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Characterization of Locality in Spin States and Forced Moves for Optimizations

Yoshiki Sato, Makiko Konoshima, Hirotaka Tamura, Jun Ohkubo

TL;DR

This work develops an algorithm to get out of the local minima efficiently while it does not yield the exact samplings, and uses a feature that characterizes locality in the current state, which is easy to obtain with a type of specialized hardware.

Abstract

Ising formulations are widely utilized to solve combinatorial optimization problems, and a variety of quantum or semiconductor-based hardware has recently been made available. In combinatorial optimization problems, the existence of local minima in energy landscapes is problematic to use to seek the global minimum. We note that the aim of the optimization is not to obtain exact samplings from the Boltzmann distribution, and there is thus no need to satisfy detailed balance conditions. In light of this fact, we develop an algorithm to get out of the local minima efficiently while it does not yield the exact samplings. For this purpose, we utilize a feature that characterizes locality in the current state, which is easy to obtain with a type of specialized hardware. Furthermore, as the proposed algorithm is based on a rejection-free algorithm, the computational cost is low. In this work, after presenting the details of the proposed algorithm, we report the results of numerical experiments that demonstrate the effectiveness of the proposed feature and algorithm.

Characterization of Locality in Spin States and Forced Moves for Optimizations

TL;DR

This work develops an algorithm to get out of the local minima efficiently while it does not yield the exact samplings, and uses a feature that characterizes locality in the current state, which is easy to obtain with a type of specialized hardware.

Abstract

Ising formulations are widely utilized to solve combinatorial optimization problems, and a variety of quantum or semiconductor-based hardware has recently been made available. In combinatorial optimization problems, the existence of local minima in energy landscapes is problematic to use to seek the global minimum. We note that the aim of the optimization is not to obtain exact samplings from the Boltzmann distribution, and there is thus no need to satisfy detailed balance conditions. In light of this fact, we develop an algorithm to get out of the local minima efficiently while it does not yield the exact samplings. For this purpose, we utilize a feature that characterizes locality in the current state, which is easy to obtain with a type of specialized hardware. Furthermore, as the proposed algorithm is based on a rejection-free algorithm, the computational cost is low. In this work, after presenting the details of the proposed algorithm, we report the results of numerical experiments that demonstrate the effectiveness of the proposed feature and algorithm.
Paper Structure (22 sections, 20 equations, 3 figures, 1 algorithm)

This paper contains 22 sections, 20 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Backgrounds of Annealing Machines
  3. Ising model and QUBO formulation
  4. Annealing mechanism
  5. Simulated annealing
  6. Replica exchange method
  7. Comment on computational costs
  8. Proposals of Characterization of Locality and Algorithm with Forced Moves
  9. Outline of the proposal
  10. Characterization of locality
  11. Proposed algorithm
  12. Interpretation of the rule in forced moves
  13. Numerical Experiment
  14. Artificially generated problem
  15. Problem settings
  16. ...and 7 more sections

Figures (3)

  • Figure 1: Conceptual explanations of (a) the conventional Monte Carlo sampling and (b) our proposed algorithm with forced moves. We judge whether the state is in local minima or not. If it falls into the local minima, forced moves are applied.
  • Figure 2: Numerical results for optimization. We solved the same problem $100$ times. The horizontal axis of each figure represents the minimum energy in a trial, and the vertical axis represents the histogram. (a) shows the results of the conventional replica exchange method. (b) corresponds to the proposed method with the control parameter $\alpha = 0.2$. (c) and (d) correspond to cases with $\alpha = 0.4$ and $\alpha = 0.8$, respectively.
  • Figure 3: Numerical results for the knapsack problem. We solved the same problem $100$ times. The horizontal axis of each figure represents the total value of items, and the vertical axis represents the histogram. Note that the bin width for the histogram is adaptively changed. (a) and (b) show results by the conventional replica exchange method for the iteration number of $5,000$ and $500,000$ times, respectively. (c) and (d) are those with the proposed method. The dotted vertical line in each figure (value $= 1024$) means the maximum total value of the problem.