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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 dd 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, and a descent direction is selected 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 5 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. No experimental validation is included here. A subsequent extended version will provide empirical results, code, and comparisons with existing methods. The algorithms are presented as heuristic tools; rigorous convergence guarantees are left for future work.