The Modern Mathematics of Deep Learning

An unbiased stochastic update

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.

Learning and parameter updates · Select a rule or update parameters using a learning rule.

The current objective supplies a gradient; training updates the model parameters.

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 Learning from examples

A stochastic gradient can vary between updates. It is unbiased when its conditional average equals the full gradient.

In the toy, D is 0 or 2 with equal probability, so its mean is 1.

The update combines a stochastic gradient with a step size.

Source formula example

\mathbb E[D^{(k)}\mid\Theta^{(k-1)}]=\nabla r(\Theta^{(k-1)}),\quad\Theta^{(k)}=\Theta^{(k-1)}-\eta_kD^{(k)}

illustration · This exactly enumerates a two-outcome toy conditional distribution; it does not establish unbiasedness or convergence for an arbitrary algorithm.

Symbols

$D^{(k)}$: Stochastic gradient at step k.
$η_k$: Step size.
$Θ^{(k-1)}$: Parameter before the step.

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.

Current parameterStep size

This example’s data flow

You choose

input.theta = 2
input.rate = 0.5

Initial values in code

gradients = [0, 2]

Calculated here

Result expression: { gradients, meanGradient, fullGradient: 1, nextParameters }

meanGradient = (gradients[0] + gradients[1]) / 2 Uses: gradients
nextParameters = gradients.map((value) => input.theta - input.rate * value) Uses: gradients, input.theta, input.rate

Returns

Example result

gradients: array (2 items): [0, 2]
meanGradient: 1
fullGradient: 1
nextParameters: array (2 items): [2, 1]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source algorithm requires the conditional expectation of D given the previous iterate to equal the full gradient. The toy chooses a full gradient of 1 and a two-outcome distribution with mean 1.

Source: Algorithm 1