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, so the parameter count is constant regardless of sequence length. Unbounded effective context (in principle), with no fixed-size window like an n-gram.

Unrolled in time

An RNN unrolled across three time steps; the same weights are reused at every step.

Setup

from d2l import mxnet as d2l
from mxnet import np, npx
npx.set_np()

The recurrence in code

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)
array([[ 0.9422816 ,  0.01977926, -2.6496055 , -2.2366421 ],
       [ 1.1883913 ,  1.8222132 , -2.8613284 , -4.743068  ],
       [-1.6715677 , -2.8657658 ,  2.348141  ,  3.5657983 ]])

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

d2l.matmul(d2l.concat((X, H), 1), d2l.concat((W_xh, W_hh), 0))
array([[ 0.94228166,  0.01977928, -2.6496055 , -2.2366421 ],
       [ 1.1883914 ,  1.8222132 , -2.8613286 , -4.743068  ],
       [-1.6715677 , -2.8657656 ,  2.348141  ,  3.5657983 ]])

The concatenate-then-multiply form is what most framework RNN implementations actually do.

Constant memory per step

  • \mathbf{h}_t has a fixed size, independent of t.
  • To form \mathbf{h}_t we need only \mathbf{h}_{t-1} and \mathbf{x}_t; everything older can be dropped.
  • Any-length input at flat per-step memory and compute.
  • The trade: the state is a lossy summary, so it must learn what to keep. Attention (later) keeps everything instead, at a cost that grows with length.

As a language model

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

Teacher forcing

Targets are the inputs shifted forward by one token; each step predicts the next token.

Train on gold prefixes; generate on the model’s own outputs. That mismatch is the rollout-error problem again.

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 step; the hidden state is a fixed-size summary of the whole past.
  • Trains by backprop through time: gradients flow from \mathbf{h}_T back to every earlier hidden state.
  • Those long products vanish or explode on long sequences, fixed by LSTM and GRU in the next chapter.