Controlled Perturbation Algorithms for Saddle Point Escape of Generic Non-convex Optimization Problems (Algorithm Description – Version 1.1)

We introduce the Controlled Perturbation Algorithm (CPA) for escaping saddle points in generic deterministic non‑convex optimization problems. CPA requires only 2 gradient computations per iteration, incurring a cost of $O(d)$ where $d$ is the number of degrees of freedom. Its key idea is elegant: for each coordinate, two adaptive perturbations are applied; their directional derivatives are evaluated; but instead of using these derivatives to update the parameters directly—as in gradient descent (GD)—their signs are used to determine the descent or ascent direction semi‑deterministically, all without computing second‑ or higher‑order derivatives. Additionally, we define the Non‑Descent Direction Approximation (NDDA) index, a computationally cheap heuristic that indicates proximity to a local minimum.

Since Version 1.1 of this preprint, building on CPA, we present its extension—the 3‑Gradient‑Probe Controlled Perturbation Algorithm (3GCPA)—which is designed for both deterministic and probabilistic optimization problems (e.g., machine learning and neural networks). 3GCPA uses 3 gradient computations per iteration, still $O(d)$, thereby preserving linear scalability. By merging gradient‑based optimization with a finite‑element‑like probing strategy, 3GCPA effectively overcomes the challenges posed by stochasticity in probabilistic models, offering robust performance where pure CPA may struggle.

This note is a preliminary algorithmic description intended to establish priority. The algorithms are presented as heuristic tools; rigorous convergence guarantees are left for future work. Some experimental results for deterministic optimization problems are included in Sections 5 and 6 of the following preprint

https://doi.org/10.5281/zenodo.20083397

while the performance of CPA and 3GCPA at experiments of probabilistic ones (like machine learning and deep learning) are still under progress.

The source code is provided with that preprint. If any errors are discovered, the author would appreciate being notified by email at khcheng920911@gmail.com