AI summaryⓘ
The authors study a way to solve black-box optimization problems, where it's hard to write down or quickly calculate the function to optimize. They improve an existing approach called FMQA by using different ways to encode variables at different stages: one-hot encoding when learning the model and domain-wall encoding when searching for solutions. Their tests show that the choice of encoding during the learning stage has the biggest impact on performance, with one-hot encoding generally working best. Using different encodings in different stages can sometimes help more, especially with more detailed problem settings. Overall, their work shows that mixing encoding methods intelligently can improve optimization results.
Black-box optimizationFactorization machineQuadratic-optimization annealingIsing machineOne-hot encodingDomain-wall encodingQUBO (Quadratic Unconstrained Binary Optimization)Surrogate modelRastrigin functionDiscrete optimization
Authors
Ryo Ogawa, Mayumi Nakano, Yuya Seki, Shu Tanaka
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.