Escaping Saddle Points via Curvature-Calibrated Perturbations: A Complete Analysis with Explicit Constants and Empirical Validation

Abstract

We present a comprehensive theoretical analysis of first-order methods for escaping strict saddle points in smooth non-convex optimization. Our main contribution is a Perturbed Saddle-escape Descent (PSD) algorithm with fully explicit constants and a rigorous separation between gradient-descent and saddle-escape phases. For a function $f:\mathbb{R}^d\to\mathbb{R}$ with $\ell$-Lipschitz gradient and $\rho$-Lipschitz Hessian, we prove that PSD finds an $(\epsilon,\sqrt{\rho\epsilon})$-approximate second-order stationary point with high probability using at most $O(\ell\Delta_f/\epsilon^2)$ gradient evaluations for the descent phase plus $O((\ell/\sqrt{\rho\epsilon})\log(d/\delta))$ evaluations per escape episode, with at most $O(\ell\Delta_f/\epsilon^2)$ episodes needed. We validate our theoretical predictions through extensive experiments across both synthetic functions and practical machine learning tasks, confirming the logarithmic dimension dependence and the predicted per-episode function decrease. We also provide complete algorithmic specifications including a finite-difference variant (PSD-Probe) and a stochastic extension (PSGD) with robust mini-batch sizing.