Power Homotopy for Zeroth-Order Non-Convex Optimizations

The existing method of GS-PowerOpt solves the non-convex optimization problem of the form $\max_{\boldsymbol{x} \in \mathbb{R}^d} f(\boldsymbol{x})$ through maximizing a Gaussian-smoothed surrogate $F_{N,\sigma}(\boldsymbol{\mu}) = \mathbb{E}_{\boldsymbol{x}\sim\mathcal{N}(\boldsymbol{\mu},\sigma^2 I_d)}[e^{N f(\boldsymbol{x})}]$. We analyze the role of the smoothing radius $\sigma>0$ and identify a limitation of the fixed-$\sigma$ design used in GS-PowerOpt. Specifically, $\sigma$ induces an inherent exploration--refinement tradeoff: a larger $\sigma$ improves global exploration and finite-time surrogate optimization, but may distort the location of the surrogate maximizer; in contrast, a smaller $\sigma$ better preserves local structure but can weaken gradient signals away from high-value regions.
To address this limitation, we propose GS-PowerHP, a power-smoothed homotopy method with an incrementally decaying $\sigma$ schedule. The proposed mechanism uses larger smoothing radii in early iterations to maintain informative gradient signals when the iterate is far from high-value regions, and gradually decreases $\sigma$ to improve local refinement near the maximizer. We provide theoretical results showing that this decaying schedule improves the exploration--refinement tradeoff of fixed-$\sigma$ power smoothing. Empirically, GS-PowerHP consistently outperforms the fixed-$\sigma$ baseline and exhibits robust performance across different optimization tasks, including adversarial attacks on ImageNet ($d=150{,}528$), where it substantially improves over other smoothing-based zeroth-order methods.