# Non-Convex Self-Concordant Functions: Practical Algorithms and Complexity Analysis

arXiv: https://arxiv.org/abs/2511.15019 · HTML: https://mlsanity.com/paper/2511.15019/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Nov 2025.

## Built on by (most cited first, 1 of 1)

- [From Non-Convex Self-Concordant Regularization to Scalable Quasi-Newton Training of PINNs](https://mlsanity.com/paper/2608.04206/) (2026)

## Abstract (arXiv)

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 -- textitweakly self-concordant functions and textitF-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.