The Modern Mathematics of Deep Learning

Activation properties in a theorem

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.

Source assumptions and notation · Interpret symbols and assumptions in a source statement.

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: 2.2 Weight size and room to decide

The theorem uses an activation with Lipschitz constant 1 and a network with output dimension 1.

These conditions restrict the network class under study.

Source formula example

1

illustration · This checks one pair for ReLU; its global Lipschitz property is not proved by the single pair.

Symbols

$1$: The stated Lipschitz constant or output dimension, depending on occurrence.
$ϱ$: The activation 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

a = -2
b = 1

Calculated here

Result expression: { a, b, outputDifference: Math.abs(relu(a) - relu(b)), inputDifference: Math.abs(a - b), outputDimension: 1, }

Returns

Example result

a: -2
b: 1
outputDifference: 1
inputDifference: 3
outputDimension: 1

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 unless a source value is specified. This checks one pair for ReLU; its global Lipschitz property is not proved by the single pair.

Source: Inline notation