The Modern Mathematics of Deep Learning

Staying near initialization

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.

Optimization theory · Study convergence and loss geometry under 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: 5.2 When parameters move only a little

Subtract initialization from the current parameter vector and check whether its Euclidean length is at most one.

The source’s local gradient estimates are used while iterates remain in this neighborhood.

Source formula example

\|\Theta^{(k)}-\Theta\|\leq 1

illustration · Checking one iterate does not establish the condition for an entire training trajectory.

Symbols

$Θ$: Initialization vector.
$Θ⁽ᵏ⁾$: Parameter vector at iteration k.
$\Vert ·\Vert$: Euclidean norm.

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.

First coordinate

This example’s data flow

You choose

input.displacement = 0.5

Initial values in code

initial = [0, 0]

Calculated here

Result expression: { initial, current, distance, withinUnitBall: distance <= 1 }

current = [input.displacement, 0] Uses: input.displacement
distance = Math.hypot(current[0] - initial[0], current[1] - initial[1]) Uses: current, initial

Returns

Example result

initial: array (2 items): [0, 0]
current: array (2 items): [0.5, 0]
distance: 0.5
withinUnitBall: true

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

This evaluates the condition at one invented iterate; the source requires it for every specified k.

Source: Source expression