The Modern Mathematics of Deep Learning

A Sobolev approximation error

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: 3.2 Simple pieces, many layers

The theorem bounds the best error in a norm that can include function values and derivatives.

On (0,1), the toy chooses g(x)=x and Φ(x)=ReLU(x). They agree everywhere on the interval, so their difference is the zero function and its Sobolev norm is zero.

An approximation guarantee can control derivatives as well as function values.

Source formula example

\inf_{\theta\in{\mathbb{R}}^{P(N)}}\|\Phi_{{a}}(\theta,\cdot)-g\|_{W^{s,p}((0,1)^{d})}\leq{\varepsilon}.

illustration · The zero error is known analytically for this pair of functions on (0,1), not inferred from one sampled x. The example does not prove the general theorem.

Symbols

$W^{s,p}$: A Sobolev-type space with error order s and integrability p.
$ε$: A positive target accuracy below 1/2.

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.

Point xAccuracy ε

This example’s data flow

You choose

input.x = 0.5
input.epsilon = 0.25

Initial values in code

exactErrorNorm = 0

Calculated here

Result expression: { target, prediction, exactErrorNorm, epsilon: input.epsilon, satisfiesBound: exactErrorNorm <= input.epsilon, }

target = input.x Uses: input.x
prediction = Math.max(0, input.x) Uses: input.x

Returns

Example result

target: 0.5
prediction: 0.5
exactErrorNorm: 0
epsilon: 0.25
satisfiesBound: true

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source assumes d,k positive integers with k≥2, p∈[1,∞], s∈[0,1], a Sobolev norm bound B>0, and a piecewise-linear activation with a breakpoint. For every ε∈(0,1/2) it constructs an architecture satisfying the stated parameter bound and approximating every function in the specified W^{k,p} ball. This toy gives only one exactly represented function.

Source: Source expression