The Ideal Expression Is Not a Local Optimum: A Revisit of EQL with Zero-Point Constraints

Symbolic Regression aims to discover interpretable mathematical expressions from data. Equation Learner (EQL) is a gradient-based method with strong fitting capability and expressive potential, yet it often activates redundant operators as model complexity grows, leading to over-complex expressions and unstable equation recovery. We analyze a gradient residual issue induced by operators that do not vanish at zero, which can prevent the ideal sparse expression from being a local optimum and bias training toward unnecessarily complex structures, making exact recovery nearly unattainable in practice. To address this, we propose EQL-Z, a structurally controllable symbolic regression framework. EQL-Z enforces zero-point constraints via zero-point consistent operator transformations to eliminate residual gradients on silent paths, and performs an incremental small-to-large structure search that grows depth/width from a compact seed under a complexity-penalized validation score. After selecting a compact structure, we optionally apply BFGS fine-tuning to refine coefficients. Experiments on synthetic and real-world datasets show that EQL-Z substantially improves exact equation recovery and in-/out-of-distribution generalization over vanilla EQL, achieving performance close to the best existing symbolic regression baselines. The code is available at https://anonymous.4open.science/r/EQL-Z-BE6C/.