The Modern Mathematics of Deep Learning

Regression 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

Squared-error risk equals the error relative to the conditional mean plus Bayes risk.

This separates avoidable prediction error from irreducible variability.

Source formula example

{\mathcal{R}}(f)={\mathbbm{E}}\big[(f(X)-{\mathbbm{E}}[Y|X])^{2}\big]+{\mathcal{R}}^{*},\quad f\in{\mathcal{M}}({\mathcal{X}},{\mathcal{Y}}),

exact · One finite distribution illustrates rather than proves the decomposition.

Symbols

$R(f)$: Expected squared-error risk.
$E[Y|X]$: Conditional mean of Y given X.
$R*$: Bayes risk.
$f$: Measurable prediction function.

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.

Prediction

This example’s data flow

You choose

input.prediction = 2

Initial values in code

outcomes = [0, 2]
conditionalMean = 1

Calculated here

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

prediction = input.prediction Uses: input.prediction
risk = outcomes.reduce((s, y) => s + (prediction - y) ** 2, 0) / 2 Uses: outcomes, prediction
bayesRisk = outcomes.reduce((s, y) => s + (conditionalMean - y) ** 2, 0) / 2 Uses: outcomes, conditionalMean
excess = (prediction - conditionalMean) ** 2 Uses: prediction, conditionalMean

Returns

Example result

conditionalMean: 1
risk: 2
bayesRisk: 1
excess: 1
sum: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

X has one value and Y is equally likely 0 or 2, so E[Y|X] = 1. The full finite distribution is enumerated. The general identity requires the relevant squared moments to be finite.

Source: Source expression