The Modern Mathematics of Deep Learning

Finite function-class cardinality

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: 1.2.3 Does it work on new examples?

For a finite function class 𝓕, |𝓕| is its number of functions. The uniform probability on the class assigns each function probability 1/|𝓕|.

A finite class has a well-defined uniform distribution.

Source formula example

p(f)=1/|{\mathcal{F}}|

illustration · The example has two distinct named rules; an infinite hypothesis class does not have this finite uniform formula.

Symbols

$\mathcal{F}$: A finite nonempty hypothesis set.
$|\mathcal{F}|$: Its number of functions.
$p(f)$: Probability assigned to f.

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

functions = ["constant zero", "identity"]

Calculated here

Result expression: { functions, cardinality: functions.length, probability: 1 / functions.length }

Returns

Example result

functions: array (2 items): [constant zero, identity]
cardinality: 2
probability: 0.5

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 example has two distinct named rules; an infinite hypothesis class does not have this finite uniform formula.

Source: Inline notation