Overview
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
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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
Overview source: Curated paper metadata.
Methods and contributions
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Findings and contributions
- Four of the six grid step sizes converged; the non-converged runs either hit the iteration cap at small α or were unstable when |1−α| ≥ 1.
- α=1.0 reached the optimum in one iteration for this unit-Hessian problem, the fastest configuration.
- The paper reports a stability boundary at α=2: α<2 converges and α≥2 diverges for the unit-Hessian problem.
- In the dimensional benchmark, iterations to convergence rose only modestly (219 to 238) across two decades of dimension.
- The author states the scientific claims are textbook material and the non-standard contribution is procedural (config-driven figures, CSV and manuscript numbers).
Methods
- Fixed-step gradient descent on a 1-D convex quadratic (A=1, b=1) — Runs fixed-step gradient descent on f(x)=½xᵀAx−bᵀx with A=[1], b=[1], analytic optimum x*=1, f(x*)=−0.5.
- Six-point step-size grid from α=0.01 to α=2.5 — Sweeps six fixed step sizes spanning conservative, near-optimal, aggressive and divergent regimes, with a gradient-norm tolerance and iteration cap.
- Comparison to scalar contraction factor ρ(α)=|1−α| — Relates empirical iteration counts and error decay to the contraction factor of the linear error recurrence for the unit-Hessian case.
- Stability grid (8 starts × 6 step sizes) and dimensional scaling benchmark — Evaluates accuracy over 48 start/step-size combinations and separately times gradient_descent() on identity-Hessian quadratics of increasing dimension.
- Zero-mock test suite with ≥90% coverage gate and variable-injected manuscript — Tests src/ without mocks under a CI coverage gate; results are injected into the manuscript from the analysis CSV via placeholders.
Summary sources: Paper metadata and evidence · Extracted source text.
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Citation
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