吳恩達 Coursera 課程 DeepLearning.ai 程式設計作業系列，本文為《序列模型》部分的第一週“迴圈神經網路”的課程作業——第二部分：字母級的語言模型：恐龍獸島。

Character level language model - Dinosaurus land

Welcome to Dinosaurus Island! 65 million years ago, dinosaurs existed, and in this assignment they are back. You are in charge of a special task. Leading biology researchers are creating new breeds of dinosaurs and bringing them to life on earth, and your job is to give names to these dinosaurs. If a dinosaur does not like its name, it might go beserk, so choose wisely!

Luckily you have learned some deep learning and you will use it to save the day. Your assistant has collected a list of all the dinosaur names they could find, and compiled them into this dataset. (Feel free to take a look by clicking the previous link.) To create new dinosaur names, you will build a character level language model to generate new names. Your algorithm will learn the different name patterns, and randomly generate new names. Hopefully this algorithm will keep you and your team safe from the dinosaurs’ wrath!

By completing this assignment you will learn:

• How to store text data for processing using an RNN
• How to synthesize data, by sampling predictions at each time step and passing it to the next RNN-cell unit
• How to build a character-level text generation recurrent neural network
• Why clipping the gradients is important

We will begin by loading in some functions that we have provided for you in rnn_utils. Specifically, you have access to functions such as rnn_forward and rnn_backward which are equivalent to those you’ve implemented in the previous assignment.

import numpy as np
from utils import *
import random

在utils包中，我們會用到的函式：

import numpy as np

def softmax(x):
    e_x = np.exp(x - np.max(x))
    return e_x / e_x.sum(axis=0)

def smooth(loss, cur_loss):
    return loss * 0.999 + cur_loss * 0.001

def print_sample(sample_ix, ix_to_char):
    txt = ''.join(ix_to_char[ix] for ix in sample_ix)
    txt = txt[0].upper() + txt[1:]  # capitalize first character 
    print ('%s' % (txt, ), end='')

def get_initial_loss(vocab_size, seq_length):
    return -np.log(1.0/vocab_size)*seq_length

def softmax(x):
    e_x = np.exp(x - np.max(x))
    return e_x / e_x.sum(axis=0)

def initialize_parameters(n_a, n_x, n_y):
    """
    Initialize parameters with small random values

    Returns:
    parameters -- python dictionary containing:
                        Wax -- Weight matrix multiplying the input, numpy array of shape (n_a, n_x)
                        Waa -- Weight matrix multiplying the hidden state, numpy array of shape (n_a, n_a)
                        Wya -- Weight matrix relating the hidden-state to the output, numpy array of shape (n_y, n_a)
                        b --  Bias, numpy array of shape (n_a, 1)
                        by -- Bias relating the hidden-state to the output, numpy array of shape (n_y, 1)
    """
    np.random.seed(1)
    Wax = np.random.randn(n_a, n_x)*0.01 # input to hidden
    Waa = np.random.randn(n_a, n_a)*0.01 # hidden to hidden
    Wya = np.random.randn(n_y, n_a)*0.01 # hidden to output
    b = np.zeros((n_a, 1)) # hidden bias
    by = np.zeros((n_y, 1)) # output bias

    parameters = {"Wax": Wax, "Waa": Waa, "Wya": Wya, "b": b,"by": by}

    return parameters

def rnn_step_forward(parameters, a_prev, x):

    Waa, Wax, Wya, by, b = parameters['Waa'], parameters['Wax'], parameters['Wya'], parameters['by'], parameters['b']
    a_next = np.tanh(np.dot(Wax, x) + np.dot(Waa, a_prev) + b) # hidden state
    p_t = softmax(np.dot(Wya, a_next) + by) # unnormalized log probabilities for next chars # probabilities for next chars 

    return a_next, p_t

def rnn_step_backward(dy, gradients, parameters, x, a, a_prev):

    gradients['dWya'] += np.dot(dy, a.T)
    gradients['dby'] += dy
    da = np.dot(parameters['Wya'].T, dy) + gradients['da_next'] # backprop into h
    daraw = (1 - a * a) * da # backprop through tanh nonlinearity
    gradients['db'] += daraw
    gradients['dWax'] += np.dot(daraw, x.T)
    gradients['dWaa'] += np.dot(daraw, a_prev.T)
    gradients['da_next'] = np.dot(parameters['Waa'].T, daraw)
    return gradients

def update_parameters(parameters, gradients, lr):

    parameters['Wax'] += -lr * gradients['dWax']
    parameters['Waa'] += -lr * gradients['dWaa']
    parameters['Wya'] += -lr * gradients['dWya']
    parameters['b']  += -lr * gradients['db']
    parameters['by']  += -lr * gradients['dby']
    return parameters

def rnn_forward(X, Y, a0, parameters, vocab_size = 27):

    # Initialize x, a and y_hat as empty dictionaries
    x, a, y_hat = {}, {}, {}

    a[-1] = np.copy(a0)

    # initialize your loss to 0
    loss = 0

    for t in range(len(X)):

        # Set x[t] to be the one-hot vector representation of the t'th character in X.
        # if X[t] == None, we just have x[t]=0. This is used to set the input for the first timestep to the zero vector. 
        x[t] = np.zeros((vocab_size,1)) 
        if (X[t] != None):
            x[t][X[t]] = 1

        # Run one step forward of the RNN
        a[t], y_hat[t] = rnn_step_forward(parameters, a[t-1], x[t])

        # Update the loss by substracting the cross-entropy term of this time-step from it.
        loss -= np.log(y_hat[t][Y[t],0])

    cache = (y_hat, a, x)

    return loss, cache

def rnn_backward(X, Y, parameters, cache):
    # Initialize gradients as an empty dictionary
    gradients = {}

    # Retrieve from cache and parameters
    (y_hat, a, x) = cache
    Waa, Wax, Wya, by, b = parameters['Waa'], parameters['Wax'], parameters['Wya'], parameters['by'], parameters['b']

    # each one should be initialized to zeros of the same dimension as its corresponding parameter
    gradients['dWax'], gradients['dWaa'], gradients['dWya'] = np.zeros_like(Wax), np.zeros_like(Waa), np.zeros_like(Wya)
    gradients['db'], gradients['dby'] = np.zeros_like(b), np.zeros_like(by)
    gradients['da_next'] = np.zeros_like(a[0])

    ### START CODE HERE ###
    # Backpropagate through time
    for t in reversed(range(len(X))):
        dy = np.copy(y_hat[t])
        dy[Y[t]] -= 1
        gradients = rnn_step_backward(dy, gradients, parameters, x[t], a[t], a[t-1])
    ### END CODE HERE ###

    return gradients, a

該部分需要用到的資料集和模型檔案，點選這裡獲取。

1 - Problem Statement

1.1 - Dataset and Preprocessing

Run the following cell to read the dataset of dinosaur names, create a list of unique characters (such as a-z), and compute the dataset and vocabulary size.

data = open('dinos.txt', 'r').read()
data= data.lower()
chars = list(set(data))
data_size, vocab_size = len(data), len(chars)
print('There are %d total characters and %d unique characters in your data.' % (data_size, vocab_size))

There are 19909 total characters and 27 unique characters in your data.

The characters are a-z (26 characters) plus the “\n” (or newline character), which in this assignment plays a role similar to the <EOS> (or “End of sentence”) token we had discussed in lecture, only here it indicates the end of the dinosaur name rather than the end of a sentence. In the cell below, we create a python dictionary (i.e., a hash table) to map each character to an index from 0-26. We also create a second python dictionary that maps each index back to the corresponding character character. This will help you figure out what index corresponds to what character in the probability distribution output of the softmax layer. Below, char_to_ix and ix_to_char are the python dictionaries.

char_to_ix = { ch:i for i,ch in enumerate(sorted(chars)) }
ix_to_char = { i:ch for i,ch in enumerate(sorted(chars)) }
print(ix_to_char)

{0: '\n', 1: 'a', 2: 'b', 3: 'c', 4: 'd', 5: 'e', 6: 'f', 7: 'g', 8: 'h', 9: 'i', 10: 'j', 11: 'k', 12: 'l', 13: 'm', 14: 'n', 15: 'o', 16: 'p', 17: 'q', 18: 'r', 19: 's', 20: 't', 21: 'u', 22: 'v', 23: 'w', 24: 'x', 25: 'y', 26: 'z'}

1.2 - Overview of the model

Your model will have the following structure:

• Initialize parameters
• Run the optimization loop
  
  • Forward propagation to compute the loss function
  • Backward propagation to compute the gradients with respect to the loss function
  • Clip the gradients to avoid exploding gradients
  • Using the gradients, update your parameter with the gradient descent update rule.
• Return the learned parameters

At each time-step, the RNN tries to predict what is the next character given the previous characters. The dataset X=(x⟨1⟩,x⟨2⟩,...,x⟨Tx⟩) is a list of characters in the training set, while Y=(y⟨1⟩,y⟨2⟩,...,y⟨Tx⟩) is such that at every time-step t, we have y⟨t⟩=x⟨t+1⟩.

2 - Building blocks of the model

In this part, you will build two important blocks of the overall model:
- Gradient clipping: to avoid exploding gradients
- Sampling: a technique used to generate characters

You will then apply these two functions to build the model.

2.1 - Clipping the gradients in the optimization loop

In this section you will implement the clip function that you will call inside of your optimization loop. Recall that your overall loop structure usually consists of a forward pass, a cost computation, a backward pass, and a parameter update. Before updating the parameters, you will perform gradient clipping when needed to make sure that your gradients are not “exploding,” meaning taking on overly large values.

In the exercise below, you will implement a function clip that takes in a dictionary of gradients and returns a clipped version of gradients if needed. There are different ways to clip gradients; we will use a simple element-wise clipping procedure, in which every element of the gradient vector is clipped to lie between some range [-N, N]. More generally, you will provide a maxValue (say 10). In this example, if any component of the gradient vector is greater than 10, it would be set to 10; and if any component of the gradient vector is less than -10, it would be set to -10. If it is between -10 and 10, it is left alone.

Exercise: Implement the function below to return the clipped gradients of your dictionary gradients. Your function takes in a maximum threshold and returns the clipped versions of your gradients. You can check out this hint for examples of how to clip in numpy. You will need to use the argument out = ....

### GRADED FUNCTION: clip

def clip(gradients, maxValue):
    '''
    Clips the gradients' values between minimum and maximum.

    Arguments:
    gradients -- a dictionary containing the gradients "dWaa", "dWax", "dWya", "db", "dby"
    maxValue -- everything above this number is set to this number, and everything less than -maxValue is set to -maxValue

    Returns: 
    gradients -- a dictionary with the clipped gradients.
    '''

    dWaa, dWax, dWya, db, dby = gradients['dWaa'], gradients['dWax'], gradients['dWya'], gradients['db'], gradients['dby']

    ### START CODE HERE ###
    # clip to mitigate exploding gradients, loop over [dWax, dWaa, dWya, db, dby]. (≈2 lines)
    for gradient in [dWax, dWaa, dWya, db, dby]:
        np.clip(gradient, -maxValue, maxValue, out=gradient)
    ### END CODE HERE ###

    gradients = {"dWaa": dWaa, "dWax": dWax, "dWya": dWya, "db": db, "dby": dby}

    return gradients

np.random.seed(3)
dWax = np.random.randn(5,3)*10
dWaa = np.random.randn(5,5)*10
dWya = np.random.randn(2,5)*10
db = np.random.randn(5,1)*10
dby = np.random.randn(2,1)*10
gradients = {"dWax": dWax, "dWaa": dWaa, "dWya": dWya, "db": db, "dby": dby}
gradients = clip(gradients, 10)
print("gradients[\"dWaa\"][1][2] =", gradients["dWaa"][1][2])
print("gradients[\"dWax\"][3][1] =", gradients["dWax"][3][1])
print("gradients[\"dWya\"][1][2] =", gradients["dWya"][1][2])
print("gradients[\"db\"][4] =", gradients["db"][4])
print("gradients[\"dby\"][1] =", gradients["dby"][1])

gradients["dWaa"][1][2] = 10.0
gradients["dWax"][3][1] = -10.0
gradients["dWya"][1][2] = 0.29713815361
gradients["db"][4] = [ 10.]
gradients["dby"][1] = [ 8.45833407]

* Expected output:*

2.2 - Sampling

Now assume that your model is trained. You would like to generate new text (characters). The process of generation is explained in the picture below:

Exercise: Implement the sample function below to sample characters. You need to carry out 4 steps:

• Step 1: Pass the network the first “dummy” input x⟨1⟩=0⃗  (the vector of zeros). This is the default input before we’ve generated any characters. We also set a⟨0⟩=0⃗ 

• Step 2: Run one step of forward propagation to get a⟨1⟩ and y^⟨1⟩. Here are the equations:

Note that y^⟨t+1⟩ is a (softmax) probability vector