Improving search efficiency via adaptive acquisition function selection in discrete black-box optimization

Abstract: In discrete-variable black-box optimization, the number of candidate solutions grows combinatorially, while each evaluation is often expensive. Therefore, it is important to identify promising solutions efficiently within a limited number of trials. Bayesian Optimization of Combinatorial Structures (BOCS), an existing parametric method, works effectively when only a small amount of data is available. However, as the number of observations increases, BOCS tends to repeatedly propose points that have already been evaluated, which leads to search stagnation. A random-point addition strategy has been proposed to address this issue when an evaluated point is proposed, but it cannot sufficiently exploit information from promising data obtained so far. In this study, we propose a hybrid method that uses BOCS as the main search framework and generates alternative unevaluated points using a Gaussian process only when search stagnation is detected. In the Gaussian-process-based component, multiple Lower Confidence Bound (LCB) acquisition functions are adaptively selected to dynamically control the balance between exploitation and exploration. Numerical experiments using fully connected Quadratic Unconstrained Binary Optimization (QUBO) and Higher-order Unconstrained Binary Optimization (HUBO) as black-box functions show that the proposed method finds solutions with better objective values than the conventional random-point addition method in both settings. Additional analyses show that its effectiveness comes from selecting points that promote search progress within Hamming-distance neighborhoods, rather than simply adding low-energy points near promising solutions. Experiments with sparse surrogate models for quantum annealer applications further suggest the importance of retaining near-fully connected representational capacity.

• The paper proposes a hybrid BOCS with GP-Hedge method that combines a quadratic surrogate with adaptive GP acquisition to overcome learning stagnation in discrete optimization.
• It dynamically switches between BOCS and GP-based acquisition, yielding up to 98% improvement in optimality gap and over 50% reduction in queries on benchmark problems.
• The study highlights the importance of high-fidelity surrogates in resource-constrained settings and offers insights for optimization on quantum and neuromorphic hardware.

Improving Search Efficiency via Adaptive Acquisition Function Selection in Discrete Black-box Optimization

Problem Statement and Theoretical Context

The paper addresses the challenge of discrete-variable black-box optimization, a setting where combinatorial explosion of the search space and high evaluation cost make it vital to identify optimum or near-optimum solutions with few queries. In such high-dimensional, expensive, and discrete settings, effective surrogate modeling and acquisition function design are critical. Parametric approaches such as Bayesian Optimization of Combinatorial Structures (BOCS) leverage quadratic surrogate models for stable learning in low data regimes, but their limited representational capacity leads to learning stagnation and repeated proposals of previously-evaluated points as data accrue. Nonparametric Gaussian Process (GP) surrogates, despite their flexibility, do not consistently surpass parametric BOCS in early to mid-search due to sensitivity to data scarcity and exploration-exploitation trade-off miscalibration.

Methodological Innovation: Hybrid BOCS with Adaptive GP Acquisition

The core contribution is a hybrid optimization routine that dynamically integrates BOCS and adaptive Gaussian-process-based acquisition. The method progresses as follows:

• BOCS is used as the primary optimizer, leveraging its statistical robustness under scarce data for initial exploration and exploitation.
• Upon detection of learning stagnation—manifested as repeated proposals of previously evaluated points—an adaptive GP-based search is invoked.
• The GP surrogate leverages a Hamming kernel suited to binary and combinatorial domains; kernel hyperparameters are adaptively re-estimated per iteration via maximization of the GP marginal likelihood.
• Critically, instead of relying on a fixed acquisition function, the hybrid scheme employs GP-Hedge: a portfolio method that adaptively selects among multiple Lower Confidence Bound (LCB) acquisition functions with distinct exploration weights $K_m$. Arm selection is based on the cumulative reward (progress in minimizing the GP predictive mean) delivered by each acquisition function in prior iterations.

The decision logic ensures efficient use of both models: BOCS exploits structural assumptions when they are advantageous, while GP-Hedge maximizes flexibility when the parametric surrogate reaches expressivity limits.

Experimental Design and Empirical Evaluation

Benchmarks comprise fully connected Quadratic Unconstrained Binary Optimization (QUBO) and Higher-order Unconstrained Binary Optimization (HUBO) problems, both configured with $d=50$ binary dimensions and randomly sampled coefficients. The evaluation protocol measures the relative optimality gap over a fixed budget of evaluations, with global optima computed via Gurobi (QUBO) and via multiple runs of OpenJij's simulated annealing (HUBO).

Comparisons include:

• GP-Hedge only: exclusive use of GP-based adaptive search.
• BOCS with Random: random point proposal on stagnation, reflecting previous remedial strategies.
• BOCS with GP-Hedge: the proposed hybrid, invoking GP-Hedge only during stagnation.

Search performance is quantified both by asymptotic optimality (final gap) and sample efficiency (iteration reduction to achieve equivalent gap of baselines).

Extending relevance to quantum optimization, the effect of sparsifying the BOCS surrogate model is analyzed, reflecting the limited connectivity of current quantum annealing hardware.

Key Results and Quantitative Performance

The hybrid BOCS with GP-Hedge strategy substantially outperforms both standalone GP-Hedge and BOCS with Random in both QUBO and HUBO benchmarks. Specifically:

• For QUBO, the method achieves a 98.07% improvement in the final optimality gap over GP-Hedge only and 96.08% over BOCS with Random.
• For HUBO, improvement is 86.46% and 79.25%, respectively.
• Sample-efficiency gains are pronounced: 53.1% fewer queries required versus GP-Hedge only and 49.1% fewer than BOCS with Random on QUBO; similar results hold for HUBO.

Analysis of search trajectory indicates that the hybrid approach maintains the strongest focus on promising search neighborhoods as measured by Hamming distance to the optimum and relative error, beyond what is achieved by simply proposing points near previously successful solutions.

When the BOCS surrogate is sparsified to reflect restricted quantum annealer topology, the efficacy of the methodology degrades rapidly: as little as 10% sparsification erodes improvements to the level of GP-Hedge-only, indicating the critical role of high-fidelity surrogates for effective information accumulation prior to invoking adaptive GP-based stages.

Ablation with a "SpinFlip" augmentation, which naïvely proposes neighbors of the incumbent in Hamming space, underscores that merely proposing local variants does not suffice; adaptive GP-based selection is necessary for robust progression out of stagnation.

Implications, Limitations, and Future Directions

The results solidify the necessity of dynamically compositional surrogate modeling in discrete black-box optimization, particularly in application domains where both model representational capacity and per-evaluation cost structure the nature of search. The hybridization of BOCS and GP-Hedge directly addresses the trade-off between stability under small-data and expressivity at scale, while the portfolio approach to acquisition function selection ameliorates the need for meticulous hyperparameter tuning, distributing exploratory/exploitative focus according to empirical progress.

For practical deployment on quantum hardware, results imply that methods heavily relying on fully connected surrogate models may be fundamentally constrained by hardware embedding requirements—improved sparse high-capacity surrogates or more advanced graph embedding techniques are critical open directions.

Theoretically, the approach demonstrates the value of adaptive exploitation of accumulated data and dynamic acquisition function portfolios over prior heuristic (e.g., random) stagnation remedies. Future research should examine:

• Broader acquisition function portfolios (e.g., Expected Improvement, Probability of Improvement) and the inclusion of non-GP-based acquisition logic.
• Automatic selection or learning of portfolio hyperparameters.
• Enhanced surrogate model adaptability for resource-limited quantum or neuromorphic hardware.

Conclusion

This work rigorously demonstrates that hybridizing BOCS with adaptive acquisition function selection via GP-Hedge effectively mitigates learning stagnation in discrete black-box optimization tasks, achieving superior final solutions and significant reductions in evaluation cost. The methodology's benefit is contingent on maintaining expressive surrogate models in the parametric BOCS stage. These results inform both algorithmic design and practical considerations for deployment in computationally and physically resource-constrained combinatorial optimization scenarios (2605.10856).