# Stage-dependent integer-binary encoding in factorization-machine black-box optimization

arXiv: https://arxiv.org/abs/2606.23188 · HTML: https://mlsanity.com/paper/2606.23188/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 64 references that are in the map.
- First posted on arXiv 22 Jun 2026.

## Builds on (most cited first, 18 of 18)

- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Combinatorial Bayesian Optimization using the Graph Cartesian Product](https://mlsanity.com/paper/1902.00448/) (2019)
- [Bayesian Optimization of Combinatorial Structures](https://mlsanity.com/paper/1806.08838/) (2018)
- [Traffic flow optimization using a quantum annealer](https://mlsanity.com/paper/1708.01625/) (2017)
- [Application of QUBO solver using black-box optimization to structural design for resonance avoidance](https://mlsanity.com/paper/2204.04906/) (2022)
- [Quantum Annealing Applied to De-Conflicting Optimal Trajectories for Air Traffic Management](https://mlsanity.com/paper/1711.04889/) (2017)
- [Portfolio Optimisation Using the D-Wave Quantum Annealer](https://mlsanity.com/paper/2012.01121/) (2020)
- [Black-box optimization for integer-variable problems using Ising machines and factorization machines](https://mlsanity.com/paper/2209.01016/) (2022)
- [A supervised learning approach involving active subspaces for an efficient genetic algorithm in high-dimensional optimization problems](https://mlsanity.com/paper/2006.07282/) (2020)
- [A black-box optimization method with polynomial-based kernels and quadratic-optimization annealing](https://mlsanity.com/paper/2501.04225/) (2025)
- [SWIFT-FMQA: Enhancing Factorization Machine with Quadratic-Optimization Annealing via Sliding Window](https://mlsanity.com/paper/2507.21024/) (2025)
- [Effectiveness of Binary Autoencoders for QUBO-Based Optimization Problems](https://mlsanity.com/paper/2602.10037/) (2026)
- [Annealing-Assisted Column Generation for Inequality-Constrained Combinatorial Optimization Problems](https://mlsanity.com/paper/2406.01887/) (2024)
- [Function Smoothing Regularization for Precision Factorization Machine Annealing in Continuous Variable Optimization Problems](https://mlsanity.com/paper/2407.04393/) (2024)
- [Subsampling Factorization Machine Annealing](https://mlsanity.com/paper/2508.08778/) (2025)
- [Factorization Machine with Quadratic-Optimization Annealing for RNA Inverse Folding and Evaluation of Binary-Integer Encoding and Nucleotide Assignment](https://mlsanity.com/paper/2602.16643/) (2026)
- [Improving FMQA via Initial Training Data Design Considering Marginal Bit Coverage in One-Hot Encoding](https://mlsanity.com/paper/2605.04825/) (2026)

## Abstract (arXiv)

Black-box optimization (BBO) deals with problems where objective functions lack explicit analytical forms and are expensive to evaluate. Factorization machine with quadratic-optimization annealing (FMQA) constructs a surrogate model using a factorization machine (FM) and optimizes it with an Ising machine. Conventional FMQA applies a single integer-binary encoding throughout the optimization process, although the encoding best suited to surrogate learning may differ from the one best suited to Ising-machine solution search. We propose a stage-dependent FMQA framework and derive conversion formulas between one-hot and domain-wall QUBO matrices that preserve the surrogate objective over feasible integer states up to an additive constant. We evaluate the OhDw variant, which employs one-hot encoding for learning and domain-wall encoding for search, on the Rastrigin function with input dimensions N = 2 and 5 and discretization levels q = 61 and 301. Across all conditions, the dominant factor governing optimization performance is the encoding used in the learning stage, with one-hot encoding consistently yielding lower residual errors than domain-wall or binary encoding. The additional benefit of switching to domain-wall encoding for solution search is condition-dependent. For N = 5 and q = 301, OhDw achieves a lower residual error and solutions closer to the global optimum than one-hot-only FMQA, whereas for N = 5 and q = 61 the latter achieves a lower residual error. These results indicate that one-hot encoding in the learning stage is the primary performance driver and that stage-dependent encoding can provide further improvement under finer discretization.