Finite Difference Gradient Estimation for Logistic Regression: Application to SARS-CoV-2 Genomic Classification

Abstract

Gradient-based optimization conventionally relies on closed-form analytical derivatives, which are unavailable for many modern or non-differentiable model architectures. This paper proposes forward finite difference (FFD) gradient estimation as a derivative-free training alternative and validates it rigorously on logistic regression—a model whose known analytical gradient enables direct verification of the numerical approximation. A formal O(h) error bound is proved and confirmed empirically, showing that gradient direction is faithfully preserved across a wide range of step sizes. The framework is applied to binary genomic classification of SARS-CoV-2 versus non-SARS-CoV-2 coronaviruses using normalized 4-mer frequency profiles. The FFD optimizer achieves classification performance statistically equivalent to analytical gradient descent (F₁ ≥ 0.999), while an ablation study demonstrates that nucleotide composition—not sequence length—drives discrimination. External validation on unseen coronavirus lineages reveals strong generalization except for MERS-CoV, whose phylogenetic proximity to SARS-CoV-2 produces overlapping k-mer signatures. These results establish FFD logistic regression as a principled derivative-free baseline and motivate its extension to architectures where analytical gradients are intractable.