The Modern Mathematics of Deep Learning

A zero output weight

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: 1.2 Learning from examples

If the final weight matrix is zero, the network output equals its final bias, regardless of its earlier layers.

This gives a constant network function.

Source formula example

W^{(L)}=0

illustration · The example has one output; the argument also applies componentwise to vector outputs.

Symbols

$W⁽ᴸ⁾$: The final weight matrix.
$b⁽ᴸ⁾$: The final bias vector.
$Φₐ$: The network output 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.

This example’s data flow

Initial values in code

hidden = [2, 3]
weights = [0, 0]
bias = 4

Calculated here

Result expression: { hidden, weights, bias, output }

output = weights.reduce((s, w, i) => s + w * hidden[i], bias) Uses: weights, hidden, bias

Returns

Example result

hidden: array (2 items): [2, 3]
weights: array (2 items): [0, 0]
bias: 4
output: 4

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. The example has one output; the argument also applies componentwise to vector outputs.

Source: Inline notation