Bayesian Optimization of Robustness Measures Using Randomized GP-UCB-based Algorithms under Input Uncertainty (2504.03172v1)

Abstract: Bayesian optimization based on Gaussian process upper confidence bound (GP-UCB) has a theoretical guarantee for optimizing black-box functions. Black-box functions often have input uncertainty, but even in this case, GP-UCB can be extended to optimize evaluation measures called robustness measures. However, GP-UCB-based methods for robustness measures include a trade-off parameter $\beta$, which must be excessively large to achieve theoretical validity, just like the original GP-UCB. In this study, we propose a new method called randomized robustness measure GP-UCB (RRGP-UCB), which samples the trade-off parameter $\beta$ from a probability distribution based on a chi-squared distribution and avoids explicitly specifying $\beta$. The expected value of $\beta$ is not excessively large. Furthermore, we show that RRGP-UCB provides tight bounds on the expected value of regret based on the optimal solution and estimated solutions. Finally, we demonstrate the usefulness of the proposed method through numerical experiments.

Bayesian Optimization of Robustness Measures Under Input Uncertainty

In the paper "Bayesian Optimization of Robustness Measures Using Randomized GP-UCB-based Algorithms under Input Uncertainty," Inatsu explores advancements in Bayesian optimization (BO) for problems involving black-box functions with inherent input uncertainty. The paper's focal point is the optimization of robustness measures which evaluate the performance of design variables in unpredictable environments. This research provides a nuanced expansion of methods that integrate Gaussian Process Upper Confidence Bound (GP-UCB) frameworks into robustness optimization.

Summary of Methods

The research introduces a pivotal modification to traditional GP-UCB methods, aimed at addressing the conservative blowout of the trade-off parameter $\beta_t$ commonly required for theoretical guarantees. The proposed Randomized Robustness Measure GP-UCB (RRGP-UCB) leverages a probabilistic sampling of $\beta_t$ from a chi-squared distribution. This stochastic aspect circumvents the necessity of manually adjusting $\beta_t$, which theoretically must scale excessively with iterations to ensure convergence.

RRGP-UCB optimizes a wide array of robustness measures, such as expected value, worst-case scenarios, and value-at-risk, by supplanting the deterministic adjustment of GP-UCB parameters with a randomized approach. The algorithm's structure allows for sublinear regrets—demonstrating improved theoretical performance by steering the expected value of regret towards none other but the optimal solution over iterative evaluations.

Theoretical Implications

The theoretical analysis conducted in this paper provides sublinear bounds on regret across several robustness measures, revealing that the expected value of cumulative regret diminishes significantly over time. The bounds established for these regret measures serve as pivotal indicators of efficiency in Bayesian Optimization, particularly in environments where both design and environmental variables are involved in defining the robustness of the solution.

Practical and Theoretical Impact

Practically, RRGP-UCB extends its applicability beyond expectation measures to encompass robustness measures such as the mean absolute deviation and conditional value-at-risk, which are often critical in fields like financial risk management and engineering reliability assessments. The versatility proposed herein supports applications requiring robust decision-making under uncertain conditions, reflecting real-world scenarios where environment variables cannot be precisely controlled.

Theoretically, the paper emboldens the paradigm of utilizing randomized parameters in optimization algorithms to reduce the conservatism inherent in deterministic settings. This suggests potential expansions into more intricate optimization problems multi-objective scenarios or higher-dimensional datasets.

Future Directions

This paper lays a foundation for future explorations into enhanced BO algorithms. One promising avenue is the application of RRGP-UCB in multi-fidelity and multi-stage optimization scenarios where decisions evolve over consecutive stages or levels of fidelity. Moreover, further optimization of parameter $\beta_t$ distributions could refine the balance between exploration and exploitation, thereby addressing high-dimensionality challenges in complex systems.

Conclusion

Inatsu’s work on RRGP-UCB makes a distinct contribution by presenting a methodology tailored to robust optimization under uncertainty—a scenario commonly encountered in practical situations. Thus, for researchers and practitioners involved in Bayesian optimization or those needing efficient methods for decision-making under stochastic circumstances, this paper is a significant step forward in advancing theoretical models and applicable strategies.