Derivative-Free Emergent Optimization via Local Rules and Cellular Automata

We introduce the Cellular Direct Search (CDS), a derivative-free optimization method inspired by the emergent dynamics of Cellular Automata. CDS partitions the parameter space into a discrete grid where a population of agents evolves according to strictly local rules: a Jealous Neighbor Rule for competitive selection and a Solitary Regeneration Rule for adaptive exploration. From a theoretical perspective, we formally ground CDS in the theory of Directional Direct Search, proving that its local neighborhood constitutes a maximal positive basis. Under Lipschitz smoothness assumptions, we establish a formal guarantee of O(parallel to h parallel to) -approximate first-order stationarity for the interior points reached by the algorithm on a fixed grid of resolution h. Moreover, we interpret the imposed hyperspherical search boundary through the lens of Lagrangian duality, providing an intuitive link to L-2-type regularization under Slater's condition, and derive a heuristic proportionality between grid resolution and gradient-descent learning rates. We evaluate CDS against a comprehensive suite of modern and classical baselines across standardized benchmark problems from BBOB and the complete ABS and Layeb benchmark families, including shifted and rotated instances, as well as on a real-world convolutional neural network hyperparameter optimization task. The experimental results characterize the strengths and limitations of CDS across diverse optimization landscapes. Additional robustness analyses indicate that the observed behavior cannot be explained solely by center-location or coordinate-alignment effects. Furthermore, CDS demonstrates robust behavior in high-dimensional settings, maintaining stable performance even under constrained evaluation budgets. This establishes CDS as a viable and interpretable paradigm for closed-box optimization tasks where gradient information is unavailable or unreliable.