In an earlier post you saw a program that used gradient descent to find polynomial logistic models for three classes. That program included a cost() function written in Python as well as the (handwritten) function d_cost_d_theta() that calculated the partial derivative with respect to any element of .

Below, you will see the same program written using TensorFlow. Let’s look at how the cost function is defined and how the partial derivative function is automatically generated.

First, be sure you have run pip install numpy tensorflow or otherwise installed the libraries so that you can import them.

The creation of the graph begins with the instantiation of placeholder objects.

example_input = tf.placeholder(tf.float32, [None, len(training_examples[0].feature_vector)])
example_output = tf.placeholder(tf.float32, [None, len(training_examples[0].output_vector)])

When the graph is executed, the placeholders will be assigned the 5-value input and 3-value output vectors from the list of examples. The assignment of placeholders to lists of actual values is made in a feed dictionary:

feeds = {
    example_input: [ex.feature_vector for ex in training_examples],
    example_output: [ex.output_vector for ex in training_examples]

is defined as a Variable matrix in the graph, initially composed of zeroes:

theta = tf.Variable(tf.zeros(theta_shape))

The following lines look a lot like we are calculating a value of the model and a cost—but don’t be fooled! In reality, these lines only extend the graph to calculate the model value and the cost from the example input, example output, and defined above. The actual calculation will happen later, during execution of the graph.

model = 1.0 / (1 + tf.exp(tf.matmul(example_input, theta)))
cost = -1.0 / len(training_examples) * tf.reduce_sum(tf.multiply(example_output, tf.log(model)) + tf.multiply((1 - example_output), tf.log(1 - model)))

Similarly, the following lines extend the graph to determine the gradients of the cost function with respect to and then update theta. The gradients() function derives the partial derivatives of the cost function and extends the graph to compute them.

Note that we don’t write something like theta -= tf.multiply(...). theta is a Python variable referring to a matrix in the graph. We don’t want to re-assign theta at this point. We want to extend the graph so that during execution, the value of the matrix is updated. update_theta refers to the portion of the graph that does this.

calculate_gradients = tf.gradients(xs=theta, ys=cost)[0]
update_theta = theta.assign(theta - tf.multiply(learning_rate, calculate_gradients))

The graph will execute in the context of a Session. We instantiate the Session and initialize any global variables in the graph:

with tf.Session() as sess:

Finally, we are ready to execute the graph. The following line executes the portion of the graph referred to by cost, feeding the examples in through the placeholders defined earlier:

    print("Starting cost: %s" %, feed_dict=feeds)) returns the result of the execution, which in this case is the initial cost, when all values are initialized to zero.

The following lines execute the portion of the graph that calculates the gradients and the portion that updates . We could ignore or discard the return values, because the execution has already updated the matrix. But we need the values in gradients to know when to stop iterating. Also, it helps debugging to keep and print the return values.

      gradients =, feed_dict=feeds)
      new_theta =, feed_dict=feeds)

You can run the complete program below, take the theta values it produces, and enter them here to verify that it has fit logistic models to the three classes of data.