The Modern Mathematics of Deep Learning

Dimension-dependent approximation 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.

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: 1.3 The questions ahead

The factors p⁻¹ᐟᵈ and p⁻²ᐟᵈ depend on the parameter count p and dimension d. Ω describes a lower bound up to a positive constant for sufficiently large p.

Such factors describe how approximation error can change as parameter count increases.

Source formula example

\Omega(p^{-1/d})

illustration · Evaluating two rates does not establish the source lower bound, its hidden constant, or a limit proof.

Symbols

$p$: Positive parameter count.
$d$: Positive input dimension.
$Ω$: An asymptotic lower bound up to a positive 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.

This example’s data flow

Initial values in code

p = 16
d = 2

Calculated here

Result expression: { p, d, first: p ** (-1 / d), second: p ** (-2 / d) }

Returns

Example result

p: 16
d: 2
first: 0.25
second: 0.0625

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. Evaluating two rates does not establish the source lower bound, its hidden constant, or a limit proof.

Source: Inline notation