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Optimization by Parallel Quasi-Quantum Annealing with Gradient-Based Sampling

Yuma Ichikawa, Yamato Arai

TL;DR

This paper introduces Parallel Quasi-Quantum Annealing (PQQA), a general-purpose solver for combinatorial optimization that fuses gradient-based continuous relaxation with Quasi-Quantum Annealing and multi-GPU parallelism. By extending the Boltzmann framework with a gamma-controlled entropy term and a parallel run communication mechanism, PQQA achieves scalable exploration and faster convergence, particularly for large-scale problems. Empirical results across MIS, Max Clique, Max Cut, Balanced Graph Partition, and Graph Coloring show PQQA performing on par with or better than iSCO and several learning-based solvers, with superior speed-quality trade-offs on large instances. The work highlights the potential of GPU-accelerated, gradient-based sampling approaches in CO and invites further exploration of learning-based methods under rigorous efficiency comparisons.

Abstract

Learning-based methods have gained attention as general-purpose solvers due to their ability to automatically learn problem-specific heuristics, reducing the need for manually crafted heuristics. However, these methods often face scalability challenges. To address these issues, the improved Sampling algorithm for Combinatorial Optimization (iSCO), using discrete Langevin dynamics, has been proposed, demonstrating better performance than several learning-based solvers. This study proposes a different approach that integrates gradient-based update through continuous relaxation, combined with Quasi-Quantum Annealing (QQA). QQA smoothly transitions the objective function, starting from a simple convex function, minimized at half-integral values, to the original objective function, where the relaxed variables are minimized only in the discrete space. Furthermore, we incorporate parallel run communication leveraging GPUs to enhance exploration capabilities and accelerate convergence. Numerical experiments demonstrate that our method is a competitive general-purpose solver, achieving performance comparable to iSCO and learning-based solvers across various benchmark problems. Notably, our method exhibits superior speed-quality trade-offs for large-scale instances compared to iSCO, learning-based solvers, commercial solvers, and specialized algorithms.

Optimization by Parallel Quasi-Quantum Annealing with Gradient-Based Sampling

TL;DR

This paper introduces Parallel Quasi-Quantum Annealing (PQQA), a general-purpose solver for combinatorial optimization that fuses gradient-based continuous relaxation with Quasi-Quantum Annealing and multi-GPU parallelism. By extending the Boltzmann framework with a gamma-controlled entropy term and a parallel run communication mechanism, PQQA achieves scalable exploration and faster convergence, particularly for large-scale problems. Empirical results across MIS, Max Clique, Max Cut, Balanced Graph Partition, and Graph Coloring show PQQA performing on par with or better than iSCO and several learning-based solvers, with superior speed-quality trade-offs on large instances. The work highlights the potential of GPU-accelerated, gradient-based sampling approaches in CO and invites further exploration of learning-based methods under rigorous efficiency comparisons.

Abstract

Learning-based methods have gained attention as general-purpose solvers due to their ability to automatically learn problem-specific heuristics, reducing the need for manually crafted heuristics. However, these methods often face scalability challenges. To address these issues, the improved Sampling algorithm for Combinatorial Optimization (iSCO), using discrete Langevin dynamics, has been proposed, demonstrating better performance than several learning-based solvers. This study proposes a different approach that integrates gradient-based update through continuous relaxation, combined with Quasi-Quantum Annealing (QQA). QQA smoothly transitions the objective function, starting from a simple convex function, minimized at half-integral values, to the original objective function, where the relaxed variables are minimized only in the discrete space. Furthermore, we incorporate parallel run communication leveraging GPUs to enhance exploration capabilities and accelerate convergence. Numerical experiments demonstrate that our method is a competitive general-purpose solver, achieving performance comparable to iSCO and learning-based solvers across various benchmark problems. Notably, our method exhibits superior speed-quality trade-offs for large-scale instances compared to iSCO, learning-based solvers, commercial solvers, and specialized algorithms.
Paper Structure (47 sections, 5 theorems, 42 equations, 3 figures, 8 tables)

This paper contains 47 sections, 5 theorems, 42 equations, 3 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Combinatorial Optimization.
  4. Energy Based Model (EBM).
  5. Metropolis--Hastings Algorithm.
  6. Simulated Annealing.
  7. Method
  8. Continuous Relaxation Strategy
  9. Quasi-Quantum Annealing
  10. Communication in Parallel Runs
  11. Optimization-Specific Accelerations
  12. Sensitive Transitions.
  13. Related Work
  14. Experiment
  15. Implementation.
  16. ...and 32 more sections

Key Result

Proposition 2.1

As the temperature $T$ approaches infinity, the Boltzmann distribution $P({\bm x}; T)$ converges to a uniform distribution over $\{0, 1\}^{N}$. When the temperature $T=0$, the Boltzmann distribution becomes a uniform distribution over the optimal solutions of Eq. (eq:penarized-loss).

Figures (3)

  • Figure 1: Approximation ratio comparison across different node sizes in train/test graphs.
  • Figure 2: Comparison of SA and single-run QQA (S=1).
  • Figure 3: Ablation study on (left) annealing schedule speed, (middle) initial $\gamma$, and (right) communication strength $\alpha$. The shaded area denotes the standard deviation across five random seeds.

Theorems & Definitions (8)

  • Proposition 2.1
  • Theorem 3.1
  • Theorem A.1
  • proof
  • Proposition A.2
  • proof
  • Proposition B.1
  • proof