Character level language model - Dinosaurus Island¶

Welcome to Dinosaurus Island! 65 million years ago, dinosaurs existed, and in this assignment, they have returned.

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 berserk, so choose wisely!

Luckily you're equipped with some deep learning now, 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 the time you complete this assignment, you'll be able to:

• Store text data for processing using an RNN
• Build a character-level text generation model using an RNN
• Sample novel sequences in an RNN
• Explain the vanishing/exploding gradient problem in RNNs
• Apply gradient clipping as a solution for exploding gradients

Begin by loading in some functions that are 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.

Important Note on Submission to the AutoGrader¶

Before submitting your assignment to the AutoGrader, please make sure you are not doing the following:

1. You have not added any extra print statement(s) in the assignment.
2. You have not added any extra code cell(s) in the assignment.
3. You have not changed any of the function parameters.
4. You are not using any global variables inside your graded exercises. Unless specifically instructed to do so, please refrain from it and use the local variables instead.
5. You are not changing the assignment code where it is not required, like creating extra variables.

If you do any of the following, you will get something like, Grader Error: Grader feedback not found (or similarly unexpected) error upon submitting your assignment. Before asking for help/debugging the errors in your assignment, check for these first. If this is the case, and you don't remember the changes you have made, you can get a fresh copy of the assignment by following these instructions.

Table of Contents¶

• Packages
• 1 - Problem Statement
  • 1.1 - Dataset and Preprocessing
  • 1.2 - Overview of the Model
• 2 - Building Blocks of the Model
  • 2.1 - Clipping the Gradients in the Optimization Loop
    • Exercise 1 - clip
  • 2.2 - Sampling
    • Exercise 2 - sample
• 3 - Building the Language Model
  • 3.1 - Gradient Descent
    • Exercise 3 - optimize
  • 3.2 - Training the Model
    • Exercise 4 - model
• 4 - Writing like Shakespeare (OPTIONAL/UNGRADED)
• 5 - References

Packages¶

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.

• The characters are a-z (26 characters) plus the "\n" (or newline character).
• In this assignment, the newline character "\n" plays a role similar to the <EOS> (or "End of sentence") token discussed in lecture.
  • Here, "\n" indicates the end of the dinosaur name rather than the end of a sentence.
• char_to_ix: In the cell below, you'll create a Python dictionary (i.e., a hash table) to map each character to an index from 0-26.
• ix_to_char: Then, you'll create a second Python dictionary that maps each index back to the corresponding character.
  • This will help you figure out which index corresponds to which character in the probability distribution output of the softmax layer.

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 parameters with the gradient descent update rule.
• Return the learned parameters

• At each time-step, the RNN tries to predict what the next character is, given the previous characters.
• $\mathbf{X} = (x^{\langle 1 \rangle}, x^{\langle 2 \rangle}, ..., x^{\langle T_x \rangle})$ is a list of characters from the training set.
• $\mathbf{Y} = (y^{\langle 1 \rangle}, y^{\langle 2 \rangle}, ..., y^{\langle T_x \rangle})$ is the same list of characters but shifted one character forward.
• At every time-step $t$, $y^{\langle t \rangle} = x^{\langle t+1 \rangle}$. The prediction at time $t$ is the same as the input at time $t + 1$.

2 - Building Blocks of the Model¶

In this part, you will build two important blocks of the overall model:

1. Gradient clipping: to avoid exploding gradients
2. 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.

Exploding gradients¶

• When gradients are very large, they're called "exploding gradients."
• Exploding gradients make the training process more difficult, because the updates may be so large that they "overshoot" the optimal values during back propagation.

Recall that your overall loop structure usually consists of:

• forward pass,
• cost computation,
• backward pass,
• parameter update.

Before updating the parameters, you will perform gradient clipping to make sure that your gradients are not "exploding."

Gradient clipping¶

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.
• You will use a simple element-wise clipping procedure, in which every element of the gradient vector is clipped to fall between some range [-N, N].
• For example, if the N=10
  • The range is [-10, 10]
  • If any component of the gradient vector is greater than 10, it is set to 10.
  • If any component of the gradient vector is less than -10, it is set to -10.
  • If any components are between -10 and 10, they keep their original values.

Exercise 1 - clip¶

Return the clipped gradients of your dictionary gradients.

• Your function takes in a maximum threshold and returns the clipped versions of the gradients.
• You can check out numpy.clip for more info.
  • You will need to use the argument "out = ...".
  • Using the "out" parameter allows you to update a variable "in-place".
  • If you don't use "out" argument, the clipped variable is stored in the variable "gradient" but does not update the gradient variables dWax, dWaa, dWya, db, dby.

Expected values

Gradients for mValue=10
gradients["dWaa"][1][2] = 10.0
gradients["dWax"][3][1] = -10.0
gradients["dWya"][1][2] = 0.2971381536101662
gradients["db"][4] = [10.]
gradients["dby"][1] = [8.45833407]

Gradients for mValue=5
gradients["dWaa"][1][2] = 5.0
gradients["dWax"][3][1] = -5.0
gradients["dWya"][1][2] = 0.2971381536101662
gradients["db"][4] = [5.]
gradients["dby"][1] = [5.]

2.2 - Sampling¶

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

Exercise 2 - sample¶

Implement the sample function below to sample characters.

You need to carry out 4 steps:

• Step 1: Input the "dummy" vector of zeros $x^{\langle 1 \rangle} = \vec{0}$.
  • This is the default input before you've generated any characters. You also set $a^{\langle 0 \rangle} = \vec{0}$

• Step 2: Run one step of forward propagation to get $a^{\langle 1 \rangle}$ and $\hat{y}^{\langle 1 \rangle}$. Here are the equations:

hidden state:
$$ a^{\langle t+1 \rangle} = \tanh(W_{ax} x^{\langle t+1 \rangle } + W_{aa} a^{\langle t \rangle } + b)\tag{1}$$

activation: $$ z^{\langle t + 1 \rangle } = W_{ya} a^{\langle t + 1 \rangle } + b_y \tag{2}$$

prediction: $$ \hat{y}^{\langle t+1 \rangle } = softmax(z^{\langle t + 1 \rangle })\tag{3}$$

• Details about $\hat{y}^{\langle t+1 \rangle }$:
  • Note that $\hat{y}^{\langle t+1 \rangle }$ is a (softmax) probability vector (its entries are between 0 and 1 and sum to 1).
  • $\hat{y}^{\langle t+1 \rangle}_i$ represents the probability that the character indexed by "i" is the next character.
  • A softmax() function is provided for you to use.

• Step 3: Sampling:

  • Now that you have $y^{\langle t+1 \rangle}$, you want to select the next letter in the dinosaur name. If you select the most probable, the model will always generate the same result given a starting letter. To make the results more interesting, use np.random.choice to select a next letter that is likely, but not always the same.
  • Pick the next character's index according to the probability distribution specified by $\hat{y}^{\langle t+1 \rangle }$.
  • This means that if $\hat{y}^{\langle t+1 \rangle }_i = 0.16$, you will pick the index "i" with 16% probability.
  • Use np.random.choice.

  Example of how to use np.random.choice():

  np.random.seed(0)
  probs = np.array([0.1, 0.0, 0.7, 0.2])
  idx = np.random.choice(range(len(probs)), p = probs)
  

  • This means that you will pick the index (idx) according to the distribution:

  $P(index = 0) = 0.1, P(index = 1) = 0.0, P(index = 2) = 0.7, P(index = 3) = 0.2$.

  • Note that the value that's set to p should be set to a 1D vector.
  • Also notice that $\hat{y}^{\langle t+1 \rangle}$, which is y in the code, is a 2D array.
  • Also notice, while in your implementation, the first argument to np.random.choice is just an ordered list [0,1,.., vocab_len-1], it is not appropriate to use char_to_ix.values(). The order of values returned by a Python dictionary .values() call will be the same order as they are added to the dictionary. The grader may have a different order when it runs your routine than when you run it in your notebook.

Additional Hints¶

• Documentation for the built-in Python function range
• Docs for numpy.ravel, which takes a multi-dimensional array and returns its contents inside of a 1D vector.

arr = np.array([[1,2],[3,4]])
print("arr")
print(arr)
print("arr.ravel()")
print(arr.ravel())

Output:

arr
[[1 2]
 [3 4]]
arr.ravel()
[1 2 3 4]

• Note that append is an "in-place" operation, which means the changes made by the method will remain after the call completes. In other words, don't do this:

fun_hobbies = fun_hobbies.append('learning')  ## Doesn't give you what you want!

• Step 4: Update to $x^{\langle t \rangle }$
  • The last step to implement in sample() is to update the variable x, which currently stores $x^{\langle t \rangle }$, with the value of $x^{\langle t + 1 \rangle }$.
  • You will represent $x^{\langle t + 1 \rangle }$ by creating a one-hot vector corresponding to the character that you have chosen as your prediction.
  • You will then forward propagate $x^{\langle t + 1 \rangle }$ in Step 1 and keep repeating the process until you get a "\n" character, indicating that you have reached the end of the dinosaur name.

Additional Hints¶

• In order to reset x before setting it to the new one-hot vector, you'll want to set all the values to zero.
  • You can either create a new numpy array: numpy.zeros
  • Or fill all values with a single number: numpy.ndarray.fill

Expected output

Sampling:
list of sampled indices:
 [23, 16, 26, 26, 24, 3, 21, 1, 7, 24, 15, 3, 25, 20, 6, 13, 10, 8, 20, 12, 2, 0]
list of sampled characters:
 ['w', 'p', 'z', 'z', 'x', 'c', 'u', 'a', 'g', 'x', 'o', 'c', 'y', 't', 'f', 'm', 'j', 'h', 't', 'l', 'b', '\n']

What you should remember:

• Very large, or "exploding" gradients updates can be so large that they "overshoot" the optimal values during back prop -- making training difficult
  • Clip gradients before updating the parameters to avoid exploding gradients
• Sampling is a technique you can use to pick the index of the next character according to a probability distribution.
  • To begin character-level sampling:
    • Input a "dummy" vector of zeros as a default input
    • Run one step of forward propagation to get 𝑎⟨1⟩ (your first character) and 𝑦̂ ⟨1⟩ (probability distribution for the following character)
    • When sampling, avoid generating the same result each time given the starting letter (and make your names more interesting!) by using np.random.choice

3 - Building the Language Model¶

It's time to build the character-level language model for text generation!

3.1 - Gradient Descent¶

In this section you will implement a function performing one step of stochastic gradient descent (with clipped gradients). You'll go through the training examples one at a time, so the optimization algorithm will be stochastic gradient descent.

As a reminder, here are the steps of a common optimization loop for an RNN:

• Forward propagate through the RNN to compute the loss
• Backward propagate through time to compute the gradients of the loss with respect to the parameters
• Clip the gradients
• Update the parameters using gradient descent

Exercise 3 - optimize¶

Implement the optimization process (one step of stochastic gradient descent).

The following functions are provided:

def rnn_forward(X, Y, a_prev, parameters):
    """ Performs the forward propagation through the RNN and computes the cross-entropy loss.
    It returns the loss' value as well as a "cache" storing values to be used in backpropagation."""
    ....
    return loss, cache
    
def rnn_backward(X, Y, parameters, cache):
    """ Performs the backward propagation through time to compute the gradients of the loss with respect
    to the parameters. It returns also all the hidden states."""
    ...
    return gradients, a

def update_parameters(parameters, gradients, learning_rate):
    """ Updates parameters using the Gradient Descent Update Rule."""
    ...
    return parameters

Recall that you previously implemented the clip function:

Parameters¶

• Note that the weights and biases inside the parameters dictionary are being updated by the optimization, even though parameters is not one of the returned values of the optimize function. The parameters dictionary is passed by reference into the function, so changes to this dictionary are making changes to the parameters dictionary even when accessed outside of the function.
• Python dictionaries and lists are "pass by reference", which means that if you pass a dictionary into a function and modify the dictionary within the function, this changes that same dictionary (it's not a copy of the dictionary).

Expected output

Loss = 126.50397572165389
gradients["dWaa"][1][2] = 0.19470931534713587
np.argmax(gradients["dWax"]) = 93
gradients["dWya"][1][2] = -0.007773876032002162
gradients["db"][4] = [-0.06809825]
gradients["dby"][1] = [0.01538192]
a_last[4] = [-1.]

3.2 - Training the Model¶

• Given the dataset of dinosaur names, you'll use each line of the dataset (one name) as one training example.
• Every 2000 steps of stochastic gradient descent, you will sample several randomly chosen names to see how the algorithm is doing.

Exercise 4 - model¶

Implement model().

When examples[index] contains one dinosaur name (string), to create an example (X, Y), you can use this:

Set the index idx into the list of examples¶

• Using the for-loop, walk through the shuffled list of dinosaur names in the list "examples."
• For example, if there are n_e examples, and the for-loop increments the index to n_e onwards, think of how you would make the index cycle back to 0, so that you can continue feeding the examples into the model when j is n_e, n_e + 1, etc.
• Hint: (n_e + 1) % n_e equals 1, which is otherwise the 'remainder' you get when you divide (n_e + 1) by n_e.
• % is the modulo operator in python.

Extract a single example from the list of examples¶

• single_example: use the idx index that you set previously to get one word from the list of examples.

Convert a string into a list of characters: single_example_chars¶

• single_example_chars: A string is a list of characters.
• You can use a list comprehension (recommended over for-loops) to generate a list of characters.

str = 'I love learning'
list_of_chars = [c for c in str]
print(list_of_chars)

['I', ' ', 'l', 'o', 'v', 'e', ' ', 'l', 'e', 'a', 'r', 'n', 'i', 'n', 'g']

• For more on list comprehensions:

Convert list of characters to a list of integers: single_example_ix¶

• Create a list that contains the index numbers associated with each character.
• Use the dictionary char_to_ix
• You can combine this with the list comprehension that is used to get a list of characters from a string.

Create the list of input characters: X¶

• rnn_forward uses the None value as a flag to set the input vector as a zero-vector.
• Prepend the list [None] in front of the list of input character indices.
• There is more than one way to prepend a value to a list. One way is to add two lists together: ['a'] + ['b']

Get the integer representation of the newline character ix_newline¶

• ix_newline: The newline character signals the end of the dinosaur name.
  • Get the integer representation of the newline character '\n'.
  • Use char_to_ix

Set the list of labels (integer representation of the characters): Y¶

• The goal is to train the RNN to predict the next letter in the name, so the labels are the list of characters that are one time-step ahead of the characters in the input X.
  • For example, Y[0] contains the same value as X[1]
• The RNN should predict a newline at the last letter, so add ix_newline to the end of the labels.
  • Append the integer representation of the newline character to the end of Y.
  • Note that append is an in-place operation.
  • It might be easier for you to add two lists together.

When you run the following cell, you should observe your model outputting random-looking characters at the first iteration. After a few thousand iterations, your model should learn to generate reasonable-looking names.

Expected output

...
Iteration: 22000, Loss: 22.728886

Onustreofkelus
Llecagosaurus
Mystolojmiaterltasaurus
Ola
Yuskeolongus
Eiacosaurus
Trodonosaurus

Conclusion¶

You can see that your algorithm has started to generate plausible dinosaur names towards the end of training. At first, it was generating random characters, but towards the end you could begin to see dinosaur names with cool endings. Feel free to run the algorithm even longer and play with hyperparameters to see if you can get even better results! Our implementation generated some really cool names like maconucon, marloralus and macingsersaurus. Your model hopefully also learned that dinosaur names tend to end in saurus, don, aura, tor, etc.

If your model generates some non-cool names, don't blame the model entirely -- not all actual dinosaur names sound cool. (For example, dromaeosauroides is an actual dinosaur name and is in the training set.) But this model should give you a set of candidates from which you can pick the coolest!

This assignment used a relatively small dataset, so that you're able to train an RNN quickly on a CPU. Training a model of the English language requires a much bigger dataset, and usually much more computation, and could run for many hours on GPUs. We ran our dinosaur name for quite some time, and so far our favorite name is the great, the fierce, the undefeated: Mangosaurus!

Congratulations!¶

You've finished the graded portion of this notebook and created a working language model! Awesome job.

By now, you've:

• Stored text data for processing using an RNN
• Built a character-level text generation model
• Explored the vanishing/exploding gradient problem in RNNs
• Applied gradient clipping to avoid exploding gradients

You've also hopefully generated some dinosaur names that are cool enough to please you and also avoid the wrath of the dinosaurs. If you had fun with the assignment, be sure not to miss the ungraded portion, where you'll be able to generate poetry like the Bard Himself. Good luck and have fun!

4 - Writing like Shakespeare (OPTIONAL/UNGRADED)¶

The rest of this notebook is optional and is not graded, but it's quite fun and informative, so you're highly encouraged to try it out!

A similar task to character-level text generation (but more complicated) is generating Shakespearean poems. Instead of learning from a dataset of dinosaur names, you can use a collection of Shakespearean poems. Using LSTM cells, you can learn longer-term dependencies that span many characters in the text--e.g., where a character appearing somewhere a sequence can influence what should be a different character, much later in the sequence. These long-term dependencies were less important with dinosaur names, since the names were quite short.

Below, you can implement a Shakespeare poem generator with Keras. Run the following cell to load the required packages and models. This may take a few minutes.

To save you some time, a model has already been trained for ~1000 epochs on a collection of Shakespearean poems called "The Sonnets."

Let's train the model for one more epoch. When it finishes training for an epoch (this will also take a few minutes), you can run generate_output, which will prompt you for an input (<40 characters). The poem will start with your sentence, and your RNN Shakespeare will complete the rest of the poem for you! For example, try, "Forsooth this maketh no sense" (without the quotation marks!). Depending on whether you include the space at the end, your results might also differ, so try it both ways, and try other inputs as well.

Congratulations on finishing this notebook!¶

The RNN Shakespeare model is very similar to the one you built for dinosaur names. The only major differences are:

• LSTMs instead of the basic RNN to capture longer-range dependencies
• The model is a deeper, stacked LSTM model (2 layer)
• Using Keras instead of Python to simplify the code

5 - References¶

• This exercise took inspiration from Andrej Karpathy's implementation: https://gist.github.com/karpathy/d4dee566867f8291f086. To learn more about text generation, also check out Karpathy's blog post.