Noisy Pairwise-Comparison Random Search for Smooth Nonconvex Optimization

TL;DR

This paper introduces NCRS, a novel direct-search algorithm for high-dimensional nonconvex optimization using noisy pairwise comparisons, which adapts to intrinsic dimension and provides convergence guarantees.

Contribution

The paper proposes NCRS, a new method that leverages random line search to efficiently optimize high-dimensional nonconvex functions with noisy comparisons, reducing dependence on ambient dimension.

Findings

NCRS achieves $ ext{O}(k/(p^{2} ext{epsilon}^{2}))$ complexity under a uniform-margin oracle.

A tie-aware variant of NCRS attains $ ext{O}(k^{2}/ ext{epsilon}^{4})$ complexity in degraded comparison settings.

The method adapts to intrinsic dimension, improving efficiency over classical zeroth-order approaches.

Abstract

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 novel non-convex convergence guarantees for approximate stationarity: under a uniform-margin oracle, 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 $\mathcal{O}(k^{2}/\epsilon^{4})$.