The Modern Mathematics of Deep Learning

A norm-constrained network class

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.

Model classes and structure · Describe families of rules, their size and structural properties.

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

A network belongs to the class only when each weight matrix satisfies the Frobenius norm bound.

This class is used for norm-based generalization bounds.

Source formula example

\widetilde{{\mathcal{F}}}_{L,C}\coloneqq\big\{\Phi_{(N,\varrho)}(\cdot,\theta)\colon N\in{\mathbb{N}}^{L+1},\ N_{0}=d,\ N_{L}=1,\ \theta=((W^{(\ell)},0))_{\ell=1}^{L}\in{\mathbb{R}}^{P(N)},\ \|W^{(\ell)}\|_{F}\leq C\big\}.

illustration · A membership check for one network does not enumerate the full class.

Symbols

$F̃L,C$: Network class with L layers and each weight Frobenius norm at most C.
$N$: Width vector with input width d and scalar output.
$W⁽ˡ⁾$: Real weight matrix in layer ℓ.
$ϱ$: Activation.
$C$: Positive norm bound.

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

Initial values in code

weights = [[[1]], [[1]]]
biases = [[0], [0]]

Calculated here

Result expression: { weights, biases, norms, inClass, output }

norms = weights.map((matrix) => Math.sqrt(matrix.flat().reduce((s, v) => s + v * v, 0)), ) Uses: weights
inClass = norms.every((v) => v <= input.C) Uses: norms, input.C
output = Math.max(0, 2)

Returns

Example result

weights: array (2 items): [array (1 items): [array (1 items)], array (1 items): [array (1 items)]]
biases: array (2 items): [array (1 items): [0], array (1 items): [0]]
norms: array (2 items): [1, 1]
inClass: true
output: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy has d=1, L=2, widths (1,1,1), zero biases and ReLU. Both 1 by 1 weight matrices contain 1, so their Frobenius norms are 1.

Source: Source expression