# Gradient descent

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

In fixed Euclidean coordinates, a positive step moves opposite the gradient; descent and convergence still require objective and step-size hypotheses.

mko-gradient-descent · algorithm · v0.10.0



## Mathematical statement

For differentiable f:R^d→R and explicit alpha_k>0, unconstrained gradient descent updates x_(k+1)=x_k−alpha_k*grad f(x_k). The exact family fixes coordinate order and a positive diagonal quadratic f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, recalculating every gradient, iterate and objective. A safe finite trace additionally checks strictly0<alpha<2/L with L=max_i d_i and audits each objective comparison. This is not a convergence certificate for arbitrary functions or an unverified infinite sequence. A positive step alone need not descend, a zero gradient may occur at a maximum or saddle, and an unconstrained step is not projected or constrained descent.





## Representative formula (exact TeX)

G(f,x,a)=y





## Assumptions


- Positive diagonal and linear vectors have the same arity and fixed coordinate order; all values are bounded exact rationals.
- The one-step family requires only alpha>0; a safe trace separately requires0<alpha<2/L.
- Only unconstrained deterministic gradient descent is covered, not projection, stochastic updates or line search.





## Symbol scope


- G — gradient-descent update
- f — differentiable objective
- x — current vector
- a — positive step size
- y — next vector





## Intuition, definition, and misconceptions

Declare coordinate order, objective and step size together.

Check each safe diagonal-quadratic step rather than substituting alpha>0 for descent conditions.

Stationarity, local/global optimality and convergence are distinct responsibilities.

alpha>0 alone does not imply descent, and neither a zero gradient nor a finite trace proves global optimality or convergence.





## Adopted derivation guide

[Safe descent steps for a positive diagonal quadratic](https://ocme.evemisslab.com/en/derivations/derivation-mko-gradient-descent-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Gradient vector](https://ocme.evemisslab.com/en/mko/mko-gradient-vector/) — Supply the negative-gradient update.
- [Optimization problem](https://ocme.evemisslab.com/en/mko/mko-optimization-problem/) — Specify the objective and minimization semantics.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — Keep the ordered iterate trace.
- [Real vector space](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/) — Use Euclidean vector updates and inner products.





## Verification boundary

No general convex/nonconvex convergence rate, projection or stochastic update is claimed.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add the L-smooth convex descent lemma, strongly convex rates and projected gradient descent.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/gradient-descent.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-gradient-descent/index.md)