curvature concept

Definition	The second derivative of a consumer, which the finite difference of equation 0.8 does not read and which sets the error of the linear model at a step. Chapter 0 section 0.5 and chapter 7 section 7.2. Also second derivative, second-order.
Example	For C(x) equal to x squared, the central difference at 0.8 with step 0.6 gives exactly 1.6, and the one-sided difference gives 2.2.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `curvature`, kind concept.
Status	measures [predicted]. Corrections: none recorded.
Defining equation	Book equation 0.8.
Assumptions and scope	The second derivative of a consumer, which the finite difference of equation 0.8 does not read. For a quadratic the second difference recovers it exactly at every step, the error of the linear model at a step is the curvature times the step squared, and the gradient at the two ends of a step differs by the curvature times the step. An affine consumer has none. The reader of a gradient is the optimizer, and its read operator is the loss curvature, which chapter 4’s flip was measured against on a real logistic model with its exact Hessian, anti three hundred of three hundred and the flip in 82 of 300.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:122`, `lean/DataMiningAsObservation/Curvature.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

The second derivative, which the finite difference does not read and which sets the linear model's error.

Equation

Book equation 0.8.

\[g_j\;\approx\;\frac{C(x+h\,e_j)-C(x-h\,e_j)}{2h},\qquad j=1,\dots,d.\]

Conditions

The second derivative of a consumer, which the finite difference of equation 0.8 does not read. For a quadratic the second difference recovers it exactly at every step, the error of the linear model at a step is the curvature times the step squared, and the gradient at the two ends of a step differs by the curvature times the step. An affine consumer has none.
The reader of a gradient is the optimizer, and its read operator is the loss curvature, which chapter 4’s flip was measured against on a real logistic model with its exact Hessian, anti three hundred of three hundred and the flip in 82 of 300.

Conditions are curated in entries.toml rather than read from a record.

Ledger

measures. GO-B-optim-D4 (034 · D4) [predicted]. Optimization — gradient compression, curvature (Hessian) read operator, on a REAL model (logistic regression); optional stretch geometric-observation/claims/LEDGER.md:122.

First stated

Chapter 0 section 0.5 and chapter 7 section 7.2 of Data Mining as Observation, with the curvature reader’s flip in geometric-observation/claims/LEDGER.md row GO-B-optim-D4.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 7 section 7.2	second-order leaf values, logistic curvature	ESL 2e 10.9 to 10.13; XGBoost introduction to boosted trees

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Curvature.lean, theorems second_quad, second_affine, linear_error, gradient_change, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation chapters 0, 3, 4, 7.

Hessian; finite difference; sensitivity; boosting; read operator.

Status

Generated 2026-09-10 by encyclopedia/generate.py; book at observation-data-mining f3914f0; the commit of every record is listed in the encyclopedia’s provenance.

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