The Modern Mathematics of Deep Learning

Regularized objective minimum

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.

Learning and parameter updates · Select a rule or update parameters using a learning rule.

The current objective supplies a gradient; training updates the model parameters.

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

Minimize empirical squared error plus α times the squared parameter.

The regularizer changes the best parameter selected by fitting.

Source formula example

\min_{\theta\in{\mathbb{R}}^{P(N)}}\widehat{\mathcal{R}}_{s}(\Phi_{{a}}(\cdot,\theta))+\alpha\|\theta\|^{2}_{2},

exact · This explicit minimizer applies only to the toy linear model and squared loss.

Symbols

$R̂s$: Empirical risk on sample s.
$Φₐ$: Network predictor.
$α$: Nonnegative regularization parameter.
$θ$: Real parameter vector.

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.

Regularizer α

This example’s data flow

You choose

input.alpha = 0.5

Initial values in code

xs = [1, 2]
ys = [2, 4]

Calculated here

Result expression: { theta, empiricalRisk, penalty, minimum: empiricalRisk + penalty }

alpha = input.alpha Uses: input.alpha
theta = 5 / (2.5 + alpha) Uses: alpha
errors = xs.map((x, i) => theta * x - ys[i]) Uses: xs, theta, ys
empiricalRisk = errors.reduce((sum, e) => sum + e * e, 0) / 2 Uses: errors
penalty = alpha * theta * theta Uses: alpha, theta

Returns

Example result

theta: ≈1.667
empiricalRisk: ≈0.2778
penalty: ≈1.389
minimum: ≈1.667

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy uses fθ(x)=θx and two invented pairs (1,2),(2,4). Differentiating its quadratic objective gives θ=5/(2.5+α), its global minimum for α ≥ 0.

Source: Source expression