# Q-learning with a state-action-state reward structure and a Q-matrix with states as rows and actions as columns

I have set up a Q-learning problem in R, and would like some help with the theoretical correctness of my approach in framing the problem.

Problem structure For this problem, the environment consists of 10 possible states. When in each state, the agent has 11 potential actions which it can choose from (these actions are the same regardless of the state which the agent is in). Depending on the particular state which the agent is in and the subsequent action which the agent then takes, there is a unique distribution for transition to a next state i.e. the transition probabilities to any next state are dependant on (only) the previous state as well as the action then taken.

Each episode has 9 iterations i.e. the agent can take 9 actions and make 9 transitions before a new episode begins. In each episode, the agent will begin in state 1.

In each episode, after each of the agent's 9 actions, the agent will get a reward which is dependant on the agent's (immediately) previous state and their (immediately) previous action as well as the state which they landed on i.e. the agent's reward structure is dependant on a state-action-state triplet (of which there will be 9 in an episode).

The transition probability matrix of the agent is static, and so is the reward matrix.

I have set up two learning algorithms. In the first, the q-matrix update happens after each action in each episode. In the second, the q-matrix is updated after each episode. The algorithm uses an epsilon greedy learning formula.

The big problem is that in my Q-learning, my agent is not learning. It gets less and less of a reward over time. I have looked into other potential problems such as simple calculation errors, or bugs in code, but I think that the problem lies with the conceptual structure of my q-learning problem.

Questions

I have set up my Q-matrix as being a 10 row by 11 column matrix i.e. all the 10 states are the rows and the 11 actions are the columns. Would this be the best way to do so? This means that an agent is learning a policy which says that "whenever you are in state x, do action y" Given this unique structure of my problem, would the standard Q-update still apply? i.e. Q[cs,act]<<-Q[cs,act]+alpha*(Reward+gamma*max(Q[ns,])-Q[cs,act]) Where cs is current state; act is action chosen; Reward is the reward given your current state, your action chosen and the next state which you will transition to; ns is the next state which you will transition to given your last state and last action (note that you transitioned to this state stochastically).

Is there an open AI gym in R? Are there Q-learning packages for problems of this structure?

• If you have the model of the environment (transition and reward matrices) why you are trying to solve the MDP with a model-free algorithm? With this small state space you can use value iteration and obtain the optimal policy. Be aware to use R(s,a,s') and not R(s,a) in your equations. Could you elaborate more why the iterations are 9? It seems to me that you are more likely interested in planning with either a simple tree search (e.g. with depth 9) or dynamic programming rather than a model-free approach. Let us know more details so we can try to help! – Constantinos Aug 2 '17 at 22:47

## 1 Answer

There is a problem in your definition of the problem.

Q(s,a) is the expected utility of taking action a in state s and following the optimal policy afterwards.

The expected utility of a certain state (based on your definition) is different after taking 1, 2 or 9 steps. That means that the reward of being in state s_0 and taking action a_0 is different in step 0 from what you get in step 9

To adequate model the problem, you should reframe it and consider the state to be both the 'position'+'step'. You will now have 90 states (10pos*9steps).

• Sorry for the confusing answer. The point is that the Q value at step 1 is different from the Q value at step 4 or 9. Imagine the Q value step 8 and the Q-value in step 9. The expected sum of rewards in both situations are different. – Juan Leni Aug 2 '17 at 19:46