Probabilistic Smoothing with Ratio-Monotone Transforms for Global Optimization

2026-05-26

AI Summary

Existing probabilistic smoothing methods suffer from sensitivity to hyperparameters and limited robustness due to their reliance on Gaussian kernels and specific transformations. This work proposes a general smoothing framework that combines symmetric unimodal kernels with ratio-based monotonic transformations, theoretically guaranteeing the preservation of global optima and concentration of stationary points near the true optimum—without requiring a decreasing smoothing schedule. We provide the first explicit complexity bound for stochastic gradient ascent under this framework and introduce a leave-one-out variance-reduced baseline. Empirical evaluations demonstrate that the proposed method achieves superior robustness and competitive performance on high-dimensional benchmark optimization and black-box adversarial attack tasks.

Abstract

Probabilistic smoothing is a standard tool for global optimization, but existing methods rely on Gaussian kernels and specific transforms, often resulting in strong hyperparameter sensitivity and limited robustness. We propose a general smoothing framework that combines flexible symmetric unimodal kernels with monotonic ratio-based transformations. Under mild conditions, we show that the smoothed objective preserves the global maximizer and that all stationary points concentrate near the true optimum for sufficiently large amplification, without requiring a decreasing smoothing schedule. We further provide explicit complexity bounds for stochastic gradient ascent and show that a leave-one-out baseline provably reduces variance. Experiments on high-dimensional benchmarks and black-box adversarial attacks demonstrate improved robustness and competitive performance.

Problem

• probabilistic smoothing
• global optimization
• hyperparameter sensitivity
• robustness
• ratio-monotone transforms

Innovation

• probabilistic smoothing
• ratio-monotone transforms
• symmetric unimodal kernels
• global optimization
• variance reduction