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Here is a commented version of the code proposed by Andrej Karpathy in his article Minimal character-level language model with a Vanilla Recurrent Neural Network, in Python/numpy .

The model is a simple RNN with one hidden layer of size hidden_size.

RNN

It is parameterized by:

Reading data

Steps are:

  • Reading of text input.
  • Recovery of the list of unique characters that compose the other words of vocabulary.
  • Construction of two dictionaries:
    1. A dictionary encoding characters in number so that the RNN works with numbers. Example: {'a':0, 'b':1, 'c':2, ...}.
    2. A dictionary decoding number into characters to translate the character output of RNN. Example: {0:'a', 1:'b', 2:'c', ...}.
import numpy as np

# data I/O
data = open('input.txt', 'r').read() # should be simple plain text file
chars = list(set(data))
data_size, vocab_size = len(data), len(chars)
print 'data has %d characters, %d unique.' % (data_size, vocab_size)
char_to_ix = { ch:i for i,ch in enumerate(chars) }
ix_to_char = { i:ch for i,ch in enumerate(chars) }

Forward pass

Let $c(t)$ the character read at $t$ and $c(t+1)$ the next character to be predicted. $x(t)$ encodes $c(t)$ as a vector of vocab_size size.

$y(t)$ encodes probabilistically the next letter to be predicted $c(t+1)$.

Using the example of previous encoding dictionary:

In accordance with the structure of the RNN, the equations relating $ x(t) $ and $ p(t) $ are the following:

We also find these equations “forward” in the sampling function.

# hyperparameters
hidden_size = 100 # size of hidden layer of neurons
seq_length = 25 # number of steps to unroll the RNN for
learning_rate = 1e-1

# model parameters
Wxh = np.random.randn(hidden_size, vocab_size) * 0.01 # input to hidden
Whh = np.random.randn(hidden_size, hidden_size) * 0.01 # hidden to hidden
Why = np.random.randn(vocab_size, hidden_size) * 0.01 # hidden to output
bh = np.zeros((hidden_size, 1)) # hidden bias
by = np.zeros((vocab_size, 1)) # output bias

def lossFun(inputs, targets, hprev):
    xs, hs, ys, ps = {}, {}, {}, {}
    hs[-1] = np.copy(hprev)
    loss = 0
    # forward pass
    for t in xrange(len(inputs)):
        xs[t] = np.zeros((vocab_size,1)) # encode in 1-of-k representation
        xs[t][inputs[t]] = 1
        hs[t] = np.tanh(np.dot(Wxh, xs[t]) + np.dot(Whh, hs[t-1]) + bh) # hidden state
        ys[t] = np.dot(Why, hs[t]) + by # unnormalized log probabilities for next chars
        ps[t] = np.exp(ys[t]) / np.sum(np.exp(ys[t])) # probabilities for next chars
        loss += -np.log(ps[t][targets[t],0]) # softmax (cross-entropy loss)
    # backward pass: compute gradients going backwards
    dWxh, dWhh, dWhy = np.zeros_like(Wxh), np.zeros_like(Whh), np.zeros_like(Why)
    dbh, dby = np.zeros_like(bh), np.zeros_like(by)
    dhnext = np.zeros_like(hs[0])

Backward pass

Let the cost function: .

Let $ \Delta y(n) = \hat{y}(n) - y(n) $.

The minimization of cost function using gradient descent gives:

where

Same for and .

# continuing def lossFun(inputs, targets, hprev)
    for t in reversed(xrange(len(inputs))):
        dy = np.copy(ps[t])
        dy[targets[t]] -= 1 # backprop into y
        dWhy += np.dot(dy, hs[t].T)
        dby += dy
        dh = np.dot(Why.T, dy) + dhnext # backprop into h
        dhraw = (1 - hs[t] * hs[t]) * dh # backprop through tanh nonlinearity
        dbh += dhraw
        dWxh += np.dot(dhraw, xs[t].T)
        dWhh += np.dot(dhraw, hs[t-1].T)
        dhnext = np.dot(Whh.T, dhraw)
    for dparam in [dWxh, dWhh, dWhy, dbh, dby]:
        np.clip(dparam, -5, 5, out=dparam) # clip to mitigate exploding gradients
    return loss, dWxh, dWhh, dWhy, dbh, dby, hs[len(inputs)-1]

Sampling

The forward propagation equation is used.

Then the probability distribution estimated by the RNN is used to select randomly the letter sent to output.

def sample(h, seed_ix, n):
    x = np.zeros((vocab_size, 1))
    x[seed_ix] = 1
    ixes = []
    for t in xrange(n):
        h = np.tanh(np.dot(Wxh, x) + np.dot(Whh, h) + bh)
        y = np.dot(Why, h) + by
        p = np.exp(y) / np.sum(np.exp(y))
        ix = np.random.choice(range(vocab_size), p=p.ravel())
        x = np.zeros((vocab_size, 1))
        x[ix] = 1
        ixes.append(ix)
    return ixes

Main code

Main code of the program that calls successively:

  • Read the input and target
  • Random drawing of Monte Carlo type
  • Update the neural network parameters
n, p = 0, 0
mWxh, mWhh, mWhy = np.zeros_like(Wxh), np.zeros_like(Whh), np.zeros_like(Why)
mbh, mby = np.zeros_like(bh), np.zeros_like(by) # memory variables for Adagrad
smooth_loss = -np.log(1.0/vocab_size) * seq_length # loss at iteration 0
while True:
    # prepare inputs (we're sweeping from left to right in steps seq_length long)
    if p+seq_length+1 >= len(data) or n == 0:
        hprev = np.zeros((hidden_size,1)) # reset RNN memory
        p = 0 # go from start of data
    inputs = [char_to_ix[ch] for ch in data[p:p+seq_length]]
    targets = [char_to_ix[ch] for ch in data[p+1:p+seq_length+1]]

    # sample from the model now and then
    if n % 100 == 0:
        sample_ix = sample(hprev, inputs[0], 200)
        txt = ''.join(ix_to_char[ix] for ix in sample_ix)
        print '----\n %s \n----' % (txt, )

    # forward seq_length characters through the net and fetch gradient
    loss, dWxh, dWhh, dWhy, dbh, dby, hprev = lossFun(inputs, targets, hprev)
    smooth_loss = smooth_loss * 0.999 + loss * 0.001
    if n % 100 == 0: print 'iter %d, loss: %f' % (n, smooth_loss) # print progress

    # perform parameter update with Adagrad
    for param, dparam, mem in zip([Wxh, Whh, Why, bh, by],
                                  [dWxh, dWhh, dWhy, dbh, dby],
                                  [mWxh, mWhh, mWhy, mbh, mby]):
        mem += dparam * dparam
        param += -learning_rate * dparam / np.sqrt(mem + 1e-8) # adagrad update

    p += seq_length # move data pointer
    n += 1 # iteration counter