Note: AlexNet — Dropout, and What Actually Changed from LeNet

The problem, from first principles

AlexNet (2012) is usually credited with "starting" deep learning's modern era, but its architecture wasn't a radical departure from LeNet (1998) — it's a deeper stack of the same convolution/pool/fully-connected pattern. What actually mattered was a handful of practical changes that made a deep network trainable and generalizable at all: ReLU instead of sigmoid/tanh (avoiding vanishing gradients — see 02-deep-learning-core's activation questions), and dropout, a regularization technique that fights overfitting in the large fully-connected layers that dominate AlexNet's parameter count. This question implements dropout, since it's the one AlexNet-era idea concrete and self-contained enough to code and test directly.

From theory to code

Theory says: during training, randomly and independently zero out each unit with some probability p, forcing the network to not rely on any single unit too heavily (since it might vanish on any given forward pass). To keep the expected value of the output unchanged (so nothing downstream needs to know or care whether dropout is active), scale every unit that survives by 1 / (1 - p).

Implement dropout_forward(x, p, rng) against that reasoning.

Constraints

• x: any-shape array of activations.
• p: drop probability, in [0, 1).
• rng: a numpy.random.Generator, passed in explicitly so tests (and callers) can control and reproduce randomness.
• Returns an array the same shape as x, where each element is either exactly 0.0 (dropped) or x[i] / (1 - p) (kept and rescaled).
• x is never modified in place.

Hints

Open one at a time. Each gives away a little more than the last.