular polynomials
  8. Definition of $t$-sos submodularity
  9. Equivalent characterizations of $t$-sos submodularity
  10. Operations that preserve $t$-sos submodularity
  11. A study of the gap between submodularity and $t$-sos submodularity
  12. Applications
  13. Submodular regression
  14. Approximate submodular maximization
  15. Difference of submodular optimization
  16. ...and 17 more sections

Key Result

Proposition 3

Let $f:2^{\Omega} \rightarrow \mathbb{R}$ be a set function and let $F$ be its multilinear extension. We have that $f$ is submodular if and only if any one of the following seven equivalent conditions holds:

Figures (4)

  • Figure 1: Comparison of root mean squared test error for various methods, fitted to noisy samples of submodular log functions.
  • Figure 2: Comparison of $\gamma^*$, $\gamma_{spectral}$ and $\gamma_{trunc,sos}^{t,k}$ for different values of $k$.
  • Figure 3: Comparison of root mean squared test error for various methods, fitted to noisy samples of facility location (right) and submodular log-determinant (left) functions.
  • Figure 4: Comparison of average relative optimality gap obtained for ten runs of the submodular-supermodular procedure for the trivial decomposition and 2-sos-submodular-irreducible decomposition for each randomly generated set function in Section \ref{['subsec:diff.submod']}

Theorems & Definitions (84)

  • Definition 1
  • Definition 2
  • Proposition 3
  • Remark 4
  • Proposition 5: gallo1989supermodularcrama1989recognitionbillionnet1985maximizing
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Proposition 10
  • ...and 74 more