The Modern Mathematics of Deep Learning

Expected feature-curve length

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.

Approximation and depth · Study approximation error and network size under stated conditions.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

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: 3.3 Other ways to measure possibility

The paper considers the average trajectory length over random network initializations.

For a simple discrete toy, choose a single weight W = −1 or +1 with equal probability, zero bias, γ(t) = t, and feature ReLU(Wt). The two possible lengths are 0 and 1.

An expected length describes the average over a specified initialization distribution.

Source formula example

{\mathbbm{E}}\big[\operatorname{Length}(\bar{\Phi}^{(\ell)}(\gamma(\cdot),\Theta_{1}))\big]=c>0.

illustration · This exactly averages a discrete toy distribution; it is not the Gaussian initialization used in the source and does not prove the positive-expectation statement.

Symbols

$Θ_1$: Random initial network parameters in the source.
$c$: The positive expected length in the source setting.

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

weights = [-1, 1]

Calculated here

Result expression: { weights, lengths, expectation: (lengths[0] + lengths[1]) / 2 }

lengths = weights.map((weight) => Math.max(0, weight)) Uses: weights

Returns

Example result

weights: array (2 items): [-1, 1]
lengths: array (2 items): [0, 1]
expectation: 0.5

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source considers a nonconstant continuous input curve and independent normally initialized parameters, with specified variances for weights and biases. This toy instead enumerates two discrete weights and uses zero bias. Its expectation is exact for that toy only.

Source: Expected trajectory length