The Modern Mathematics of Deep Learning

A mini-batch gradient step

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

Compute one gradient for each example in a mini-batch, average them, scale by the step size, and subtract.

A mini-batch step can update parameters without using every training example.

Source formula example

\Theta^{(k)}\coloneqq\Theta^{(k-1)}-\frac{\eta_{k}}{m^{\prime}}\sum_{z\in{S}^{\prime}}\nabla_{\theta}{\mathcal{L}}(\Phi_{{a}}(\cdot,\Theta^{(k-1)}),z),

illustration · The two supplied gradients illustrate the arithmetic; the code does not compute them from a network or establish convergence.

Symbols

$m'$: Positive mini-batch size.
$η_k$: Step size at iteration k.

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.

First gradientCurrent parameterStep size

This example’s data flow

You choose

input.firstGradient = 2
input.theta = 2
input.rate = 0.1

Calculated here

Result expression: { gradients, meanGradient, next: input.theta - input.rate * meanGradient }

gradients = [input.firstGradient, 4] Uses: input.firstGradient
meanGradient = gradients.reduce((a, b) => a + b, 0) / gradients.length Uses: gradients

Returns

Example result

gradients: array (2 items): [2, 4]
meanGradient: 3
next: 1.7

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Change the input and run the same code again. The displayed result uses the values shown above.

Source: Source expression