A Parameter-Free Zeroth-Order Method with Covariance Matrix Adaptation and Effective Dimension

Zeroth-order optimization methods are essential for solving black-box problems where gradient information is unavailable or expensive to compute. This paper presents POEM-CMA, a novel parameter-free stochastic zeroth-order algorithm that extends the recent POEM method by integrating covariance matrix alignment and the notion of effective dimension. In contrast to traditional zeroth-order approaches that rely on isotropic random directions, POEM-CMA performs anisotropic sampling by constructing a covariance matrix from gradient estimates. This enables the algorithm to focus sampling efforts on the most informative directions. We introduce the use of the empirical effective dimension $d^* = \frac{\operatorname{tr}(\hatΣ)}{λ_{\max}(\hatΣ)}$, which reflects the intrinsic dimensionality of the problem and replaces the ambient dimension in both sampling and complexity analysis. We prove that POEM-CMA achieves a near-optimal convergence rate, requiring only $\tilde{\mathcal{O}}\left(\frac{d^* κ(\hatΣ) L^2 D_{\mathcal{X}}^2}{\varepsilon^2}\right)$ stochastic zeroth-order oracle queries. The method remains fully parameter-free and demonstrates significant improvements over the original POEM in problems with low-rank structure where $d^* \ll d$. Numerical experiments on hinge-loss binary classification tasks using LibSVM datasets confirm the practical superiority of the proposed approach.