The Modern Mathematics of Deep Learning

A network hypothesis class

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 hypothesis class contains all functions generated as the network parameters vary.

The class specifies which predictors the learning algorithm may select.

Source formula example

{\mathcal{F}}_{{{a}}}=\big\{\Phi_{{a}}(\cdot,\theta)\colon\theta\in{\mathbb{R}}^{P(N)}\big\}

illustration · Three sampled functions are not the entire infinite class.

Symbols

$Fₐ$: Set of functions from architecture a.
$Φₐ$: Network function with architecture a.
$θ$: Real network parameter vector.

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.

Input

This example’s data flow

You choose

input.x = 2

Initial values in code

weights = [0, 1, 2]

Calculated here

Result expression: { weights, outputs: weights.map((w) => w * x) }

x = input.x Uses: input.x

Returns

Example result

weights: array (3 items): [0, 1, 2]
outputs: array (3 items): [0, 2, 4]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy lists three members of the real-parameter class fθ(x) = θx. The source class includes all real parameter vectors.

Source: Source expression