Noisy Pairwise-Comparison Random Search for Smooth Nonconvex Optimization

We consider minimizing high-dimensional smooth nonconvex objectives using only noisy pairwise comparisons. Unlike classical zeroth-order methods limited by the ambient dimension $d$, we propose Noisy-Comparison Random Search (NCRS), a direct-search method that exploits random line search to adapt to the intrinsic dimension $k \le d$. We establish a novel nonconvex analysis for approximate stationarity: under a uniform-margin oracle with advantage $p$, NCRS attains $\epsilon$-stationarity with complexity $\mathcal{O}(k/(p^{2}\epsilon^{2}))$, explicitly replacing ambient dependence with the intrinsic dimension. Furthermore, we introduce a general tie-aware noise model where comparison quality degrades near ties; for this setting, we prove that a majority-vote variant of NCRS achieves $\epsilon$-stationarity with complexity $\mathcal{O}(k^{2}/\epsilon^{4})$.