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 tensorflow as d2l
import tensorflow as tf

The naive form: two matrix multiplies, summed:

X, W_xh = d2l.normal((3, 1)), d2l.normal((1, 4))
H, W_hh = d2l.normal((3, 4)), d2l.normal((4, 4))
d2l.matmul(X, W_xh) + d2l.matmul(H, W_hh)
<tf.Tensor: shape=(3, 4), dtype=float32, numpy=
array([[-2.190528 , -1.6804276, -2.6916413,  0.0785881],
       [-1.4864042,  1.7159821, -1.6885962,  0.6511531],
       [-1.1071088, -2.559943 , -1.0138488,  0.7972275]], dtype=float32)>

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))
<tf.Tensor: shape=(3, 4), dtype=float32, numpy=
array([[-2.1905282 , -1.6804273 , -2.6916416 ,  0.0785881 ],
       [-1.4864042 ,  1.7159821 , -1.6885962 ,  0.6511531 ],
       [-1.1071088 , -2.559943  , -1.0138489 ,  0.79722756]],
      dtype=float32)>

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.