The Modern Mathematics of Deep Learning

A limit behind convergence

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.

Vectors and matrices · Combine numbers using vector and matrix operations.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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

As t increases, (1 − x/t) raised to t approaches exp(−x).

The source uses this limit to simplify a repeated decrease expression.

Source formula example

\lim_{t\to\infty}(1-x/t)^{t}=e^{-x}

illustration · A finite comparison is not a proof of the limit.

Symbols

$x$: Fixed real value.
$t$: Positive real argument tending to infinity; the toy uses integers.
$e$: Base of the natural exponential.

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.

xPositive integer t

This example’s data flow

You choose

input.x = 1
input.t = 10

Calculated here

Result expression: { finiteValue: (1 - x / t) ** t, limit: Math.exp(-x) }

x = input.x Uses: input.x
t = input.t Uses: input.t

Returns

Example result

finiteValue: ≈0.3487
limit: ≈0.3679

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy compares one finite t with the limiting value. The source limit is over real t tending to infinity, with x fixed.

Source: Source expression