The Modern Mathematics of Deep Learning

A Lipschitz function

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.

Functions and derivatives · Describe maps and how their outputs change.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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: 3.1 A rule based on distance

C is a Lipschitz constant when the change in function value is at most C times the change in input.

The constant controls how quickly a function can change.

Source formula example

C

illustration · The linear function has this bound for all real inputs; checking one pair cannot establish it for a general function.

Symbols

$C$: A nonnegative Lipschitz constant.
$g̃$: A real-valued function.

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

C = 2
a = 1
b = 3

Calculated here

Result expression: { C, a, b, outputDifference: Math.abs(g(a) - g(b)), bound: C * Math.abs(a - b) }

Returns

Example result

C: 2
a: 1
b: 3
outputDifference: 4
bound: 4

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. The linear function has this bound for all real inputs; checking one pair cannot establish it for a general function.

Source: Inline notation