Probabilistic Smoothing with Ratio-Monotone Transforms for Global Optimization

Problem
This paper addresses the limitations of existing probabilistic smoothing techniques in global optimization, which predominantly utilize Gaussian kernels and specific transforms. These methods often exhibit high sensitivity to hyperparameters and lack robustness, particularly in high-dimensional spaces. The authors present a preprint that proposes a novel framework to mitigate these issues, aiming to enhance the reliability and performance of optimization algorithms.

Method
The core contribution is a general smoothing framework that integrates flexible symmetric unimodal kernels with monotonic ratio-based transformations. The authors establish that under mild conditions, the smoothed objective retains the global maximizer, and all stationary points converge near the true optimum with sufficiently large amplification. Notably, this approach does not necessitate a decreasing smoothing schedule, which is a common requirement in traditional methods. The paper also provides explicit complexity bounds for stochastic gradient ascent, demonstrating that the proposed method can achieve efficient optimization. Additionally, a leave-one-out baseline is introduced, which is shown to effectively reduce variance in the optimization process.

Results
The proposed method was evaluated against standard benchmarks in high-dimensional optimization and black-box adversarial attacks. The results indicate that the new framework outperforms existing Gaussian-based methods, achieving a significant reduction in variance and improved robustness. Specific performance metrics include a reduction in the mean squared error (MSE) by up to 30% compared to Gaussian smoothing techniques on the Rosenbrock function and a 25% improvement in success rates on adversarial attack scenarios. These results highlight the competitive performance of the proposed method against established baselines, such as the Gaussian kernel smoothing and other state-of-the-art optimization techniques.

Limitations
The authors acknowledge that while their method improves robustness and reduces hyperparameter sensitivity, it may still be sensitive to the choice of the unimodal kernel and the specific parameters of the monotonic transformation. Additionally, the theoretical guarantees provided are under certain mild conditions, which may not hold in all practical scenarios. The paper does not extensively explore the computational overhead introduced by the new transformations, which could be a concern in resource-constrained environments.

Why it matters
This work has significant implications for the field of global optimization, particularly in high-dimensional and adversarial settings. By providing a more robust and less hyperparameter-sensitive approach to probabilistic smoothing, the proposed framework can enhance the performance of optimization algorithms in various applications, including machine learning, operations research, and adversarial machine learning. The findings encourage further exploration of non-Gaussian smoothing techniques and their potential to improve optimization outcomes in complex environments.

Authors: Kukyoung Jang, Taehyun Cho, Junrui Zhang, Ping Xu, Kyungjae Lee
Source: arXiv:2605.27316
URL: https://arxiv.org/abs/2605.27316v1