Reinforcement learning cost function

Question

Newb question I am writing a OpenAI Gym pong player with TensorFlow and thus far have been able to create the network based on a random initialization so that it would randomly return to move the player paddle up or down.

After the epoch is over (21 games played where the computer won) I collected a set of observations, moves and scores. The final observation of a game receives a score and each preceding observation can be scored based on Bellman equation.

Now my questions what I do not understand yet: How do I calculate the cost function so that it would be propagated as a start gradient for backward propagation? I totally get it with supervised learning, but here we do not have any labels to score agains.

How would I start optimizing the network?

Maybe a pointer to existing code or some literature would help.

Here's where I compute the rewards:

def compute_observation_rewards(self, gamma, up_score_probabilities):
        """
        Applies Bellman equation and determines reward for each stored observation
        :param gamma: Learning decay
        :param up_score_probabilities: Probabilities for up score
        :returns: List of scores for each move
        """

        score_sum = 0
        discounted_rewards = []
        # go backwards through all observations
        for i, p in enumerate(reversed(self._states_score_action)):
            o = p[0]
            s = p[1]

            if s != 0:
                score_sum = 0

            score_sum = score_sum * gamma + s
            discounted_rewards.append(score_sum)

        # # normalize scores
        discounted_rewards = np.array(discounted_rewards)
        discounted_rewards -= np.mean(discounted_rewards)
        discounted_rewards /= np.std(discounted_rewards)

        return discounted_rewards

Below is my network:

with tf.variable_scope('NN_Model', reuse=tf.AUTO_REUSE):

        layer1 = tf.layers.conv2d(inputs,
                                3,
                                3,
                                strides=(1, 1),
                                padding='valid',
                                data_format='channels_last',
                                dilation_rate=(1, 1),
                                activation= tf.nn.relu, 
                                use_bias=True,
                                bias_initializer=tf.zeros_initializer(),
                                trainable=True,
                                name='layer1'
                            )
        # (N - F + 1) x (N - F + 1)
        # => layer1 should be 
        # (80 - 3 + 1) * (80 - 3 + 1) = 78 x 78

        pool1 = tf.layers.max_pooling2d(layer1,
                                        pool_size=5,
                                        strides=2,
                                        name='pool1')

        # int((N - f) / s +1) 
        # (78 - 5) / 2 + 1 = 73/2 + 1 = 37

        layer2 = tf.layers.conv2d(pool1,
                                5,
                                5,
                                strides=(2, 2),
                                padding='valid',
                                data_format='channels_last',
                                dilation_rate=(1, 1),
                                activation= tf.nn.relu, 
                                use_bias=True,
                                kernel_initializer=tf.random_normal_initializer(),
                                bias_initializer=tf.zeros_initializer(),
                                trainable=True,
                                name='layer2',
                                reuse=None
                            )

        # ((N + 2xpadding - F) / stride + 1) x ((N + 2xpadding - F) / stride + 1)
        # => layer1 should be 
        # int((37 + 0 - 5) / 2) + 1 
        # 16 + 1 = 17

        pool2 = tf.layers.max_pooling2d(layer2,
                                        pool_size=3,
                                        strides=2,
                                        name='pool2')

        # int((N - f) / s +1) 
        # (17 - 3) / 2 + 1 = 7 + 1 = 8

        flat1 = tf.layers.flatten(pool2, 'flat1')

        # Kx64

        full1 = tf.contrib.layers.fully_connected(flat1,
                                            num_outputs=1,
                                            activation_fn=tf.nn.sigmoid,
                                            weights_initializer=tf.contrib.layers.xavier_initializer(),
                                            biases_initializer=tf.zeros_initializer(),
                                            trainable=True,
                                            scope=None
                                        )

Answer 1

The algorithm you're looking for is called REINFORCE. I would suggest reading chapter 13 of Sutton and Barto's RL book .

Here's pseudocode from the book.

Here, theta is the set of weights of your neural net. If you're unfamiliar with some of the rest of the notation, I'd suggest reading Chapter 3 of the above-mentioned book. It covers the basic problem formulation.