Temporal difference learning

Temporal difference (TD) learning refers to a class of model-free reinforcement learning methods which learn by bootstrapping from the current estimate of the value function. These methods sample from the environment, like Monte Carlo methods, and perform updates based on current estimates, like dynamic programming methods. While Monte Carlo methods only adjust their estimates once the final outcome is known, TD methods adjust predictions to match later, more accurate, predictions about the future before the final outcome is known. This is a form of bootstrapping, as illustrated with the following example: Suppose you wish to predict the weather for Saturday, and you have some model that predicts Saturday's weather, given the weather of each day in the week. In the standard case, you would wait until Saturday and then adjust all your models. However, when it is, for example, Friday, you should have a pretty good idea of what the weather would be on Saturday – and thus be able to change, say, Saturday's model before Saturday arrives. Temporal difference methods are related to the temporal difference model of animal learning. The tabular TD(0) method is one of the simplest TD methods. It is a special case of more general stochastic approximation methods. It estimates the state value function of a finite-state Markov decision process (MDP) under a policy . Let denote the state value function of the MDP with states , rewards and discount rate under the policy : We drop the action from the notation for convenience. satisfies the Hamilton-Jacobi-Bellman Equation: so is an unbiased estimate for . This observation motivates the following algorithm for estimating . The algorithm starts by initializing a table arbitrarily, with one value for each state of the MDP. A positive learning rate is chosen. We then repeatedly evaluate the policy , obtain a reward and update the value function for the old state using the rule: where and are the old and new states, respectively. The value is known as the TD target.