Datasets:

jkdkr2439
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optimization-curve-convexity-heuristic

This repository contains a technical note proposing a conservative step-size heuristic for gradient descent on convex, L-smooth objectives, evaluated under a discrete optimization-curve convexity diagnostic.

This is not a theorem-driven work.
It is a heuristic + diagnostic + reproducible evaluation protocol.

Core idea (plain language)

Gradient descent guarantees monotone decrease of the objective when
[ 0 < \eta < \frac{2}{L}, ] but this does not guarantee that the sequence of objective values looks “convex” or smooth when viewed as a discrete curve.

This repo introduces a simple diagnostic: [ \Delta^2 g_k = g_{k-1} - 2g_k + g_{k+1}, \quad g_k = f(x_k), ] and empirically studies how often this quantity becomes negative under different step sizes.

Based on experiments, we propose: [ \eta \approx \frac{\varphi}{L}, \quad \varphi = \frac{1+\sqrt{5}}{2}, ] as a conservative, memorable engineering heuristic.

Explicit scope and non-claims (important)

What is claimed

A well-defined discrete diagnostic on the optimization trajectory.
An empirically conservative step-size heuristic relative to more aggressive normalized step sizes.
A reproducible evaluation protocol.

What is not claimed

❌ No universal guarantee for all convex L-smooth functions.
❌ No claim that the golden ratio is optimal.
❌ No claim about convergence rates or global optimality.
❌ No replacement for classical convergence theory.

If you are looking for a theorem, this is not it.

paper.pdf
Golden-Ratio Step Size as a Conservative Heuristic for Discrete Optimization-Curve Convexity in Gradient Descent
(optional) paper.tex
LaTeX source for transparency and reuse.