The Modern Mathematics of Deep Learning

VC dimension on two inputs

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 Learning from examples

A class shatters m inputs when it realizes all 2ᵐ binary labelings.

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

Source formula example

\operatorname{VCdim}({\mathcal{F}})\coloneqq\sup\big\{m\in{\mathbb{N}}\cup\{0\}\colon\operatorname{growth}(m,{\mathcal{F}})=2^{m}\big\},

illustration · The result applies to the chosen finite class, not every neural network class.

Symbols

$VCdim(F)$: Largest shattered-set size, or supremum if unbounded.
$growth(m,F)$: Maximum number of binary patterns realized on m inputs.
$m$: Nonnegative integer set size.

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

classPatterns = [ [1, 1], [1, -1], [-1, 1], [-1, -1], ]

Calculated here

Result expression: { growth1, growth2, vcDimension: 2 }

growth1 = new Set(classPatterns.map((p) => p[0])).size Uses: classPatterns
growth2 = new Set(classPatterns.map((p) => p.join(","))).size Uses: classPatterns

Returns

Example result

growth1: 2
growth2: 4
vcDimension: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy domain has exactly two inputs and the class contains all four labelings. It shatters both inputs; the domain has no three distinct inputs. This proves its VC dimension is 2.

Source: Source expression