Imports¶

In [1]:
import numpy as np
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from IPython import display
plt.style.use('seaborn-white')


In [2]:
data = open('input.txt', 'r').read()


Process data and calculate indexes

In [3]:
chars = list(set(data))
data_size, X_size = len(data), len(chars)
print("data has %d characters, %d unique" % (data_size, X_size))
char_to_idx = {ch:i for i,ch in enumerate(chars)}
idx_to_char = {i:ch for i,ch in enumerate(chars)}

data has 1115402 characters, 65 unique


Parameters¶

In [4]:
H_size = 100 # Size of the hidden layer
T_steps = 25 # Number of time steps (length of the sequence) used for training
learning_rate = 1e-1 # Learning rate
weight_sd = 0.1 # Standard deviation of weights for initialization
z_size = H_size + X_size # Size of concatenate(H, X) vector


Activation Functions and Derivatives¶

Sigmoid¶

\begin{align} \sigma(x) &= \frac{1}{1 + e^{-x}}\\ \frac{d\sigma(x)}{dx} &= \sigma(x) \cdot (1 - \sigma(x)) \end{align}

Tanh¶

\begin{align} \frac{d\text{tanh}(x)}{dx} &= 1 - \text{tanh}^2(x) \end{align}
In [5]:
def sigmoid(x):
return 1 / (1 + np.exp(-x))

def dsigmoid(y):
return y * (1 - y)

def tanh(x):
return np.tanh(x)

def dtanh(y):
return 1 - y * y


Initialize weights¶

We use random weights with normal distribution (0, weight_sd) for $tanh$ activation function and (0.5, weight_sd) for $sigmoid$ activation function.

Biases are initialized to zeros.

Formulae for LSTM are shown below.

In [6]:
W_f = np.random.randn(H_size, z_size) * weight_sd + 0.5
b_f = np.zeros((H_size, 1))

W_i = np.random.randn(H_size, z_size) * weight_sd + 0.5
b_i = np.zeros((H_size, 1))

W_C = np.random.randn(H_size, z_size) * weight_sd
b_C = np.zeros((H_size, 1))

W_o = np.random.randn(H_size, z_size) * weight_sd + 0.5
b_o = np.zeros((H_size, 1))

#For final layer to predict the next character
W_y = np.random.randn(X_size, H_size) * weight_sd
b_y = np.zeros((X_size, 1))


In [7]:
dW_f = np.zeros_like(W_f)
dW_i = np.zeros_like(W_i)
dW_C = np.zeros_like(W_C)

dW_o = np.zeros_like(W_o)
dW_y = np.zeros_like(W_y)

db_f = np.zeros_like(b_f)
db_i = np.zeros_like(b_i)
db_C = np.zeros_like(b_C)

db_o = np.zeros_like(b_o)
db_y = np.zeros_like(b_y)


Forward pass¶

Image taken from Understanding LSTM Networks. Please read the article for a good explanation of LSTMs.

Concatenation of $h_{t-1}$ and $x_t$¶

\begin{align} z & = [h_{t-1}, x_t] \\ \end{align}

LSTM functions¶

\begin{align} f_t & = \sigma(W_f \cdot z + b_f) \\ i_t & = \sigma(W_i \cdot z + b_i) \\ \bar{C}_t & = tanh(W_C \cdot z + b_C) \\ C_t & = f_t * C_{t-1} + i_t * \bar{C}_t \\ o_t & = \sigma(W_o \cdot z + b_t) \\ h_t &= o_t * tanh(C_t) \\ \end{align}

Logits¶

\begin{align} y_t &= W_y \cdot h_t + b_y \\ \end{align}

Softmax¶

\begin{align} \hat{p_t} &= \text{softmax}(y_t) \end{align}

$\mathbf{\hat{p_t}}$ is p in code and $\mathbf{p_t}$ is targets.

In [8]:
def forward(x, h_prev, C_prev):
assert x.shape == (X_size, 1)
assert h_prev.shape == (H_size, 1)
assert C_prev.shape == (H_size, 1)

z = np.row_stack((h_prev, x))
f = sigmoid(np.dot(W_f, z) + b_f)
i = sigmoid(np.dot(W_i, z) + b_i)
C_bar = tanh(np.dot(W_C, z) + b_C)

C = f * C_prev + i * C_bar
o = sigmoid(np.dot(W_o, z) + b_o)
h = o * tanh(C)

y = np.dot(W_y, h) + b_y
p = np.exp(y) / np.sum(np.exp(y))

return z, f, i, C_bar, C, o, h, y, p


Backward pass¶

Loss¶

\begin{align} \mathcal{L} &= -\sum p_{t,j} log \hat{p_{t,j}} \\ \end{align}

\begin{align} dy_t &= \hat{p_t} - p_t \\ dh_t &= dh'_{t+1} + W_y^T \cdot d_y \\ do_t &= dh_t * \text{tanh}(C_t) \\ dC_t &= dC'_{t+1} + dh_t * o_t * (1 - \text{tanh}^2(C_t))\\ d\bar{C}_t &= dC_t * i_t \\ di_t &= dC_t * \bar{C}_t \\ df_t &= dC_t * C_{t-1} \\ dz_t &= W_f^T \cdot df_t + W_i^T \cdot di_t + W_C^T \cdot d\bar{C}_t + W_o^T \cdot do_t \\ [dh'_t, dx_t] &= dz_t \\ dC'_t &= f * dC_t \end{align}
• target is target character index $\mathbf{p_t}$
• dh_next is $\mathbf{dh_{t+1}}$ (size H x 1)
• dC_next is $\mathbf{dC_{t+1}}$ (size H x 1)
• C_prev is $\mathbf{C_{t-1}}$ (size H x 1)
• Returns $\mathbf{dh_t}$ and $\mathbf{dC_t}$
In [9]:
def backward(target, dh_next, dC_next, C_prev, z, f, i, C_bar, C, o, h, y, p):

global dW_f, dW_i, dW_C, dW_o, dW_y
global db_f, db_i, db_C, db_o, db_y

assert z.shape == (X_size + H_size, 1)
assert y.shape == (X_size, 1)
assert p.shape == (X_size, 1)

for param in [dh_next, dC_next, C_prev, f, i, C_bar, C, o, h]:
assert param.shape == (H_size, 1)

dy = np.copy(p)
dy[target] -= 1

dW_y += np.dot(dy, h.T)
db_y += dy

dh = np.dot(W_y.T, dy)
dh += dh_next
do = dh * tanh(C)
do = dsigmoid(o) * do
dW_o += np.dot(do, z.T)
db_o += do

dC = np.copy(dC_next)
dC += dh * o * dtanh(tanh(C))
dC_bar = dC * i
dC_bar = dC_bar * dtanh(C_bar)
dW_C += np.dot(dC_bar, z.T)
db_C += dC_bar

di = dC * C_bar
di = dsigmoid(i) * di
dW_i += np.dot(di, z.T)
db_i += di

df = dC * C_prev
df = dsigmoid(f) * df
dW_f += np.dot(df, z.T)
db_f += df

dz = np.dot(W_f.T, df) \
+ np.dot(W_i.T, di) \
+ np.dot(W_C.T, dC_bar) \
+ np.dot(W_o.T, do)
dh_prev = dz[:H_size, :]
dC_prev = f * dC

return dh_prev, dC_prev


Forward Backward Pass¶

Calculate and store the values in forward pass. Accumulate gradients in backward pass and clip gradients to avoid exploding gradients.

• input, target are list of integers, with character indexes.
• h_prev is the array of initial h at $\mathbf{h_-1}$ (size H x 1)
• C_prev is the array of initial C at $\mathbf{C_-1}$ (size H x 1)
• Returns loss, final $\mathbf{h_T}$ and $\mathbf{C_T}$
In [10]:
def forward_backward(inputs, targets, h_prev, C_prev):
# To store the values for each time step
x_s, z_s, f_s, i_s, C_bar_s, C_s, o_s, h_s, y_s, p_s = {}, {}, {}, {}, {}, {}, {}, {}, {}, {}

# Values at t - 1
h_s[-1] = np.copy(h_prev)
C_s[-1] = np.copy(C_prev)

loss = 0
# Loop through time steps
assert len(inputs) == T_steps
for t in range(len(inputs)):
x_s[t] = np.zeros((X_size, 1))
x_s[t][inputs[t]] = 1 # Input character

z_s[t], f_s[t], i_s[t], C_bar_s[t], C_s[t], o_s[t], h_s[t], y_s[t], p_s[t] \
= forward(x_s[t], h_s[t - 1], C_s[t - 1]) # Forward pass

loss += -np.log(p_s[t][targets[t], 0]) # Loss for at t

for dparam in [dW_f, dW_i, dW_C, dW_o, dW_y, db_f, db_i, db_C, db_o, db_y]:
dparam.fill(0)

dh_next = np.zeros_like(h_s[0]) #dh from the next character
dC_next = np.zeros_like(C_s[0]) #dh from the next character

for t in reversed(range(len(inputs))):
# Backward pass
dh_next, dC_next = backward(target = targets[t], dh_next = dh_next, dC_next = dC_next, C_prev = C_s[t-1],
z = z_s[t], f = f_s[t], i = i_s[t], C_bar = C_bar_s[t], C = C_s[t], o = o_s[t],
h = h_s[t], y = y_s[t], p = p_s[t])

for dparam in [dW_f, dW_i, dW_C, dW_o, dW_y, db_f, db_i, db_C, db_o, db_y]:
np.clip(dparam, -1, 1, out=dparam)

return loss, h_s[len(inputs) - 1], C_s[len(inputs) - 1]


Sample the next character¶

In [11]:
def sample(h_prev, C_prev, first_char_idx, sentence_length):
x = np.zeros((X_size, 1))
x[first_char_idx] = 1

h = h_prev
C = C_prev

indexes = []

for t in range(sentence_length):
_, _, _, _, C, _, h, _, p = forward(x, h, C)
idx = np.random.choice(range(X_size), p=p.ravel())
x = np.zeros((X_size, 1))
x[idx] = 1
indexes.append(idx)

return indexes


\begin{align} w = w - \eta\frac{dw}{\sum dw_{\tau}^2} \end{align}
In [12]:
def update_status(inputs, h_prev, C_prev):
#initialized later
global plot_iter, plot_loss
global smooth_loss

# Get predictions for 200 letters with current model
display.clear_output(wait=True)

sample_idx = sample(h_prev, C_prev, inputs[0], 200)
txt = ''.join(idx_to_char[idx] for idx in sample_idx)

# Clear and plot
plt.plot(plot_iter, plot_loss)
display.display(plt.gcf())

#Print prediction and loss
print("----\n %s \n----" % (txt, ))
print("iter %d, loss %f" % (iteration, smooth_loss))


In [13]:
mW_f = np.zeros_like(W_f)
mW_i = np.zeros_like(W_i)
mW_C = np.zeros_like(W_C)
mW_o = np.zeros_like(W_o)
mW_y = np.zeros_like(W_y)

mb_f = np.zeros_like(b_f)
mb_i = np.zeros_like(b_i)
mb_C = np.zeros_like(b_C)
mb_o = np.zeros_like(b_o)
mb_y = np.zeros_like(b_y)

In [14]:
# Exponential average of loss
# Initialize to a error of a random model
smooth_loss = -np.log(1.0 / X_size) * T_steps

iteration, p = 0, 0

# For the graph
plot_iter = np.zeros((0))
plot_loss = np.zeros((0))

In [15]:
while True:
# Try catch for interruption
try:
# Reset
if p + T_steps >= len(data) or iteration == 0:
g_h_prev = np.zeros((H_size, 1))
g_C_prev = np.zeros((H_size, 1))
p = 0

inputs = [char_to_idx[ch] for ch in data[p: p + T_steps]]
targets = [char_to_idx[ch] for ch in data[p + 1: p + T_steps + 1]]

loss, g_h_prev, g_C_prev =  forward_backward(inputs, targets, g_h_prev, g_C_prev)
smooth_loss = smooth_loss * 0.999 + loss * 0.001

# Print every hundred steps
if iteration % 100 == 0:
update_status(inputs, g_h_prev, g_C_prev)

# Update weights
for param, dparam, mem in zip([W_f, W_i, W_C, W_o, W_y, b_f, b_i, b_C, b_o, b_y],
[dW_f, dW_i, dW_C, dW_o, dW_y, db_f, db_i, db_C, db_o, db_y],
[mW_f, mW_i, mW_C, mW_o, mW_y, mb_f, mb_i, mb_C, mb_o, mb_y]):
mem += dparam * dparam # Calculate sum of gradients
#print(learning_rate * dparam)
param += -(learning_rate * dparam / np.sqrt(mem + 1e-8))

plot_iter = np.append(plot_iter, [iteration])
plot_loss = np.append(plot_loss, [loss])

p += T_steps
iteration += 1
except KeyboardInterrupt:
update_status(inputs, g_h_prev, g_C_prev)
break

----
tingul---
Bue chapced
gleaphar beia allidi:
Whee,, fort, we lades you coold him
We lave
And the bogiend toinctes inscieniud.

SOCINIUS:
What velyed
! mand his where,
your beer; in his one shangs! Cove
----
iter 5173, loss 46.208255


Approximate the numerical gradients by changing parameters and running the model. Check if the approximated gradients are equal to the computed analytical gradients (by backpropagation).

Try this on num_checks individual paramters picked randomly for each weight matrix and bias vector.

In [16]:
from random import uniform

In [17]:
def gradient_check(inputs, target, h_prev, C_prev):
global W_f, W_i, W_C, W_o, W_y, b_f, b_i, b_C, b_o, b_y
global dW_f, dW_i, dW_C, dW_o, dW_y, db_f, db_i, db_C, db_o, db_y

num_checks = 10 # Number of parameters to test
delta = 1e-5 # The change to make on the parameter

_, _, _ =  forward_backward(inputs, targets, h_prev, C_prev)

for param, dparam, name in zip([W_f, W_i, W_C, W_o, W_y, b_f, b_i, b_C, b_o, b_y],
[dW_f, dW_i, dW_C, dW_o, dW_y, db_f, db_i, db_C, db_o, db_y],
['W_f', 'W_i', 'W_C', 'W_o', 'W_y', 'b_f', 'b_i', 'b_C', 'b_o', 'b_y']):
assert param.shape == dparam.shape
dparam_copy = np.copy(dparam) #Make a copy because this will get modified

# Test num_checks times
for i in range(num_checks):
# Pick a random index
rnd_idx = int(uniform(0,param.size))

# evaluate cost at [x + delta] and [x - delta]
old_val = param.flat[rnd_idx]
param.flat[rnd_idx] = old_val + delta
loss_plus_delta, _, _ = forward_backward(inputs, targets, h_prev, C_prev)
param.flat[rnd_idx] = old_val - delta
loss_mins_delta, _, _ = forward_backward(inputs, targets, h_prev, C_prev)
param.flat[rnd_idx] = old_val

grad_numerical = (loss_plus_delta - loss_mins_delta) / (2 * delta)
# Clip numerical error because grad_analytical is clipped


gradient_check(inputs, targets, g_h_prev, g_C_prev)