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

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.

Optimization Performance of Factorization Machine with Annealing under Limited Training Data

Black-box (BB) optimization problems aim to identify an input that minimizes the output of a function (the BB function) whose input-output relationship is unknown. Factorization machine with annealing (FMA) is a promising approach to this task, employing a factorization machine (FM) as a surrogate model to iteratively guide the solution search via an Ising machine. Although FMA has demonstrated strong optimization performance across various applications, its performance often stagnates as the number of optimization iterations increases. One contributing factor to this stagnation is the growing number of data points in the dataset used to train FM. It is hypothesized that as more data points are accumulated, the contribution of newly added data points becomes diluted within the entire dataset, thereby reducing their impact on improving the prediction accuracy of FM. To address this issue, we propose a novel method for sequential dataset construction that retains at most a specified number of the most recently added data points. This strategy is designed to enhance the influence of newly added data points on the surrogate model. Numerical experiments demonstrate that the proposed FMA achieves lower-cost solutions with fewer BB function evaluations compared to the conventional FMA.