The Modern Mathematics of Deep Learning

Population risk of network parameters

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.

Risk and assessment · Distinguish loss on a sample from expected loss.

The fitted rule is evaluated on 102 held-out observations, not used for fitting.

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: 5.1 The shape of training loss

Evaluate a parameterized network on every possible example, compute its loss, and take the probability-weighted mean.

This is the population-risk version of the parameter objective r.

Source formula example

{r}(\theta)={\mathcal{R}}(\Phi(\cdot,\theta))

illustration · The probabilities specify an invented fully enumerated population; a finite training sample alone would not determine population risk.

Symbols

$r(θ)$: Population risk as a function of network parameters.
$R$: Expectation of the loss over the example distribution.
$Φ(·,θ)$: Network prediction function.
$Z$: Random training or test example.

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.

Network parameter

This example’s data flow

You choose

input.theta = 1

Initial values in code

examples = [ { x: 1, y: 2 }, { x: 2, y: 4 }, ]
probabilities = [0.5, 0.5]

Calculated here

Result expression: { examples, probabilities, predictions, losses, risk }

theta = input.theta Uses: input.theta
predictions = examples.map((z) => theta * z.x) Uses: examples, theta
losses = examples.map((z, i) => (predictions[i] - z.y) ** 2) Uses: examples, predictions
risk = losses.reduce((sum, v, i) => sum + probabilities[i] * v, 0) Uses: losses, probabilities

Returns

Example result

examples: array (2 items): [object (2 items), object (2 items)]
probabilities: array (2 items): [0.5, 0.5]
predictions: array (2 items): [1, 2]
losses: array (2 items): [1, 4]
risk: 2.5

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy distribution assigns equal probability to two input-target pairs and uses Φ(x,θ)=θx with squared loss. The weighted sum is the exact population expectation for this finite distribution. The source footnote allows either population or empirical risk as the objective.

Source: Source expression