The Modern Mathematics of Deep Learning

Expected Hessian operator norm

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.

Optimization theory · Study convergence and loss geometry under 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: 5.2 When parameters move only a little

Compute the squared operator norm of each Hessian and average over a finite initialization distribution.

Bounding curvature supports a local gradient stability argument.

Source formula example

{\mathbbm{E}}\big[\|\nabla^{2}_{\theta}\Phi(x,\Theta)\|_{\mathrm{op}}^{2}\big]\lesssim 1

illustration · The two-point initialization is not the source Gaussian distribution, and this example does not prove its Hessian estimate.

Symbols

$∇²θΦ$: Hessian of network output with respect to parameters.
$\Vert ·\Vert op$: Largest singular value of the Hessian.
$𝔼$: Expectation over initialization.
$≲$: Inequality up to a constant.

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.

Input

This example’s data flow

You choose

input.x = 1

Calculated here

Result expression: { operatorNorm, squares, expectation: (squares[0] + squares[1]) / 2, chosenConstant: 11, }

squaredX = input.x ** 2 Uses: input.x
operatorNorm = (1 + Math.sqrt(5)) * squaredX Uses: squaredX
squares = [operatorNorm ** 2, operatorNorm ** 2] Uses: operatorNorm

Returns

Example result

squares: array (2 items): [≈10.47, ≈10.47]
expectation: ≈10.47
chosenConstant: 11
operatorNorm: ≈3.236

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Use Φ = a(wx)² with w = 1 and a equally likely ±1. Each Hessian is x²[[0,2],[2,2a]], whose operator norm is (1+√5)x². The finite expectation is exact. For |x| ≤ 1 the explicitly chosen toy constant 11 bounds it; this is not the source constant. For the perturbed-point notation in the source, the toy uses displacement zero.

Source: Source expression