The Modern Mathematics of Deep Learning

Approximating a random quantity by steps

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.

Approximation and depth · Study approximation error and network size under stated conditions.

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: 4.3 Structure from an equation

The source describes an approximation built by composing L transformations. Each step receives the result of the previous one.

In the toy, Yₓ is the deterministic value x². The two steps copy x and square it.

Representing a computation as steps allows a network to emulate those steps.

Source formula example

\mathcal{Y}_{x}^{(L)}\approx Y_{x}\quad\text{and}\quad\mathcal{Y}_{x}^{(\ell)}=T_{\ell}(\mathcal{Y}_{x}^{(\ell-1)}),\quad\ell=1,\dots,L,

illustration · This is an exact deterministic toy, not a stochastic numerical scheme or an error bound for general Yₓ.

Symbols

$Y_x$: The source random variable to approximate.
$T_ℓ$: A transformation at step ℓ.
$L$: Positive integer number of steps.

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.

x

This example’s data flow

You choose

input.x = 2

Calculated here

Result expression: { start, first, second, target, difference: second - target }

start = input.x Uses: input.x
first = start Uses: start
second = first ** 2 Uses: first
target = input.x ** 2 Uses: input.x

Returns

Example result

start: 2
first: 2
second: 4
target: 4
difference: 0

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source discusses approximating random variables through iterated transformations, including time-stepping constructions under appropriate assumptions. This toy has two trivial transformations and does not simulate a stochastic differential equation.

Source: Composed approximation