<![CDATA[Gradient-based optimization conventionally relies on closed-form analytical derivatives, which are unavailable for many modern or non-differentiable model architectures. This paper proposes \emph{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 $\mathcal{O}(h)$ 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 ($F_1 \geq 0.999$), 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.]]>
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