The Modern Mathematics of Deep Learning

A complexity 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.

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

The right side is a ceiling on a measure of how flexibly a class of networks can fit random signs.

Enter positive C, positive integer depth L, and positive integer sample size m to calculate that ceiling.

This source theorem connects the bound to layer norms and sample size.

Source formula example

{\mathfrak{R}}_{m}(\widetilde{{\mathcal{F}}}_{L,C})\leq\frac{C(2C)^{L-1}}{\sqrt{m}}.

illustration · The code computes the theorem’s right side. It does not compute Rademacher complexity or prove the inequality.

Symbols

$ℜ_m$: Rademacher complexity of the specified network class.
$C$: Positive bound on each weight matrix Frobenius norm.
$L$: Positive integer network depth.
$m$: Positive integer sample size.

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 bound CDepth LSample size m

This example’s data flow

You choose

input.C = 1
input.depth = 2
input.m = 4

Calculated here

Result expression: { upperBound: (input.C * (2 * input.C) ** (input.depth - 1)) / Math.sqrt(input.m), }

Returns

Example result

upperBound: 1

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The theorem assumes inputs in the unit ball in ℝᵈ; a positively homogeneous activation with Lipschitz constant 1; biasless networks with scalar output; and every weight matrix having Frobenius norm at most C. It applies for each positive integer m. The evaluated expression is a bound under those assumptions.

Source: Source expression