Last week, I promised to have a blog post about my experiments on TensorFlow. I don’t want to repeat whatever is out there about it like another tutorial but I don’t want to make it unreadable for general audience who just want to know basics. Here it is what I will do: First a very brief review of TensorFlow (what and why, how), then I will give you a simplified version of my experiments as an example.

Very brief review

What is TensorFlow? TensorFlow is a numerical computation tool with interfaces in python and C++.

Why is it useful? Its architecture has flexibility to perform computation on several CPUs or GPUs of any desktop or server.

How does it work? It has a concept of graph which has operations as its nodes and the inputs and outputs of these operations are directed edges of graph. Each line of code for a TensorFlow operation (e.g. multiplication of matrixes) adds a node in this graph. Then you can run a session and with in this session you can push your information and ask for outputs at any nodes of this graph. Let’s see an example.

A simple example (matrix multiplication)

Let’s try this matrix multiplication in TensorFlow (in python):

import tensorflow as tf

Here it is our graph:

A = tf.placeholder(tf.float32, shape=(2, 2))
B = tf.placeholder(tf.float32, shape=(2, 2))
AB = tf.matmul(A, B)

The tf.placeholder basically define inputs with their size (shape 2 by 2) and types of float32. The last line, tf.matmul does the multiplication and assign it to AB. Now, we can fed it with matrixes and ask for results.

sess = tf.Session()
result = sess.run(AB, {A: [[2, -2], [5, 3]], B: [[-1, 4], [7, -6]]})
sess.close()

Yes, as simple as this. Maybe this part is a bit strange for python programmers, A and B in session part are pointing at A and B in graph.

My toy project

Do you remember my post about meaningful language? I quoted Harnad (1990):

I am trying out different models which learn compositionality from data. One of these models is a recurrent neural network called LSTM, which basically can be used as an encoder-decoder (Cho et al. 2014) which takes a sequence of inputs and produce outputs. In this example, if sequence of ["horse", "stripes"] is our input, ["zebra"] should be the output. We encode each word with a vector to be able to do our magical mathematical operations, I will talk about word embeddings in future but lets jump to my toy example.

What I will test here in my toy experiment is much simpler than my zebra example. The compositionality is much more clear in simple math operations such as adding two natural numbers :

In my experiment, I will generate random examples of add operation: A sequence of words and their expected equivalence (e.g. ["1", "+", "2", "</eos>"] as input and ["3"] as target). I treated each operand (“1”, “2”, …) and operator (e.g. “+”) as symbols with no value. Basically, I used a simple one-hot encoding to vectorize them. This simple code using preprocessing of Scikit-learn would do this for me:

from sklearn.preprocessing import OneHotEncoder

input_vocab = ["", "0","1", "2", "+", "="]
input_vocab_ids = [[0], [1], [2], [3], [4], [5]]
output_vocab = ["0", "1","2", "3", "4"]
output_vocab_ids = [[0], [1], [2], [3], [4]]

MAX_VOCABULARY_SIZE = 10
input_encoder = OneHotEncoder(n_values=MAX_VOCABULARY_SIZE)
input_encoder.fit(input_vocab_ids)

output_encoder = OneHotEncoder(n_values=MAX_VOCABULARY_SIZE)
output_encoder.fit(output_vocab_ids)

You can test it by printing the vector representation of “1” in this example:

print("1 =>", input_encoder.transform([input_vocab_ids[input_vocab.index("1")]]).toarray())
# 1 => [[ 0.  0.  1.  0.  0.  0.  0.  0.  0.  0.]]

The LSTM model needs its own blog post I will explain that later. Story short, with a larger set of vocabulary the LSTM training process will take time but it converges to a state which can take string inputs and give you equivalence string outputs.

This model is naively simple and definitely doesn’t learn meaning of number and addition as humans do. A simple test on unseen words shows that how weak is our design. But the interesting aspect is the ability to learn ordered composition from examples. using this model. I trained a model with a corpus of numbers between 0 and 49. Well, the error rate on unseen combination is high as 99%. But error rate doesn’t say all sides of the story. For example, an unseen combination “33 + 44 =” produces “78” (which is wrong but very close to correct answer).

In this single LSTM model we don’t learn addition as a mathematical operation. All words in this model carry some parts of the meaning of addition because our corpus projects such relation between words. Even if I remove the sign “+” the result of "1 2" is "3". In my future experiments, I will try to break this simple composition and make my sample data more complicated with other operators to see if we can learn more than this.

I would like to comeback to my underlying claim that one-hot vector representations are just treating words as symbols with no values. Clearly, in my model I assigned list of numbers to each word. This is why I call it symbolic, I don’t assign any numbers to any words by using extra knowledge than these words are different from each other, even though they can have same values (e.g. “” and “0” can have same value 0). I even treated numbers and operators similarly. I could use my knowledge about the corpus and make a better loss function based on regression instead of cross-entropy but again I didn’t because I simply didn’t want use any math to learn math. But isn’t it exciting that this naive model can say: “33 + 44 = 78”? It is wrong but it’s not that bad :D.