The Modern Mathematics of Deep Learning

Peeling norm bounds

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.

Generalization and risk bounds · Relate sample behavior to population risk under stated 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

Compute the two complexity terms and the upper estimate in a one-sample scalar network.

The argument iterates norm and activation bounds across layers.

Source formula example

{\mathfrak{R}}_{m}(\widetilde{{\mathcal{F}}}_{L,C})=\frac{C}{m}{\mathbbm{E}}\big[\sup_{f\in\widetilde{{\mathcal{F}}}_{L-1,C}}\big\|\sum_{i=1}^{m}\tau_{i}\varrho(f(X^{(i)}))\big\|_{2}\big]\leq\frac{C(2C)^{L-1}}{m}{\mathbbm{E}}\big[\big\|\sum_{i=1}^{m}\tau_{i}X^{(i)}\big\|_{2}\big],

illustration · This exact one-dimensional example does not prove the general layer-peeling inequality.

Symbols

$Rₘ$: Rademacher complexity.
$F̃L,C$: Biasless network class with per-layer Frobenius norm bound C.
$τᵢ$: Independent uniform random signs ±1.
$X⁽ⁱ⁾$: Random input.
$ϱ$: Activation.
$m,L$: Sample count and network depth.

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 limit

This example’s data flow

You choose

input.C = 1

Calculated here

Result expression: { fullComplexity, peeledComplexity, upperBound }

C = input.C Uses: input.C
fullComplexity = C * C Uses: C
peeledComplexity = C * C Uses: C
upperBound = 2 * C * C Uses: C

Returns

Example result

fullComplexity: 1
peeledComplexity: 1
upperBound: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Set m=1, L=2, X=1 and use ReLU. For either τ sign, the full class supremum is C² by choosing hidden and output scalar weights at their bounds. The corresponding peeled supremum is C, and the expectation enumerates τ=±1 equally. The final upper bound is 2C². The source theorem assumes unit-ball inputs and a 1-Lipschitz positively homogeneous activation.

Source: Source expression