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Function Smoothing Regularization for Precision Factorization Machine Annealing in Continuous Variable Optimization Problems

Katsuhiro Endo, Kazuaki Z. Takahashi

TL;DR

The paper addresses noise in the Hamiltonian surface generated by FMQA when continuous variables are embedded as binary variables. It introduces Function Smoothing Regularization (FSR), yielding the loss $L' = L + \lambda_{\rm SR} \left[ \sum_{(p,q) \in \mathcal{A}} \| \boldsymbol{v}_p - \boldsymbol{v}_q \|^2 + (b_p - b_q)^2 \right]$, which enforces smooth updates across adjacent parameter blocks and stabilizes learning of $H_{ m FM}$. Empirically, FSRFM improves both the fidelity of $H_{ m FM}$ to simple toy Hamiltonians and the generalization to interactive CV problems (as quantified by $R^2$), and it enables more efficient parameter estimation in a Marmottant-based nanophysics model via FMQA, achieving near-minimum values in roughly half the steps of naive FMQA. The approach preserves the advantages of quantum annealing for black-box optimization and extends FMQA to a wider class of continuous-variable optimization tasks, with practical impact on material design and nanophysics applications.

Abstract

Solving continuous variable optimization problems by factorization machine quantum annealing (FMQA) demonstrates the potential of Ising machines to be extended as a solver for integer and real optimization problems. However, the details of the Hamiltonian function surface obtained by factorization machine (FM) have been overlooked. This study shows that in the widely common case where real numbers are represented by a combination of binary variables, the function surface of the Hamiltonian obtained by FM can be very noisy. This noise interferes with the inherent capabilities of quantum annealing and is likely to be a substantial cause of problems previously considered unsolvable due to the limitations of FMQA performance. The origin of the noise is identified and a simple, general method is proposed to prevent its occurrence. The generalization performance of the proposed method and its ability to solve practical problems is demonstrated.

Function Smoothing Regularization for Precision Factorization Machine Annealing in Continuous Variable Optimization Problems

TL;DR

The paper addresses noise in the Hamiltonian surface generated by FMQA when continuous variables are embedded as binary variables. It introduces Function Smoothing Regularization (FSR), yielding the loss , which enforces smooth updates across adjacent parameter blocks and stabilizes learning of . Empirically, FSRFM improves both the fidelity of to simple toy Hamiltonians and the generalization to interactive CV problems (as quantified by ), and it enables more efficient parameter estimation in a Marmottant-based nanophysics model via FMQA, achieving near-minimum values in roughly half the steps of naive FMQA. The approach preserves the advantages of quantum annealing for black-box optimization and extends FMQA to a wider class of continuous-variable optimization tasks, with practical impact on material design and nanophysics applications.

Abstract

Solving continuous variable optimization problems by factorization machine quantum annealing (FMQA) demonstrates the potential of Ising machines to be extended as a solver for integer and real optimization problems. However, the details of the Hamiltonian function surface obtained by factorization machine (FM) have been overlooked. This study shows that in the widely common case where real numbers are represented by a combination of binary variables, the function surface of the Hamiltonian obtained by FM can be very noisy. This noise interferes with the inherent capabilities of quantum annealing and is likely to be a substantial cause of problems previously considered unsolvable due to the limitations of FMQA performance. The origin of the noise is identified and a simple, general method is proposed to prevent its occurrence. The generalization performance of the proposed method and its ability to solve practical problems is demonstrated.
Paper Structure (13 sections, 15 equations, 7 figures)

This paper contains 13 sections, 15 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Methodology
  3. Factorization Machine Quantum Annealing
  4. One-hot representation for continuous values
  5. Continuous Variable Optimization Problems
  6. Simple Toy Hamiltonian
  7. Interactive Toy Hamiltonian
  8. Hamiltonian to predict nanophysical properties of contrast agents
  9. Results and Discussion
  10. Noisy energy surface and the function smoothing regularization to resolve it
  11. Generalization performance of the function smoothing regularization
  12. Prediction of nanophysical properties of contrast agents
  13. Conclusions

Figures (7)

  • Figure 1: FMQA flow.
  • Figure 2: Function surfaces of (a) $H_1$, and (b) $H_{\rm FM}$ after 16 steps by naive FM.
  • Figure 3: Function surfaces obtained by (a)-(d) naive FM immediately after steps 1, 2, 8, and 16, respectively, and by (e)-(h) FSRFM immediately after steps 1, 2, 8, and 16, respectively.
  • Figure 4: Function surfaces obtained by minimizing the loss functions (a) $L'$ and (b) $L"$ over 16 steps.
  • Figure 5: $H_2$-$H_{\rm FM}$ plots of test dataset for FM rank of 16.
  • ...and 2 more figures