The Modern Mathematics of Deep Learning

Classification risk decomposition

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

Add Bayes risk to the conditional-label margin where the classifier disagrees with its sign.

This shows how disagreement with the Bayes classifier creates excess risk.

Source formula example

{\mathcal{R}}(f)={\mathbbm{E}}\big[|{\mathbbm{E}}[Y|X]|\mathds{1}_{(-\infty,0)}({\mathbbm{E}}[Y|X]f(X))\big]+{\mathcal{R}}^{*},\quad f\in{\mathcal{M}}({\mathcal{X}},{\mathcal{Y}}),

exact · One fully enumerated finite distribution illustrates the decomposition; it does not prove the general identity.

Symbols

$R(f)$: Expected binary classification risk.
$E[Y|X]$: Conditional mean of the binary label.
$R*$: Bayes classification risk.
$f$: Classifier with outputs ±1.

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.

Predicted label

This example’s data flow

You choose

input.prediction = -1

Initial values in code

conditionalMean = 0.5
bayesRisk = 0.25

Calculated here

Result expression: { conditionalMean, risk, bayesRisk, excess, sum: bayesRisk + excess }

prediction = input.prediction Uses: input.prediction
risk = prediction === 1 ? 0.25 : 0.75 Uses: prediction
excess = Math.abs(conditionalMean) * (conditionalMean * prediction < 0 ? 1 : 0) Uses: conditionalMean, prediction

Returns

Example result

conditionalMean: 0.5
risk: 0.75
bayesRisk: 0.25
excess: 0.5
sum: 0.75

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

X has one value, P(Y=1)=3/4 and P(Y=−1)=1/4. Therefore E[Y|X]=1/2 and the Bayes label is +1.

Source: Source expression