The Modern Mathematics of Deep Learning

Network VC bound factor

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: 1.2.3 Does it work on new examples?

Evaluate the known width and depth factor in the VC upper bound.

This example makes the source statement computable on a finite model.

Source formula example

\operatorname{VCdim}({\mathcal{F}}_{(N,\varrho),{\operatorname{sgn}}})\leq c\big(P(N)L\log(P(N))+P(N)L^{2}\big).

illustration · The code evaluates only the known factor, not the unknown constant or VC dimension.

Symbols

$P(N)$: Number of network parameters.
$L$: Positive number of layers.
$c$: Positive constant whose existence is asserted.
$VCdim$: VC dimension of the sign-network class.

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.

Parameter count PLayers L

This example’s data flow

You choose

input.parameters = 2
input.layers = 1

Calculated here

Result expression: { factor: P * L * Math.log(P) + P * L * L, theoremConstant: null, upperBound: null, }

P = input.parameters Uses: input.parameters
L = input.layers Uses: input.layers

Returns

Example result

factor: ≈3.386
theoremConstant: null
upperBound: null

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

the source theorem assumes a piecewise polynomial activation and asserts existence of c > 0 for every positive L and layer-width vector N. The source does not provide a numeric c; therefore the actual numerical bound is unavailable.

Source: Source expression