The Modern Mathematics of Deep Learning

Replacing unused hidden weights

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.

Prediction rules and layers · Describe or apply prediction rules, layers and parameters.

The project uses a linear rule with two features. The fitted weights produce scores or normalized mass.

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

A hidden unit with zero output weight can change its incoming parameters without changing the network output.

These quantities connect network parameters to loss and optimization.

Source formula example

\hat{\theta}^{(1)}_{j}\coloneqq\begin{cases}\theta^{(1)}_{j}&\text{for }j\in[n/2],\\ (\theta_{\min}^{(1)})_{j}&\text{for }j\in[n]\setminus[n/2].\end{cases}

illustration · The invented finite inputs illustrate this expression. They do not prove a general identity, bound, or existence claim.

Symbols

$θ̂⁽¹⁾$: Replacement hidden-parameter vector.
$θmin⁽¹⁾$: First-layer parameters of the chosen minimum.
$n$: Even number of hidden units.
$j$: Hidden-unit index.

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.

Replacement feature

This example’s data flow

You choose

input.feature = 3

Initial values in code

feature1 = 2
oldFeature2 = 1

Calculated here

Result expression: { before: feature1 + 0 * oldFeature2, after: feature1 + 0 * newFeature2 }

newFeature2 = input.feature Uses: input.feature

Returns

Example result

before: 2
after: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

This toy explains the zero-output-weight step used in the source path construction. The source replacement in the source expression depends on its chosen minimum and duplicated features.

Source: Source expression