```
In [1]:
```# As usual, a bit of setup
import time, os, json
import numpy as np
import matplotlib.pyplot as plt
from cs231n.gradient_check import eval_numerical_gradient, eval_numerical_gradient_array
from cs231n.rnn_layers import *
from cs231n.captioning_solver import CaptioningSolver
from cs231n.classifiers.rnn import CaptioningRNN
from cs231n.coco_utils import load_coco_data, sample_coco_minibatch, decode_captions
from cs231n.image_utils import image_from_url
%matplotlib inline
plt.rcParams['figure.figsize'] = (10.0, 8.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'
# for auto-reloading external modules
# see http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython
%load_ext autoreload
%autoreload 2
def rel_error(x, y):
""" returns relative error """
return np.max(np.abs(x - y) / (np.maximum(1e-8, np.abs(x) + np.abs(y))))

For this exercise we will use the 2014 release of the Microsoft COCO dataset which has become the standard testbed for image captioning. The dataset consists of 80,000 training images and 40,000 validation images, each annotated with 5 captions written by workers on Amazon Mechanical Turk.

To download the data, change to the `cs231n/datasets`

directory and run the script `get_coco_captioning.sh`

.

We have preprocessed the data and extracted features for you already. For all images we have extracted features from the fc7 layer of the VGG-16 network pretrained on ImageNet; these features are stored in the files `train2014_vgg16_fc7.h5`

and `val2014_vgg16_fc7.h5`

respectively. To cut down on processing time and memory requirements, we have reduced the dimensionality of the features from 4096 to 512; these features can be found in the files `train2014_vgg16_fc7_pca.h5`

and `val2014_vgg16_fc7_pca.h5`

.

The raw images take up a lot of space (nearly 20GB) so we have not included them in the download. However all images are taken from Flickr, and URLs of the training and validation images are stored in the files `train2014_urls.txt`

and `val2014_urls.txt`

respectively. This allows you to download images on the fly for visualization. Since images are downloaded on-the-fly, **you must be connected to the internet to view images**.

Dealing with strings is inefficient, so we will work with an encoded version of the captions. Each word is assigned an integer ID, allowing us to represent a caption by a sequence of integers. The mapping between integer IDs and words is in the file `coco2014_vocab.json`

, and you can use the function `decode_captions`

from the file `cs231n/coco_utils.py`

to convert numpy arrays of integer IDs back into strings.

There are a couple special tokens that we add to the vocabulary. We prepend a special `<START>`

token and append an `<END>`

token to the beginning and end of each caption respectively. Rare words are replaced with a special `<UNK>`

token (for "unknown"). In addition, since we want to train with minibatches containing captions of different lengths, we pad short captions with a special `<NULL>`

token after the `<END>`

token and don't compute loss or gradient for `<NULL>`

tokens. Since they are a bit of a pain, we have taken care of all implementation details around special tokens for you.

You can load all of the MS-COCO data (captions, features, URLs, and vocabulary) using the `load_coco_data`

function from the file `cs231n/coco_utils.py`

. Run the following cell to do so:

```
In [2]:
```# Load COCO data from disk; this returns a dictionary
# We'll work with dimensionality-reduced features for this notebook, but feel
# free to experiment with the original features by changing the flag below.
data = load_coco_data(pca_features=True)
# Print out all the keys and values from the data dictionary
for k, v in data.iteritems():
if type(v) == np.ndarray:
print k, type(v), v.shape, v.dtype
else:
print k, type(v), len(v)

```
```

It is always a good idea to look at examples from the dataset before working with it.

You can use the `sample_coco_minibatch`

function from the file `cs231n/coco_utils.py`

to sample minibatches of data from the data structure returned from `load_coco_data`

. Run the following to sample a small minibatch of training data and show the images and their captions. Running it multiple times and looking at the results helps you to get a sense of the dataset.

Note that we decode the captions using the `decode_captions`

function and that we download the images on-the-fly using their Flickr URL, so **you must be connected to the internet to viw images**.

```
In [3]:
```# Sample a minibatch and show the images and captions
batch_size = 3
captions, features, urls = sample_coco_minibatch(data, batch_size=batch_size)
for i, (caption, url) in enumerate(zip(captions, urls)):
plt.imshow(image_from_url(url))
plt.axis('off')
caption_str = decode_captions(caption, data['idx_to_word'])
plt.title(caption_str)
plt.show()

```
```

As discussed in lecture, we will use recurrent neural network (RNN) language models for image captioning. The file `cs231n/rnn_layers.py`

contains implementations of different layer types that are needed for recurrent neural networks, and the file `cs231n/classifiers/rnn.py`

uses these layers to implement an image captioning model.

We will first implement different types of RNN layers in `cs231n/rnn_layers.py`

.

Open the file `cs231n/rnn_layers.py`

. This file implements the forward and backward passes for different types of layers that are commonly used in recurrent neural networks.

First implement the function `rnn_step_forward`

which implements the forward pass for a single timestep of a vanilla recurrent neural network. After doing so run the following to check your implementation.

```
In [4]:
```N, D, H = 3, 10, 4
x = np.linspace(-0.4, 0.7, num=N*D).reshape(N, D)
prev_h = np.linspace(-0.2, 0.5, num=N*H).reshape(N, H)
Wx = np.linspace(-0.1, 0.9, num=D*H).reshape(D, H)
Wh = np.linspace(-0.3, 0.7, num=H*H).reshape(H, H)
b = np.linspace(-0.2, 0.4, num=H)
next_h, _ = rnn_step_forward(x, prev_h, Wx, Wh, b)
expected_next_h = np.asarray([
[-0.58172089, -0.50182032, -0.41232771, -0.31410098],
[ 0.66854692, 0.79562378, 0.87755553, 0.92795967],
[ 0.97934501, 0.99144213, 0.99646691, 0.99854353]])
print 'next_h error: ', rel_error(expected_next_h, next_h)

```
```

```
In [5]:
```from cs231n.rnn_layers import rnn_step_forward, rnn_step_backward
N, D, H = 4, 5, 6
x = np.random.randn(N, D)
h = np.random.randn(N, H)
Wx = np.random.randn(D, H)
Wh = np.random.randn(H, H)
b = np.random.randn(H)
out, cache = rnn_step_forward(x, h, Wx, Wh, b)
dnext_h = np.random.randn(*out.shape)
fx = lambda x: rnn_step_forward(x, h, Wx, Wh, b)[0]
fh = lambda prev_h: rnn_step_forward(x, h, Wx, Wh, b)[0]
fWx = lambda Wx: rnn_step_forward(x, h, Wx, Wh, b)[0]
fWh = lambda Wh: rnn_step_forward(x, h, Wx, Wh, b)[0]
fb = lambda b: rnn_step_forward(x, h, Wx, Wh, b)[0]
dx_num = eval_numerical_gradient_array(fx, x, dnext_h)
dprev_h_num = eval_numerical_gradient_array(fh, h, dnext_h)
dWx_num = eval_numerical_gradient_array(fWx, Wx, dnext_h)
dWh_num = eval_numerical_gradient_array(fWh, Wh, dnext_h)
db_num = eval_numerical_gradient_array(fb, b, dnext_h)
dx, dprev_h, dWx, dWh, db = rnn_step_backward(dnext_h, cache)
print 'dx error: ', rel_error(dx_num, dx)
print 'dprev_h error: ', rel_error(dprev_h_num, dprev_h)
print 'dWx error: ', rel_error(dWx_num, dWx)
print 'dWh error: ', rel_error(dWh_num, dWh)
print 'db error: ', rel_error(db_num, db)

```
```

Now that you have implemented the forward and backward passes for a single timestep of a vanilla RNN, you will combine these pieces to implement a RNN that process an entire sequence of data.

In the file `cs231n/rnn_layers.py`

, implement the function `rnn_forward`

. This should be implemented using the `rnn_step_forward`

function that you defined above. After doing so run the following to check your implementation. You should see errors less than `1e-7`

.

```
In [6]:
```N, T, D, H = 2, 3, 4, 5
x = np.linspace(-0.1, 0.3, num=N*T*D).reshape(N, T, D)
h0 = np.linspace(-0.3, 0.1, num=N*H).reshape(N, H)
Wx = np.linspace(-0.2, 0.4, num=D*H).reshape(D, H)
Wh = np.linspace(-0.4, 0.1, num=H*H).reshape(H, H)
b = np.linspace(-0.7, 0.1, num=H)
h, _ = rnn_forward(x, h0, Wx, Wh, b)
expected_h = np.asarray([
[
[-0.42070749, -0.27279261, -0.11074945, 0.05740409, 0.22236251],
[-0.39525808, -0.22554661, -0.0409454, 0.14649412, 0.32397316],
[-0.42305111, -0.24223728, -0.04287027, 0.15997045, 0.35014525],
],
[
[-0.55857474, -0.39065825, -0.19198182, 0.02378408, 0.23735671],
[-0.27150199, -0.07088804, 0.13562939, 0.33099728, 0.50158768],
[-0.51014825, -0.30524429, -0.06755202, 0.17806392, 0.40333043]]])
print 'h error: ', rel_error(expected_h, h)

```
```

```
In [7]:
```N, D, T, H = 2, 3, 10, 5
x = np.random.randn(N, T, D)
h0 = np.random.randn(N, H)
Wx = np.random.randn(D, H)
Wh = np.random.randn(H, H)
b = np.random.randn(H)
out, cache = rnn_forward(x, h0, Wx, Wh, b)
dout = np.random.randn(*out.shape)
dx, dh0, dWx, dWh, db = rnn_backward(dout, cache)
fx = lambda x: rnn_forward(x, h0, Wx, Wh, b)[0]
fh0 = lambda h0: rnn_forward(x, h0, Wx, Wh, b)[0]
fWx = lambda Wx: rnn_forward(x, h0, Wx, Wh, b)[0]
fWh = lambda Wh: rnn_forward(x, h0, Wx, Wh, b)[0]
fb = lambda b: rnn_forward(x, h0, Wx, Wh, b)[0]
dx_num = eval_numerical_gradient_array(fx, x, dout)
dh0_num = eval_numerical_gradient_array(fh0, h0, dout)
dWx_num = eval_numerical_gradient_array(fWx, Wx, dout)
dWh_num = eval_numerical_gradient_array(fWh, Wh, dout)
db_num = eval_numerical_gradient_array(fb, b, dout)
print 'dx error: ', rel_error(dx_num, dx)
print 'dh0 error: ', rel_error(dh0_num, dh0)
print 'dWx error: ', rel_error(dWx_num, dWx)
print 'dWh error: ', rel_error(dWh_num, dWh)
print 'db error: ', rel_error(db_num, db)

```
```

In deep learning systems, we commonly represent words using vectors. Each word of the vocabulary will be associated with a vector, and these vectors will be learned jointly with the rest of the system.

In the file `cs231n/rnn_layers.py`

, implement the function `word_embedding_forward`

to convert words (represented by integers) into vectors. Run the following to check your implementation. You should see error around `1e-8`

.

```
In [8]:
```N, T, V, D = 2, 4, 5, 3
x = np.asarray([[0, 3, 1, 2], [2, 1, 0, 3]])
W = np.linspace(0, 1, num=V*D).reshape(V, D)
out, _ = word_embedding_forward(x, W)
expected_out = np.asarray([
[[ 0., 0.07142857, 0.14285714],
[ 0.64285714, 0.71428571, 0.78571429],
[ 0.21428571, 0.28571429, 0.35714286],
[ 0.42857143, 0.5, 0.57142857]],
[[ 0.42857143, 0.5, 0.57142857],
[ 0.21428571, 0.28571429, 0.35714286],
[ 0., 0.07142857, 0.14285714],
[ 0.64285714, 0.71428571, 0.78571429]]])
print 'out error: ', rel_error(expected_out, out)

```
```

```
In [9]:
```N, T, V, D = 50, 3, 5, 6
x = np.random.randint(V, size=(N, T))
W = np.random.randn(V, D)
out, cache = word_embedding_forward(x, W)
dout = np.random.randn(*out.shape)
dW = word_embedding_backward(dout, cache)
f = lambda W: word_embedding_forward(x, W)[0]
dW_num = eval_numerical_gradient_array(f, W, dout)
print 'dW error: ', rel_error(dW, dW_num)

```
```

At every timestep we use an affine function to transform the RNN hidden vector at that timestep into scores for each word in the vocabulary. Because this is very similar to the affine layer that you implemented in assignment 2, we have provided this function for you in the `temporal_affine_forward`

and `temporal_affine_backward`

functions in the file `cs231n/rnn_layers.py`

. Run the following to perform numeric gradient checking on the implementation.

```
In [10]:
```# Gradient check for temporal affine layer
N, T, D, M = 2, 3, 4, 5
x = np.random.randn(N, T, D)
w = np.random.randn(D, M)
b = np.random.randn(M)
out, cache = temporal_affine_forward(x, w, b)
dout = np.random.randn(*out.shape)
fx = lambda x: temporal_affine_forward(x, w, b)[0]
fw = lambda w: temporal_affine_forward(x, w, b)[0]
fb = lambda b: temporal_affine_forward(x, w, b)[0]
dx_num = eval_numerical_gradient_array(fx, x, dout)
dw_num = eval_numerical_gradient_array(fw, w, dout)
db_num = eval_numerical_gradient_array(fb, b, dout)
dx, dw, db = temporal_affine_backward(dout, cache)
print 'dx error: ', rel_error(dx_num, dx)
print 'dw error: ', rel_error(dw_num, dw)
print 'db error: ', rel_error(db_num, db)

```
```

In an RNN language model, at every timestep we produce a score for each word in the vocabulary. We know the ground-truth word at each timestep, so we use a softmax loss function to compute loss and gradient at each timestep. We sum the losses over time and average them over the minibatch.

However there is one wrinke: since we operate over minibatches and different captions may have different lengths, we append `<NULL>`

tokens to the end of each caption so they all have the same length. We don't want these `<NULL>`

tokens to count toward the loss or gradient, so in addition to scores and ground-truth labels our loss function also accepts a `mask`

array that tells it which elements of the scores count towards the loss.

Since this is very similar to the softmax loss function you implemented in assignment 1, we have implemented this loss function for you; look at the `temporal_softmax_loss`

function in the file `cs231n/rnn_layers.py`

.

Run the following cell to sanity check the loss and perform numeric gradient checking on the function.

```
In [11]:
```# Sanity check for temporal softmax loss
from cs231n.rnn_layers import temporal_softmax_loss
N, T, V = 100, 1, 10
def check_loss(N, T, V, p):
x = 0.001 * np.random.randn(N, T, V)
y = np.random.randint(V, size=(N, T))
mask = np.random.rand(N, T) <= p
print temporal_softmax_loss(x, y, mask)[0]
check_loss(100, 1, 10, 1.0) # Should be about 2.3
check_loss(100, 10, 10, 1.0) # Should be about 23
check_loss(5000, 10, 10, 0.1) # Should be about 2.3
# Gradient check for temporal softmax loss
N, T, V = 7, 8, 9
x = np.random.randn(N, T, V)
y = np.random.randint(V, size=(N, T))
mask = (np.random.rand(N, T) > 0.5)
loss, dx = temporal_softmax_loss(x, y, mask, verbose=False)
dx_num = eval_numerical_gradient(lambda x: temporal_softmax_loss(x, y, mask)[0], x, verbose=False