The Modern Mathematics of Deep Learning

A finite learning task

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 Learning from examples

A prediction rule maps an input to an output. A loss takes such a rule and an example, and returns a real score. A learning algorithm takes a sample and returns a rule from a class.

The toy has one input and two constant binary rules; its learning algorithm chooses the first training label.

The domains tell us what each learning operation accepts and returns.

Source formula example

{\mathcal{L}}\colon{\mathcal{M}}({\mathcal{X}},{\mathcal{Y}})\times{\mathcal{Z}}\to{\mathbb{R}}

illustration · The toy uses a finite input and output set. General learning tasks can have infinite spaces and more complex algorithms.

Symbols

$\mathcal{X},\mathcal{Y},\mathcal{Z}$: Input, output and example spaces.
$\mathcal{M}(\mathcal{X},\mathcal{Y})$: A class of measurable maps from input space to output space.
$ℒ$: A loss mapping a rule and an example to a real number.
$\mathcal{A}$: A learning algorithm mapping a sample to a rule.

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.

Training label

This example’s data flow

You choose

input.label = 0

Initial values in code

freshLabel = 1

Calculated here

Result expression: { sample, availableRules: [0, 1], chosenRule: prediction, freshLabel, loss }

sample = [input.label] Uses: input.label
prediction = sample[0] Uses: sample
loss = prediction === freshLabel ? 0 : 1 Uses: prediction, freshLabel

Returns

Example result

sample: array (1 items): [0]
availableRules: array (2 items): [0, 1]
chosenRule: 0
freshLabel: 1
loss: 1

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

A sample S is drawn independently according to the product distribution P_Z^m in the source setting. The toy gives a fixed sample and does not simulate that distribution. A function class may be a subset of all measurable rules.

Source: Source expression