The Modern Mathematics of Deep Learning

Smallest eigenvalue bound

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

The smallest eigenvalue of a positive semidefinite Gram matrix gives a lower bound for its quadratic form.

The smallest eigenvalue connects the residual length to a lower bound on the gradient.

Source formula example

\|\nabla_{\theta}{r}(\Theta)\|_{2}^{2}\geq\frac{4}{m^{2}}\lambda_{\min}(\bar{K}_{\Theta})\|(\Phi(x^{(i)},\Theta)-y^{(i)})_{i=1}^{m}\|_{2}^{2}\gtrsim\frac{n}{m}{r}(\Theta).

illustration · The chosen constant is not asserted to be a source theorem constant. This one example does not establish the theorem.

Symbols

$∇θr$: Gradient of training loss with respect to all network parameters.
$λmin$: Smallest eigenvalue of the symmetric Gram matrix.
$m,n$: Numbers of training examples and hidden units.
$≳$: Inequality up to a positive constant under the source assumptions.

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.

Output weight

This example’s data flow

You choose

input.outputWeight = 1

Initial values in code

minEigenvalue = 1

Calculated here

Result expression: { risk, fullSquaredNorm, eigenLowerBound, widthScaledRisk, chosenConstant: 4, scaledLowerBound: 4 * widthScaledRisk, }

a = input.outputWeight Uses: input.outputWeight
error = a Uses: a
risk = error ** 2 Uses: error
fullGradient = [2 * error, 2 * error * a] Uses: error, a
fullSquaredNorm = fullGradient.reduce((sum, v) => sum + v * v, 0) Uses: fullGradient
eigenLowerBound = 4 * minEigenvalue * error ** 2 Uses: minEigenvalue, error
widthScaledRisk = risk Uses: risk

Returns

Example result

risk: 1
fullSquaredNorm: 8
eigenLowerBound: 4
widthScaledRisk: 1
chosenConstant: 4
scaledLowerBound: 4

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy network Φ = awx has a single input x = 1, target zero, and fixed w = 1; m = n = 1. Its 1 by 1 Gram matrix has smallest eigenvalue 1. The final comparison uses an explicitly chosen constant 4 for this toy only. The source asymptotic constant and high-probability eigenvalue statement depend on its assumptions. 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