The Modern Mathematics of Deep Learning

A sample of examples

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.

Classification predicts three species; Ridge predicts body mass in grams using the same observations. These are separate models. Training predictions feed the objective; 102 separate test predictions never update the weights.

Observations and sampling · Describe examples, targets and samples.

The selected rows supply measurements and known answers to training; held-out IDs remain separate.

TaskTraining rowsTraining stepsPenalty αHeld-out row ID

Observations

Input: Training: 12 / 240
Output: #19: Bill length 34.4 mm, Flipper length 184 mm

Project operation

The project calculation uses these records; the source formula and example follow below. Code edits run locally; project settings travel between pages.

Observations → Observations → Features

Project input

Showing first three entries; the operation receives the complete array. Rounded preview; the operation uses full precision.

input.rows · Observations: [Id: 19 · Bill length (mm): 34.4 · Flipper length (mm): 184 · Mass (g): 3325 · Species: Adélie, Id: 307 · Bill length (mm): 40.9 · Flipper length (mm): 187 · Mass (g): 3200 · Species: Chinstrap, Id: 259 · Bill length (mm): 41.7 · Flipper length (mm): 210 · Mass (g): 4700 · Species: Gentoo] … (12 entries)

JavaScript

Result

[Row ID: 19 · Bill length (mm): 34.4 · Flipper length (mm): 184 · Mass (g): 3325 · Species: Adélie, Row ID: 307 · Bill length (mm): 40.9 · Flipper length (mm): 187 · Mass (g): 3200 · Species: Chinstrap, Row ID: 259 · Bill length (mm): 41.7 · Flipper length (mm): 210 · Mass (g): 4700 · Species: Gentoo] … (12 entries)

Used in these source sections: 1.2 Learning from examples · 1.2.1 Finding better settings · 1.2.3 Does it work on new examples? · More source sections

A sample s is an ordered sequence of m examples from the data space 𝒵. 𝒵ᵐ is the set of all such sequences of length m.

The learning algorithm receives the sample.

Source formula example

{s}=(z^{(i)})_{i=1}^{m}\in{\mathcal{Z}}^{m}

illustration · The finite example checks one sample; the mathematical data space can be infinite.

Symbols

$𝒵$: The data space.
$s$: An ordered sample from 𝒵.
$m$: Its positive integer length.
$z⁽ⁱ⁾$: Its i-th example.

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

Z = [0, 1]
s = [0, 1, 1]

Calculated here

Result expression: { Z, s, m: s.length, valid: s.every((z) => Z.includes(z)) }

Returns

Example result

Z: array (2 items): [0, 1]
s: array (3 items): [0, 1, 1]
m: 3
valid: 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. The finite example checks one sample; the mathematical data space can be infinite.

Source: Inline notation