The Modern Mathematics of Deep Learning

Training time and width factors

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

Evaluate the source time scale and exponential factor for chosen parameters.

This calculation explains the corresponding source expression.

Source formula example

k\lesssim\sqrt{m/(\eta^{2}n)}

illustration · The numerical comparison does not verify the assumptions, the source neighborhood result, or a convergence guarantee.

Symbols

$n,m$: Positive width and sample count.
$η$: Positive step size.
$c$: Positive constant in the source estimate.
$k$: Iteration number.
$t$: Time scale √(m/(η²n)).
$≲$: Comparison up to a constant.

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.

Width n

This example’s data flow

You choose

input.width = 4

Initial values in code

m = 1
eta = 0.25
c = 1
k = 1

Calculated here

Result expression: { n, m, eta, t, k, chosenConstant: 1, withinChosenMultiple: k <= t, exponential, forbiddenStep, }

n = input.width Uses: input.width
t = Math.sqrt(m / (eta * eta * n)) Uses: m, eta, n
exponential = Math.exp(-c * Math.sqrt(n / m)) Uses: c, n, m
forbiddenStep = m / (c * n) Uses: m, c, n

Returns

Example result

n: 4
m: 1
eta: 0.25
t: 2
k: 1
chosenConstant: 1
withinChosenMultiple: true
exponential: ≈0.1353

… (9 items)

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source assumes the additional local gradient upper estimate and a sufficiently small step size. This code uses chosen toy constants to evaluate the factors; it does not determine the source constants. The source warns that choosing η=m/(cn) cannot justify one-step convergence because the first-order approximation requires sufficiently small η.

Source: Source expression