The Modern Mathematics of Deep Learning

Growth on a finite input space

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.

Generalization and risk bounds · Relate sample behavior to population risk under stated assumptions.

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: 1.2 Learning from examples · 1.2.3 Does it work on new examples?

For each ordered collection of m inputs, count the different label sequences produced by the function class. Take the largest count.

The toy input space is {0,1}; the two functions output [−1,−1] and [−1,1] on those points.

Growth counts how many label patterns a class can produce.

Source formula example

\operatorname{growth}(m,{\mathcal{F}})\coloneqq\max_{(x^{(i)})_{i=1}^{m}\in{\mathcal{X}}^{m}}|\{f|_{(x^{(i)})_{i=1}^{m}}\colon f\in{\mathcal{F}}\}|.

exact · This is an exact exhaustive maximum for the specified finite input space and m = 1. Sampling a few inputs does not generally compute the maximum over an infinite input space.

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

functions = [ [-1, -1], [-1, 1], ]
collections = [[0], [1]]

Calculated here

Result expression: { counts, growth: Math.max(...counts), maximumPatterns: 2 ** 1 }

counts = collections.map( (points) => new Set(functions.map((values) => points.map((point) => values[point]).join(","))) .size, ) Uses: collections, functions

Returns

Example result

counts: array (2 items): [1, 2]
growth: 2
maximumPatterns: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Change the input and run the same code again. The displayed result uses the values shown above.

Source: Source expression