The Modern Mathematics of Deep Learning

Finite supremum

Penguin learning project

The project uses 342 complete Palmer penguin records: 240 training-pool and 102 held-out rows. The selected training rows determine feature scaling and model parameters. Source CSV

This project trains weights with softmax, unlike the homepage lab, which uses nearest species averages. Same penguin measurements; different classification methods. Predictions can differ.

Model classes and structure · Describe families of rules, their size and structural properties.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

Consider this concept alongside the project; its source assumptions may differ. These observations do not prove a theorem.

Input: 12 × 2; Held-out evaluation: 102 · #58 · 40.6 mm, 193 mm. Known answer: Adélie; Prediction: Adélie · [≈0.7383, ≈0.2444, ≈0.01737].

Used in these source sections: 6.5 Removing connections

The supremum is the least upper bound of the values. For a finite nonempty set, it is the largest value.

A uniform bound controls every function in the family.

Source formula example

\sup_{f\in F}\|f\|\leq C

illustration · A general supremum can exist without being attained; this finite example has a maximum.

Symbols

$F$: A set of functions.
$\Vert f\Vert$: The norm of a function f.
$C$: A bound.

Source formula example

This source example uses its own inputs and assumptions. Editing it does not change the project.

Edit a value or the code, then run the example. This is a finite demonstration, not a proof of the paper’s theorem.

This example’s data flow

Initial values in code

norms = [1, 2, 3]
C = 3

Calculated here

Result expression: { norms, supremum: Math.max(...norms), C, bounded: norms.every((v) => v <= C) }

Returns

Example result

norms: array (3 items): [1, 2, 3]
supremum: 3
C: 3
bounded: true

JavaScript

Read numbers from input and return the result. JavaScript basics.

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. A general supremum can exist without being attained; this finite example has a maximum.

Source: Inline notation