The Modern Mathematics of Deep Learning

A pruning error bound

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.

Pruning · Describe removal of network weights and related error bounds.

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: 6.5 Removing connections

The source gives a bound on the error between a two-layer ReLU network and a sparse network. The code evaluates its right side.

The upper bound decreases as the number of original hidden units grows, when the output-weight norm is held fixed.

Source formula example

\|\Phi_{a}(\cdot,\theta)-\Phi_{a}(\cdot,\tilde{\theta})\|_{L^{2}(\mu)}\leq\frac{15\|w^{(2)}\|_{1}}{\sqrt{n}}.

illustration · The code evaluates the bound, not the approximation error, and does not construct the sparse network or prove existence.

Symbols

$n$: Original number of hidden units, at least 100.
$\Vert w^{(2)}\Vert _1$: Sum of absolute values of output weights.

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.

Original hidden units nOutput weight L1 norm

This example’s data flow

You choose

input.n = 100
input.norm = 2

Calculated here

Result expression: { upperBound: (15 * input.norm) / Math.sqrt(input.n) }

Returns

Example result

upperBound: 3

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source requires a probability measure on the unit ball, a two-layer ReLU network with n ≥ 100, each first-layer weight norm at most 1/2 and each first-layer bias at most 1/2. It asserts existence of a parameter vector with at least 99% zero entries, together with the stated error bound. The toy values do not establish those hypotheses.

Source: Source expression