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

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:ℝd→ℝ with ℓ-Lipschitz gradient and ρ-Lipschitz Hessian, we prove that PSD finds an (ε,√ρε)-approximate second-order stationary point with high probability using at most O(ℓDeltaf/ε2) gradient evaluations for the descent phase plus O((ℓ/√ρε)log(d/δ)) evaluations per escape episode, with at most O(ℓDeltaf/ε2) 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.