The recurrence in code

Recurrent Neural Networks

Recurrent Neural Networks

A recurrent neural network carries a hidden state \mathbf{h}_t across time steps — a learned summary of all input seen so far:

\mathbf{h}_t = \phi(\mathbf{W}_{xh}\mathbf{x}_t + \mathbf{W}_{hh}\mathbf{h}_{t-1} + \mathbf{b}).

Same weights at every step → constant parameter count regardless of sequence length. Unbounded effective context (in principle), no fixed-size window like n-grams.

Stateful by design

An RNN with a hidden state.

Setup

from d2l import torch as d2l
import torch

The naive form: two matrix multiplies, summed:

X, W_xh = d2l.randn(3, 1), d2l.randn(1, 4)
H, W_hh = d2l.randn(3, 4), d2l.randn(4, 4)
d2l.matmul(X, W_xh) + d2l.matmul(H, W_hh)
tensor([[-5.8173, -2.5874,  0.1608, -1.1563],
        [-0.2778,  0.2293, -3.7569, -2.0525],
        [ 0.7006, -0.6336, -1.7256, -0.7766]])

Equivalently — concatenate input and hidden, multiply by the concatenated weight matrix — same result, one matmul:

d2l.matmul(d2l.concat((X, H), 1), d2l.concat((W_xh, W_hh), 0))
tensor([[-5.8173, -2.5874,  0.1608, -1.1563],
        [-0.2778,  0.2293, -3.7569, -2.0525],
        [ 0.7006, -0.6336, -1.7256, -0.7766]])

The “concat then multiply” form is what most framework RNN implementations actually do.

As a language model

  • Embedding maps token id → vector \mathbf{x}_t.
  • RNN updates the hidden state \mathbf{h}_t.
  • Linear head projects \mathbf{h}_t to vocab logits; softmax → P(x_{t+1} \mid x_{\le t}).
  • Loss = cross-entropy with the next-token target.

Character LM training

Input “machin”, target “achine” — same RNN, target shifted by one.

The next two sections build this end-to-end (from scratch + concise).

Recap

  • RNN: \mathbf{h}_t = \phi(\mathbf{W}_{xh}\mathbf{x}_t + \mathbf{W}_{hh}\mathbf{h}_{t-1} + \mathbf{b}).
  • Same parameters at every time step; hidden state carries arbitrarily long context (in theory).
  • Trains by backprop through time — gradients flow from \mathbf{h}_T back to every earlier hidden state.
  • Vanilla RNNs suffer from vanishing/exploding gradients on long sequences — fixed by LSTM and GRU in the next chapter.