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

Black-box optimization (BBO) often relies on surrogate models to approximate expensive objective functions. FMQA is a popular BBO framework that uses Factorization Machines (FM) to build these surrogates and Ising machines to search for optimal solutions. A critical challenge is that the binary encoding used to represent integer or continuous variables affects both the surrogate's learning accuracy and the Ising machine's search efficiency. This paper investigates whether using different encodings for the learning and search stages—a stage-dependent approach—can outperform the conventional method of using a single, fixed encoding.

The authors propose a framework that employs one-hot encoding during the machine learning stage and domain-wall encoding during the solution search stage. To enable this, they derive mathematical conversion formulas between one-hot and domain-wall QUBO matrices that preserve the surrogate objective landscape over feasible integer states. They evaluate this "OhDw" (One-hot for learning, Domain-wall for search) variant against standard approaches using the Rastrigin function, testing various input dimensions and discretization levels to isolate the impact of encoding on each stage of the optimization process.

The study reveals that the encoding choice in the machine learning stage is the dominant factor for overall performance, with one-hot encoding consistently producing lower residual errors than domain-wall or binary encoding. While domain-wall encoding is known for its search efficiency, the benefit of switching to it during the search stage is not universal. The proposed stage-dependent strategy (OhDw) yields superior results—achieving lower residual errors and solutions closer to the global optimum—specifically when the problem involves finer discretization levels. This suggests that while one-hot encoding is essential for building an accurate surrogate, stage-dependent switching can refine the search process in complex, high-resolution landscapes.