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A black-box optimization method with polynomial-based kernels and quadratic-optimization annealing

Yuki Minamoto, Yuya Sakamoto

TL;DR

Kernel-QA presents a black-box optimization approach that builds analytic surrogates using a second-order polynomial kernel within a QUBO framework, enabling efficient use of Ising machines. By employing a representer-theorem-based surrogate, Woodbury updates, a priori domain-wall encoding for variable conversion, and an exponential transformation of outputs, the method targets high-dimensional, multi-modal landscapes while avoiding overfitting. Across Rosenbrock and Rastrigin benchmarks in real and binary variables up to $d=80$ and $d_B=640$, kernel-QA demonstrates strong final performance and reduced sensitivity to dimensionality compared with Bayesian optimization, with occasional gains from a carefully tuned exploration term. The work highlights a robust, scalable BBO path that leverages Ising-machine optimization, offering practical impact for expensive black-box problems with large decision spaces.

Abstract

We introduce kernel-QA, a black-box optimization (BBO) method that constructs surrogate models analytically using low-order polynomial kernels within a quadratic unconstrained binary optimization (QUBO) framework, enabling efficient utilization of Ising machines. The method has been evaluated on artificial landscapes, ranging from uni-modal to multi-modal, with input dimensions extending to 80 for real variables and 640 for binary variables. The results demonstrate that kernel-QA is particularly effective for optimizing black-box functions characterized by local minima and high-dimensional inputs, showcasing its potential as a robust and scalable BBO approach.

A black-box optimization method with polynomial-based kernels and quadratic-optimization annealing

TL;DR

Kernel-QA presents a black-box optimization approach that builds analytic surrogates using a second-order polynomial kernel within a QUBO framework, enabling efficient use of Ising machines. By employing a representer-theorem-based surrogate, Woodbury updates, a priori domain-wall encoding for variable conversion, and an exponential transformation of outputs, the method targets high-dimensional, multi-modal landscapes while avoiding overfitting. Across Rosenbrock and Rastrigin benchmarks in real and binary variables up to and , kernel-QA demonstrates strong final performance and reduced sensitivity to dimensionality compared with Bayesian optimization, with occasional gains from a carefully tuned exploration term. The work highlights a robust, scalable BBO path that leverages Ising-machine optimization, offering practical impact for expensive black-box problems with large decision spaces.

Abstract

We introduce kernel-QA, a black-box optimization (BBO) method that constructs surrogate models analytically using low-order polynomial kernels within a quadratic unconstrained binary optimization (QUBO) framework, enabling efficient utilization of Ising machines. The method has been evaluated on artificial landscapes, ranging from uni-modal to multi-modal, with input dimensions extending to 80 for real variables and 640 for binary variables. The results demonstrate that kernel-QA is particularly effective for optimizing black-box functions characterized by local minima and high-dimensional inputs, showcasing its potential as a robust and scalable BBO approach.
Paper Structure (20 sections, 18 equations, 11 figures, 1 table)

This paper contains 20 sections, 18 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Method
  3. Expected value of the black-box function
  4. Standard deviation
  5. Variable conversion
  6. Exponential transformation of output values
  7. Assessments
  8. Artificial landscapes
  9. Assessment conditions
  10. Results
  11. General features
  12. Mitigation of curse of dimensionality
  13. Real-variable problems
  14. Binary-variable problems
  15. Effect of $\alpha_{exp}$
  16. ...and 5 more sections

Figures (11)

  • Figure 1: A flow of a typical serial optimization for a black-box objective function.
  • Figure 2: The two-dimensional ($d=2$) landscapes of the test functions, (a) Rosenbrock and (b) Rastrigin functions, in logarithm scale.
  • Figure 3: Evolution of $f(\mathbf{x}_{best, i})$ for (a) Rosenbrock and (b) Rastrigin functions with the real-variable dimension $d=5$ (r5h). Optimization uses kernel-QA (red) and Bayesian optimization (black). Corresponding results of FMQA are also shown in blue as a reference. Note that plots in the negative cycles evaluate $f(\mathbf{x})$ for the initial training dataset.
  • Figure 4: Evolution of the distance between the found and true solutions obtained for Fig. \ref{['fig:rastrigin_kq_bo_fm_d5']}.
  • Figure 5: Evolution of $f(\mathbf{x}_{best, i})$ with various real-variable dimensions $d=5$ (thin solid), $10$ (dashed), $20$ (dotted), $40$ (dash-dotted) and $80$ (solid line), for (a) Rosenbrock and (b) Rastrigin functions. Optimization uses kernel-QA (red) and Bayesian optimization (black). Note that plots in the negative cycles evaluate $f(\mathbf{x})$ for the initial training dataset.
  • ...and 6 more figures