The Modern Mathematics of Deep Learning

A limiting variance curve

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: 2.4 A second descent

The source gives this expression as an asymptotic variance term when the input dimension divided by sample size tends to κ.

The expression is undefined at κ = 1. On either side, it grows without bound as κ approaches 1.

In this specific model, the expression helps explain a peak near the interpolation threshold.

Source formula example

{\operatorname{Tr}}\big(\Sigma^{+}{\mathbbm{E}}[XX^{T}]\big)\to\frac{1-\max\{1-\kappa,0\}}{|1-\kappa|}\quad\text{for}\quad d,m\to\infty\quad\text{with}\quad\frac{d}{m}\to\kappa,

illustration · This evaluates a limiting formula, not the variance of a finite data set. It is undefined at κ = 1 and is not a universal double-descent curve.

Symbols

$κ$: Positive limiting ratio d/m; the displayed formula is finite only away from 1.
$d,m$: Input dimension and sample size, both tending to infinity in the limit.

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.

Dimension/sample ratio κ

This example’s data flow

You choose

input.kappa = 2

Calculated here

Result expression: { limitingVariance: (1 - Math.max(1 - kappa, 0)) / Math.abs(1 - kappa) }

kappa = input.kappa Uses: input.kappa

Returns

Example result

limitingVariance: 1

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source considers a centered linear model with independent centered noise and the minimum-Euclidean-norm empirical risk minimizer. For the variance expression it sets the true parameter to zero and noise variance to one. The entries of X are i.i.d. with unit variance and bounded fifth moment. The stated limit is almost sure as d,m→∞ with d/m→κ; this calculator evaluates the right side away from κ=1.

Source: Limiting variance