SARSA Implementation

Question

I am learning about SARSA algorithm implementation and had a question. I understand that the general "learning" step takes the form of:

Robot (r) is in state s. There are four actions available:

North (n), East (e), West (w) and South (s)

such that the list of Actions,

a = {n,w,e,s}

The robot randomly picks an action, and updates as follows:

Q(a,s) = Q(a,s) + L[r + DQ(a',s1) - Q(a,s)]

Where L is the learning rate, r is the reward associated to (a,s), Q(s',a') is the expected reward from an action a' in the new state s' and D is the discount factor.

Firstly, I don't undersand the role of the term - Q(a,s), why are we re-subtracting the current Q-value?

Secondly, when picking actions a and a' why do these have to be random? I know in some implementations or SARSA all possible Q(s', a') are taken into account and the highest value is picked. (I believe this is Epsilon-Greedy?) Why not to this also to pick which Q(a,s) value to update? Or why not update all Q(a,s) for the current s?

Finally, why is SARSA limited to one-step lookahead? Why, say, not also look into an hypothetical Q(s'',a'')?

I guess overall my questions boil down to what makes SARSA better than another breath-first or depth-first search algorithm?

Answer 1

Why do we subtract Q(a,s)? r + DQ(a',s1) is the reward that we got on this run through from getting to state s by taking action a. In theory, this is the value that Q(a,s) should be set to. However, we won't always take the same action after getting to state s from action a, and the rewards associated with going to future states will change in the future. So we can't just set Q(a,s) equal to r + DQ(a',s1). Instead, we just want to push it in the right direction so that it will eventually converge on the right value. So we look at the error in prediction, which requires subtracting Q(a,s) from r + DQ(a',s1). This is the amount that we would need to change Q(a,s) by in order to make it perfectly match the reward that we just observed. Since we don't want to do that all at once (we don't know if this is always going to be the best option), we multiply this error term by the learning rate, l, and add this value to Q(a,s) for a more gradual convergence on the correct value.`

Why do we pick actions randomly? The reason to not always pick the next state or action in a deterministic way is basically that our guess about which state is best might be wrong. When we first start running SARSA, we have a table full of 0s. We put non-zero values into the table by exploring those areas of state space and finding that there are rewards associated with them. As a result, something not terrible that we have explored will look like a better option than something that we haven't explored. Maybe it is. But maybe the thing that we haven't explored yet is actually way better than we've already seen. This is called the exploration vs exploitation problem - if we just keep doing things that we know work, we may never find the best solution. Choosing next steps randomly ensures that we see more of our options.

Why can't we just take all possible actions from a given state? This will force us to basically look at the entire learning table on every iteration. If we're using something like SARSA to solve the problem, the table is probably too big to do this for in a reasonable amount of time.

Why can SARSA only do one-step look-ahead? Good question. The idea behind SARSA is that it's propagating expected rewards backwards through the table. The discount factor, D, ensures that in the final solution you'll have a trail of gradually increasing expected rewards leading to the best reward. If you filled in the table at random, this wouldn't always be true. This doesn't necessarily break the algorithm, but I suspect it leads to inefficiencies.

Why is SARSA better than search? Again, this comes down to an efficiency thing. The fundamental reason that anyone uses learning algorithms rather than search algorithms is that search algorithms are too slow once you have too many options for states and actions. In order to know the best action to take from any other state action pair (which is what SARSA calculates), you would need to do a search of the entire graph from every node. This would take O(s*(s+a)) time. If you're trying to solve real-world problems, that's generally too long.