The Modern Mathematics of Deep Learning

Rescaled pruning feature

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.

Features · Transform inputs into features used by a rule.

The project scales bill and flipper measurements using training-row means and scales; its feature vector has two entries.

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: 6.5 Removing connections

Multiply a ReLU hidden feature by the sum of absolute output weights.

The rescaling puts features into the form used by the sparse approximation argument.

Source formula example

f_{i}\coloneqq\|w^{(2)}\|_{1}{\varrho_{R}}(\langle w_{i}^{(1)},\cdot\rangle+b^{(1)}_{i}),\quad i\in[n],

exact · Computing a feature does not establish the pruning error bound.

Symbols

$fᵢ$: Rescaled hidden feature i.
$w⁽²⁾$: Vector of output weights.
$wᵢ⁽¹⁾,bᵢ⁽¹⁾$: Incoming weight and bias of unit i.
$ϱR$: ReLU activation.

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.

Input

This example’s data flow

You choose

input.x = 1

Initial values in code

outputWeights = [1, -2]

Calculated here

Result expression: { norm, hidden, feature: norm * hidden }

norm = outputWeights.reduce((sum, w) => sum + Math.abs(w), 0) Uses: outputWeights
hidden = Math.max(0, 0.5 * input.x) Uses: input.x

Returns

Example result

norm: 3
hidden: 0.5
feature: 1.5

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy has a one-dimensional input, hidden weight 1/2, and bias zero.

Source: Source expression