The Modern Mathematics of Deep Learning

Gradient change and curvature

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.

Functions and derivatives · Describe maps and how their outputs change.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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

A Hessian describes how a gradient changes. For this quadratic, the gradient changes by the Hessian times the parameter displacement.

These quantities connect network parameters to loss and optimization.

Source formula example

\left\|\nabla_{\theta}\Phi(x,\Theta)-\nabla_{\theta}\Phi(x,\Theta+\bar{\theta})\right\|_{2}^{2}\lesssim\sup_{\hat{\theta}\in B_{1}(0)}\big\|\nabla^{2}_{\theta}\Phi(x,\Theta+\hat{\theta})\big\|_{\mathrm{op}}^{2},

illustration · The invented finite inputs illustrate this expression. They do not prove a general identity, bound, or existence claim.

Symbols

$Θ$: Network parameter vector at initialization.
$θ̄$: Parameter displacement in the unit ball.
$∇²θΦ$: Hessian with respect to network parameters.
$\Vert ·\Vert op$: Matrix operator norm.
$sup$: Least upper bound over the full unit ball, not a sample maximum.

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.

Displacement

This example’s data flow

You choose

input.displacement = 0.5

Calculated here

Result expression: { gradientChange, squaredChange: gradientChange ** 2, squaredHessianNorm: 4 }

displacement = input.displacement Uses: input.displacement
gradientChange = 2 * displacement Uses: displacement

Returns

Example result

gradientChange: 1
squaredChange: 1
squaredHessianNorm: 4

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The example is f(θ) = θ² and |displacement| ≤ 1. Its Hessian is constantly 2, so the supremum of its squared operator norm on the unit interval is exactly 4. This does not prove the source bound for neural networks. The source’s training discussion assumes a smooth, non-affine activation, distinct training inputs, quadratic loss, and independent Gaussian hidden and output initial parameters with variance 1/n. Its high-probability and convergence statements require the additional width, neighborhood, and step-size conditions stated in that discussion.

Source: Source expression