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

2026-06-22

AI Summary

This work addresses the challenge in black-box optimization that a single integer-to-binary encoding struggles to simultaneously satisfy the requirements of surrogate model learning and Ising machine-based search. To resolve this, the authors propose a two-stage encoding framework: during the learning phase, one-hot encoding is employed to enhance surrogate model fidelity, while domain-wall encoding is adopted in the search phase to improve the efficiency of the Ising solver. Crucially, they derive, for the first time, transformation formulas between one-hot and domain-wall QUBO matrices that preserve objective function equivalence, enabling seamless coordination between the two stages. Experimental results demonstrate that the proposed method significantly reduces learning residuals on the Rastrigin function and achieves solution quality closer to the global optimum under high-dimensional, fine-grained settings (e.g., N=5, q=301).

Abstract

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.

Problem

• black-box optimization
• factorization machine
• integer-binary encoding
• stage-dependent encoding
• QUBO

Innovation

• stage-dependent encoding
• factorization machine
• black-box optimization
• QUBO conversion
• Ising machine