The Modern Mathematics of Deep Learning

Covering numbers

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.

Generalization and risk bounds · Relate sample behavior to population risk under stated assumptions.

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?

A covering number Nα(𝓜) is the smallest number of radius-α balls needed to cover 𝓜 under the specified metric.

Covering numbers quantify the size of a set at a chosen scale.

Source formula example

\log(N_{\alpha}({\mathcal{M}}))

illustration · The listed centers cover the finite set, but this code does not prove that the count is minimal when arbitrary centers are allowed.

Symbols

$\mathcal{M}$: A metric space or set being covered.
$α$: Positive ball radius.
$Nα(\mathcal{M})$: The smallest number of radius-α balls needed to cover 𝓜.

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

M = [0, 1, 2]
alpha = 0.75
centers = [0, 1, 2]

Calculated here

Result expression: { M, alpha, centers, covers: M.every((x) => centers.some((c) => Math.abs(x - c) < alpha)), count: centers.length, logCount: Math.log(3), }

Returns

Example result

M: array (3 items): [0, 1, 2]
alpha: 0.75
centers: array (3 items): [0, 1, 2]
covers: true
count: 3
logCount: ≈1.099

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 listed centers cover the finite set, but this code does not prove that the count is minimal when arbitrary centers are allowed.

Source: Inline notation