The Modern Mathematics of Deep Learning

A union over network widths

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: 1.2.2 Can the network express the answer?

A union over widths includes every shallow network with any positive integer number of units.

This example makes the source statement computable on a finite model.

Source formula example

\widetilde{{\mathcal{F}}}\coloneqq\bigcup_{n\in{\mathbb{N}}}{\mathcal{F}}_{((d,n,1),\varrho)}

illustration · No finite list or slider enumerates the entire infinite union.

Symbols

$F̃$: Union of shallow-network classes over all widths.
$n$: Positive hidden-unit count.
$d$: Input dimension.
$ϱ$: 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.

Hidden unitsInput

This example’s data flow

You choose

input.width = 2
input.x = 1

Calculated here

Result expression: { width: n, weights, output: weights.reduce((s, w, i) => s + w * features[i], 0), }

n = input.width Uses: input.width
weights = Array.from({ length: n }, (_, i) => (i === 0 ? 1 : 0)) Uses: n
features = Array.from({ length: n }, () => Math.max(0, input.x)) Uses: n, input.x

Returns

Example result

width: 2
weights: array (2 items): [1, 0]
output: 1

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy uses one-dimensional ReLU features and selects one finite width; changing the slider selects another member of the union.

Source: Source expression