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

💡 Research Summary

The paper tackles the problem of minimizing high‑dimensional smooth non‑convex functions when the only information available is noisy pairwise comparisons (i.e., ordinal feedback). Classical zeroth‑order (ZO) methods that rely on function evaluations or finite‑difference gradients typically require a number of queries that scales linearly with the ambient dimension d, which becomes prohibitive for modern models with billions of parameters. The authors observe that many such models actually vary significantly only along a low‑dimensional active subspace of dimension k ≤ d (e.g., an intrinsic manifold or a linear embedding). Leveraging this observation, they propose a simple direct‑search algorithm called Noisy‑Comparison Random Search (NCRS) and analyze its performance under two increasingly realistic noise models.

Algorithm (NCRS).
At iteration t, the algorithm samples a random direction sₜ ∼ N(0, Iₙ) and a step size αₜ. It queries the comparison oracle with the pair (θₜ, θₜ + αₜ sₜ). If the oracle reports that the candidate is not worse (output +1), the candidate is accepted; otherwise the iterate stays unchanged. This “improve‑or‑stay” rule uses only a single bit of information per iteration.

Intrinsic‑dimension model.
The objective is assumed to have the form f(x) = g(Ax), where A ∈ ℝ^{k×d} has full row rank and g : ℝ^{k}→ℝ is smooth. Hence ∇f(x) always lies in the range of Aᵀ, i.e., the active subspace V = range(Aᵀ). The orthogonal projector onto V is P = Aᵀ(AAᵀ)⁻¹A. The key observation is that for any direction s, the true function difference f(x + αs) − f(x) depends only on the projected direction P s. Consequently, the comparison outcome depends only on P s, and components of s in the nullspace of A are invisible to the oracle.

Uniform‑margin oracle (Assumption 1.1).
The first noise model assumes a uniform bias p > 0: for any distinct pair (x, y), the oracle returns the correct ordering with probability at least ½ + p, regardless of the function gap. Under this model, Lemma 2.1 establishes a one‑step expected descent inequality: \