The Modern Mathematics of Deep Learning

A supremum equality

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.

Optimization theory · Study convergence and loss geometry under assumptions.

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: 2.2 Weight size and room to decide

Compare suprema over the full weight intervals in a one-dimensional case.

This identity supports peeling the last network layer from a complexity bound.

Source formula example

\sup_{\|W^{(\ell)}\|_{F}\leq C}\big\|\sum_{i=1}^{m}\tau_{i}\varrho(W^{(\ell)}x)\big\|_{2}=C\sup_{\|w\|_{2}\leq 1}\big|\sum_{i=1}^{m}\tau_{i}\varrho(\langle w,x\rangle)\big|.

illustration · Checking one scalar case does not prove the general equality.

Symbols

$W⁽ˡ⁾$: Weight matrix constrained by Frobenius norm C.
$w$: Vector constrained by Euclidean norm 1.
$τᵢ$: Rademacher signs ±1.
$x$: Input vector.
$ϱ$: Positively homogeneous activation.
$sup$: Least upper bound over the complete constraint set.

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.

Norm limitInput

This example’s data flow

You choose

input.C = 2
input.x = 1

Initial values in code

signSum = 2

Calculated here

Result expression: { signSum, leftSupremum, rightSupremum }

C = input.C Uses: input.C
x = input.x Uses: input.x
leftSupremum = signSum * C * x Uses: signSum, C, x
rightSupremum = C * signSum * x Uses: C, signSum, x

Returns

Example result

signSum: 2
leftSupremum: 4
rightSupremum: 4

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The scalar toy uses τ=[1,1], x≥0, and ReLU. Analytically, the left maximum occurs at W=C and the right at w=1; both full-interval suprema equal 2Cx. The source treats the corresponding matrix/vector norm domains.

Source: Source expression