The Modern Mathematics of Deep Learning

A no-free-lunch comparison

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: 1.2 Learning from examples

Compare expected learned risk with Bayes risk plus a positive sequence term.

This example makes the source statement computable on a finite model.

Source formula example

{\mathbbm{E}}\big[{\mathcal{R}}({\mathcal{A}}({S}))\big]\geq{\mathcal{R}}^{*}+a_{m}.

illustration · This example checks the bound for one deliberately poor algorithm; it does not prove the theorem or show the same distribution works for all algorithms.

Symbols

$A$: Learning algorithm.
$S$: Random training sample of size m.
$R*$: Bayes classification risk.
$aₘ$: Positive nonincreasing sequence term.

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.

Sample count

This example’s data flow

You choose

input.m = 1

Initial values in code

expectedLearnedRisk = 1
bayesRisk = 0

Calculated here

Result expression: { expectedLearnedRisk, bayesRisk, a, lowerBound: bayesRisk + a }

a = 1 / (16 * input.m) Uses: input.m

Returns

Example result

expectedLearnedRisk: 1
bayesRisk: 0
a: 0.0625
lowerBound: 0.0625

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The invented population has one input with label +1; the algorithm always predicts −1. Its learned risk is 1 and Bayes risk is 0. We choose aₘ = 1/(16m). The theorem states: for every positive nonincreasing sequence with a₁ ≤ 1/16, and every classification learning algorithm, there exists a data distribution for which the inequality holds for every m, with S independently sampled from that distribution.

Source: Source expression