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A Compressive Sensing Inspired Monte-Carlo Method for Combinatorial Optimization

Baptiste Chevalier, Shimpei Yamaguchi, Wojciech Roga, Masahiro Takeoka

TL;DR

The Monte-Carlo Compressive Optimization algorithm is presented, a new method to solve a combinatorial optimization problem that is assumed compressible and the practicality of the algorithm is enhanced by the ability to tune heuristic parameters to available computational resources.

Abstract

In this paper, we present the Monte-Carlo Compressive Optimization algorithm, a new method to solve a combinatorial optimization problem that is assumed compressible. The method relies on random queries to the objective function in order to estimate generalized moments. Next, a greedy algorithm from compressive sensing is repurposed to find the global optimum when not overfitting to samples. We provide numerical results giving evidences that our methods overcome state-of-the-art dual annealing. Moreover, we also give theoretical justification of the algorithm success and analyze its properties. The practicality of our algorithm is enhanced by the ability to tune heuristic parameters to available computational resources.

A Compressive Sensing Inspired Monte-Carlo Method for Combinatorial Optimization

TL;DR

The Monte-Carlo Compressive Optimization algorithm is presented, a new method to solve a combinatorial optimization problem that is assumed compressible and the practicality of the algorithm is enhanced by the ability to tune heuristic parameters to available computational resources.

Abstract

In this paper, we present the Monte-Carlo Compressive Optimization algorithm, a new method to solve a combinatorial optimization problem that is assumed compressible. The method relies on random queries to the objective function in order to estimate generalized moments. Next, a greedy algorithm from compressive sensing is repurposed to find the global optimum when not overfitting to samples. We provide numerical results giving evidences that our methods overcome state-of-the-art dual annealing. Moreover, we also give theoretical justification of the algorithm success and analyze its properties. The practicality of our algorithm is enhanced by the ability to tune heuristic parameters to available computational resources.

Paper Structure

This paper contains 24 sections, 14 theorems, 128 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Optimization and Compressible Combinatorial Problems
  3. Monte-Carlo Compressive Optimization Algorithm
  4. Numerical results: Comparison with other optimization methods
  5. Justification of Algorithm 1
  6. Stability and uniqueness
  7. Moment concentration---Stability
  8. Rules distinguishability---Uniqueness
  9. Final arguments
  10. Discussion on Applications and Quantum Computers
  11. Reinforcement Learning
  12. Ising Hamiltonians and Quantum Computers
  13. Proof of Proposition 1---Larger momemnts of substrings of the rules
  14. Probability of realizing the rules at a given position knowing it appears $q$ times
  15. Counting the string that contains a single $001$
  16. ...and 9 more sections

Key Result

Proposition 1

Let $\varphi^{i,i+1}_{x}=\Phi^{i,i+1}_xf_{\mathcal{R}}$,

Figures (4)

  • Figure 1: Distribution of the distance between estimate solutions $f(x)$ and the optimum $f(x^*)$. On each plot, the percentage of estimate solutions that lies in given distance interval. The methods used are: (a) annealing and (b)(c)(d) Monte-Carlo Compressive Optimization with different sketch functions. $N=12$ and $n=250$
  • Figure 2: Distribution of the Hamming distance between estimate solutions $x$ and the optimum $x^*$. On each plot, the percentage of estimate solutions for a given hamming distance. The methods used are: (a) annealing and (b)(c)(d) Monte-Carlo Compressive Optimization with different sketch functions.$N=12$ and $n=250$
  • Figure 3: Example of $f_\mathcal{R}$. On the $x$ axis, we list all binary sequences. Their respective rewards are shown on the $y$ axis.
  • Figure 4: Estimate distance to the minimum when the sample size increases. Comparison of dual annealing (dark blue) with our methods using 3 different sketch functions: (orange) random sketch, (green) quadruplets sketch, (light blue) quintuplets sketch.

Theorems & Definitions (41)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Claim 1
  • Lemma 1
  • Lemma 2
  • Theorem 4
  • proof
  • ...and 31 more