{"ID":2839320,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.15019","arxiv_id":"2511.15019","title":"Non-Convex Self-Concordant Functions: Practical Algorithms and Complexity Analysis","abstract":"We extend the standard notion of self-concordance to non-convex optimization and develop a family of second-order algorithms with global convergence guarantees. In particular, two function classes -- \\textit{weakly self-concordant} functions and \\textit{$F$-based self-concordant} functions -- generalize the self-concordant framework beyond convexity, without assuming the Lipschitz continuity of the gradient or Hessian. For these function classes, we propose a regularized Newton method and an adaptive regularization method that achieve an $ε$-approximate first-order stationary point in $O(ε^{-2})$ iterations. Equipped with an oracle capable of detecting negative curvature, the adaptive algorithm can further attain convergence to an approximate second-order stationary point. Our experimental results demonstrate that the proposed methods offer superior robustness and computational efficiency compared to cubic regularization and trust-region approaches, underscoring the broad potential of self-concordant regularization for large-scale and neural network optimization problems.","short_abstract":"We extend the standard notion of self-concordance to non-convex optimization and develop a family of second-order algorithms with global convergence guarantees. In particular, two function classes -- \\textit{weakly self-concordant} functions and \\textit{$F$-based self-concordant} functions -- generalize the self-concor...","url_abs":"https://arxiv.org/abs/2511.15019","url_pdf":"https://arxiv.org/pdf/2511.15019v2","authors":"[\"Donald Goldfarb\",\"Lexiao Lai\",\"Tianyi Lin\",\"Jiayu Zhang\"]","published":"2025-11-19T01:36:14Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}