Truncated Riemannian (1+1)-ES for Black-Box Optimization with Intrinsic Dimension Guarantees

$(1+1)$-evolution strategy (ES) is a classic variant of evolution strategy for black-box optimization via direct and random search while keeping only a single solution per generation in $\mathbb{R}^n$. Despite convergence with different step-size adaptations, it often remains impractical due to dependence on the unfavorable high dimension of the search space $\mathbb{R}^n$. In fact, many modern optimization problems are naturally constrained to low-dimensional manifolds embedded in very high-dimensional ambient spaces. To see the potential of intrinsic dimensionality reduction for improving practicality, we consider the truncated Riemannian $(1+1)$-ES on a compact Riemannian submanifold $\mathcal{M}\subset\mathbb{R}^n$ and establish its non-asymptotic stationarity guarantee at the standard nonconvex zeroth-order rate of $\widetilde{\mathcal{O}}(\varepsilon^{-4})$ function evaluations. The explicit Gaussian second-moment term in the bound scales with the intrinsic tangent dimension $d=\dim\mathcal{M}$ rather than the ambient dimension $n$, while the remaining truncation dependence is captured by an explicit small-ball drift constant $\alpha_d(R/\sigma_0)$. To the best of our knowledge, this gives the first non-asymptotic stationarity analysis for a comparison-based ES-type method on compact nonconvex Riemannian submanifolds under localized pullback smoothness. Numerical illustrations on sparse PCA over spheres and smooth-surrogate stationarity diagnostics illustrate behavior consistent with the theory and its limitations, while additional experimental studies on truncation radius diagnostics, black-box tuning, and classical evolutionary computation benchmarks support the practical relevance of geometry-aware comparison-based search under limited function-evaluation budgets.