The Modern Mathematics of Deep Learning

Barron approximation bound

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: 4.2 Approximating with random pieces

The right side bounds the best approximation error among networks of the specified form. The constants summarize properties of the function and activation.

This bound has an inverse-square-root dependence on hidden units.

Source formula example

\inf_{\theta\in{\mathbb{R}}^{P((d,n,1))}}\|\Phi_{((d,n,1),\varrho)}(\cdot,\theta)-g\|_{L^{2}(\mu)}\leq\frac{c}{\sqrt{n}}C_{g},

illustration · The code evaluates the bound using supplied constants; it does not find a best network, calculate the Fourier integral, or prove the theorem.

Symbols

$C_g$: Finite integral of the Fourier magnitude weighted by frequency length.
$c$: A positive constant whose existence is stated by the theorem.
$n$: Positive integer number of hidden units.
$μ$: Probability measure supported on the unit ball.

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.

Theorem constant cFunction constant CgHidden units n

This example’s data flow

You choose

input.c = 2
input.Cg = 3
input.n = 4

Calculated here

Result expression: { upperBound: (input.c * input.Cg) / Math.sqrt(input.n) }

Returns

Example result

upperBound: 3

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source requires ReLU or a sigmoidal activation, positive integer d and n, a probability measure supported on the unit ball in ℝᵈ, g integrable on ℝᵈ, and finite Cg = ∫‖ξ‖₂|ĝ(ξ)| dξ. The theorem states that a positive c exists; the toy c is chosen for arithmetic, not a claimed theorem constant.

Source: Source expression