Complexity guarantees and polling strategies for Riemannian direct-search methods

TL;DR

The paper develops a complexity framework for direct-search methods on smooth Riemannian manifolds, extending the Euclidean cosine measure to tangent spaces and establishing an $\mathcal{O}(\varepsilon^{-2})$ evaluation bound under appropriate PSS quality. It introduces two PSS construction paradigms—intrinsic PSSs built in tangent spaces and projected PSSs from ambient space—and analyzes their impact on the complexity measure, with a detailed sphere-manifold study showing intrinsic PSSs offer provable advantages. Empirical experiments on sphere and subspace manifolds validate the theoretical results, showing intrinsic directions yield better performance, particularly at higher codimensions, and that random rotations can enhance efficiency. The work provides practical guidance for constructing high-quality polling sets on manifolds and points to future extensions to other manifolds and probabilistic direct-search variants. Code and experiments are available on GitHub, enabling reproducibility and broader adoption.

Abstract

Direct-search algorithms are derivative-free optimization techniques that operate by polling the variable space along specific directions forming positive spanning sets (PSSs). When the problem variables are constrained to lie on a Riemannian manifold, polling must be performed along tangent directions. Although Riemannian variants of direct search have already been proposed and endowed with asymptotic guarantees, a proper generalization of PSSs on manifolds remains to be investigated. In particular, a measure of quality for those PSSs is required to obtain complexity bounds for direct search. In this paper, we derive complexity guarantees for a class of Riemannian direct-search techniques, and study two ways of generating positive spanning sets in tangent spaces. We pay particular attention the unit hypersphere case, for which we establish that generating directions directly within the tangent space leads to better complexity properties than projecting PSSs from the ambient space onto the tangent space. Our numerical experiments highlight the impact of dimension and codimension in more general settings.

Figures (7)

• Figure 1: Projection of a positive spanning set $D^{\mathrel{} }$ in $\mathbb{R}^3$ to a tangent space $\mathrm{T}_x\mathcal{M}$. The projection produces close or redundant directions, that affect the cardinality but may not increase the cosine measure significantly.
• Figure 2: Cosine measure of the projected PSS $P_{}(D^{\mathrel{} }() )$ on the sphere $\mathbb{S}^{2} \subset \mathbb{R}^3$.
• Figure 3: Plot of the cosine measure of the projected PSS $P_{}(D^{\mathrel{} }() )$ at various point on the sphere $\mathbb{S}^{n-1} \subset \mathbb{R}^n$
• Figure 4: Data Profiles of PSSs $D^{\mathrel{} }$, $D^{\mathrel{} }$ and $D^{\mathrel{} }$, intrinsic variant (dotted) or projected variant (plain). $m = 4$. Each subplot corresponds t o a different codimension $n-m$. No rotation is applied. Tolerance: $\tau = 10^{-2}$.
• Figure 5: Data Profiles of PSSs $D^{\mathrel{} }$, $D^{\mathrel{} }$ and $D^{\mathrel{} }$, intrinsic variant (dotted) or projected variant (plain). $m = 4$. Each subplot corresponds to a different manifold codimension $n-m$. Rotation is applied. Tolerance: $\tau = 10^{-2}$.

Theorems & Definitions (25)

• Remark 3.1
• Lemma 3.1
• Lemma 3.2
• Proposition 3.1
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Definition 4.1
• Example 4.1: Positive and negative basis
• Example 4.2: Basis and sum of negatives