Convergence Analysis of Gradient Descent Optimization

This paper presents a convergence study of fixed-step gradient descent on a convex quadratic, framed as the computational exemplar of the Research Project Template (https://github.com/docxology/template). The implementation lives in projects/templates/template_code_project/src/optimizer.py; experiments and figures are orchestrated by projects/templates/template_code_project/scripts/optimization_analysis.py and hydrated into the manuscript through scripts/z_generate_manuscript_variables.py, so tables and prose track output/data/optimization_results.csv after every pipeline run. We evaluate 6 step sizes from $\alpha = 0.01$ to $\alpha = 2.5$, spanning conservative, near-optimal, aggressive, and divergent regimes for a unit Hessian model. The build chain exercises template infrastructure end-to-end: scientific helpers (infrastructure.scientific.stability, infrastructure.scientific.benchmarking), validation, rendering (infrastructure/rendering/pdf_renderer.py), and reporting. Accessibility-oriented plotting defaults (colourblind-safe palette, 300 dpi exports) are centralized in src/figures/ and src/analysis/. Contributions are methodological and architectural. On the methods side, we relate empirical iteration counts and error decay to the scalar contraction factor $\rho(\alpha) = |1-\alpha|$ and document cases where runs hit $N_{\max} = 1000$ before meeting the gradient tolerance. On the architecture side, we demonstrate a zero-mock test suite on project src/ (see test_optimizer.py (https://github.com/docxology/template/blob/main/projects/templates/template_code_project/tests/test_optimizer.py)), automated six-figure analysis, and reproducibility metadata (configuration hash, artifact counts) injected into . Results (this configuration): 4 of 6 grid points report converged=True in the CSV; non-convergent rows flag either slow progress at small $\alpha$ under the iteration cap or instability when $|1-\alpha| \geq 1$. The analytical minimizer remains $x^\ast = 1.0$ with $f(x^\ast) = -0.5$ for the configured $(A,b)$. Keywords: optimization algorithms, gradient descent, convergence analysis, numerical methods, mathematical programming, reproducible research, infrastructure automation --- Associated artifacts GitHub release: v2.5.2 (https://github.com/docxology/template_code_project/releases/tag/v2.5.2) DOI: https://doi.org/10.5281/zenodo.20417136 Zenodo: https://zenodo.org/records/20417136 PDF SHA-256: cd54b95893501467503fab2c4b432573306bc94f7040085550beb87d094b4e50