The Modern Mathematics of Deep Learning

Finite approximation inequality

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: 6.5 Removing connections

Compare an approximation error with C divided by the square root of the number of terms.

This is the form of the generic approximation result used in pruning.

Source formula example

\left\|g-\sum_{i=1}^{n}c_{i}f_{i}\right\|\leq\frac{C}{\sqrt{n}},

illustration · This one exact fit checks the inequality for its invented values; it does not establish the source existence theorem.

Symbols

$g$: Target element in the normed space.
$fᵢ$: Selected approximating elements.
$cᵢ$: Real coefficients.
$C$: Nonnegative constant in the bound.
$n$: Positive number of terms.

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.

Chosen C

This example’s data flow

You choose

input.C = 2

Initial values in code

g = 1
functions = [1, 1, 1, 1]
coefficients = [0.25, 0.25, 0.25, 0.25]

Calculated here

Result expression: { approximation, error, upperBound, holds: error <= upperBound }

approximation = functions.reduce((sum, v, i) => sum + v * coefficients[i], 0) Uses: functions, coefficients
error = Math.abs(g - approximation) Uses: g, approximation
upperBound = input.C / Math.sqrt(functions.length) Uses: input.C, functions

Returns

Example result

approximation: 1
error: 0
upperBound: 1
holds: true

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source lemma takes a Hilbert space, a set F with sup over f in F of ‖f‖ ≤ C, and g in its convex hull. For each positive integer n it asserts there exist n elements of F and nonnegative coefficients summing to 1 with the displayed error bound. The toy uses F = {1}, g = 1, four equal coefficients, and a chosen C at least 1.

Source: Source expression