Convergence Analysis of Gradient Descent Optimization

Abstract 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/template code project/src/optimizer.py; experiments and figures are orchestrated by projects/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), valid