{"ID":2879530,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.16540","arxiv_id":"2508.16540","title":"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 $ρ$-Lipschitz Hessian, we prove that PSD finds an $(ε,\\sqrt{ρε})$-approximate second-order stationary point with high probability using at most $O(\\ellΔ_f/ε^2)$ gradient evaluations for the descent phase plus $O((\\ell/\\sqrt{ρε})\\log(d/δ))$ evaluations per escape episode, with at most $O(\\ellΔ_f/ε^2)$ 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.","short_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...","url_abs":"https://arxiv.org/abs/2508.16540","url_pdf":"https://arxiv.org/pdf/2508.16540v1","authors":"[\"Faruk Alpay\",\"Hamdi Alakkad\"]","published":"2025-08-22T17:06:28Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.OC\",\"stat.ML\"]","methods":"[]","has_code":false}