15-780: Graduate Artificial Intelligence. Reinforcement learning (RL)
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1 15-780: Graduate Artificial Intelligence Reinforcement learning (RL)
2 From MDPs to RL We still use the same Markov model with rewards and actions But there are a few differences: 1. We do not assume we know the Markov model 2. We adapt to new observations (online vs. offline) Examples: - Game playing - Robot interacting with enviroment - Agents
3 RL No actions With actions
4 Scenario You wonder the world At each time point you see a state and a reward Your goal is to compute the sum of discounted rewards for each state S 1, 4 S 2, 0 S 3, 2 S 2, 2 S 4, 0
5 Scenario You wonder the world At each time point you see a state and a reward Your goal is to compute the sum of discounted rewards for each state Once again we will denote these by J (S i ) S 1, 4 S 2, 0 S 3, 2 S 2, 2 S 4, 0
6 Discounted rewards Lets compute the discounted rewards for each time point: t1: * *2 + = 7.1 t2: *2 +.. = 3.4 t3: 2 + = 3.8 t4: = 2 t5: 0 = 0 State Observations Mean S S 2 3.4, S S S 1, 4 S 2, 0 S 3, 2 S 2, 2 S 4, 0
7 Supervised learning for RL Observe set of states and rewards: (s(0),r(0)) (s(t),r(t)) For t=0 T compute discounted sum: Compute J (s i ) = (mean of J(t) for t such that s(t) = s i )! = " = T t i i t i r t J # ] [ i s t s t i s t s t J s J i = =! = ] [ # ] [ ] [ ] [ We assume that we observe each state frequently enough and that we have many observations so that the final observations do not have a big impact on our prediction
8 Algorithm for supervised learning 1. Initialize Counts(s i ) = J(s i )= Disc(s i ) = 0 2. Observe a state s i and a reward r 3. Counts(s i ) = Counts(s i ) Disc(s i ) = Disc(s i ) For all states j 6. Go to 2 J(s j )= J(s j ) + r*disc(s j ) Disc(s j ) = γ*disc(s j ) At any time we can imate J* by setting: J (s i )= J(s i ) / Counts(s i )
9 Running time and space Each update takes O(n) where n is the number of states, since we are updating vectors containing entries for all states Space is also O(n) 1. Convergences to true J* can be proven 2. Can be more efficient by ignoring states for which Disc() is very low already.
10 Problems with supervised learning Takes a long time to converge Does not use all available data - We can learn transition probabilities as well!
11 Certainty-Equivalent (CE) Learning Lets try to learn the underlying Markov system s parameters S 1, 4 S 2, 0 S 3, 2 S 2, 2 S 4, 0 S 1 R 1 =???? S 3 R 3 =? S 2 R 2 =?? S 4 R 4 =?
12 CE learning We keep track of three vectors: Counts(s): number of times we visited state s J(s): sum of rewards from state s Trans(i,j): number of time we transtiioned from state s i to state s j When we visit state s i, receive reward r and move to state s j we do the following: Counts(s i ) = Counts(s i ) +1 J(s i ) =J(s i ) + r Trans(i,j) = Trans(i,j) +1
13 CE learning When we visit state s i, receive reward r and move to state s j we do the following: Counts(s i ) = Counts(s i ) +1 J(s i ) =J(s i ) + r Trans(i,j) = Trans(i,j) +1 Using this we can imate at any time the following parameters: R (s i ) = J(s i )/Counts(s i ) P (j i) = Trans(i,j) / Counts(s i )
14 CE learning We can imate at any time the following parameters: R (s i ) = J(s i )/Counts(s i ) P (j i) = Trans(i,j) / Counts(s i ) We now can solve the MDP by setting, for all states s k : J ( sk ) = r ( sk ) + "! p ( s j sk ) J ( s j ) j
15 CE: Running time and space Running time Updates: O(1) Solving MDP: - O(n 3 ) using matrix inversion - O(n 2 *#it) when using value iteration Space O(n 2 ) for transition probabilities
16 Improving CE: One backup We do the same updates and imates as the original CE: Counts(s i ) = Counts(s i ) +1 J(s i ) =J(s i ) + r Trans(i,j) = Trans(i,j) +1 R (s i ) = J(s i )/Counts(s i ) P (j i) = Trans(i,j) / Counts(s i ) But we do not carry out the full value iteration Instead, we only update J (s i ) for the current state: J ( si ) = r ( si ) + "! p ( s j si ) J ( s j ) j
17 CE one backup: Running time and space Running time Updates: O(1) Solving MDP: - O(1) just update current state Space O(n 2 ) for transition probabilities Still a lot of memory, but much more efficient Can prove convergence to optimal solution (but slower than CE)
18 Summary so far Three methods Method Time Space Supervised learning O(n) O(n) CE learning O(n 2 *#it) O(n 2 ) One backup CE O(1) O(n 2 )
19 Temporal difference (TD) learning Goal: Same efficiency as one backup CE while much less space We only maintain the J array. Assume we have J (s 1 ) J (s n ). If we observe a transition from state s i to state s j and a reward r, we update using the following rule: J ( si ) = (1 #") J ( si ) + "( r +! j ( s j ))
20 Temporal difference (TD) learning Assume we have J (s 1 ) J (s n ). If we observe a transition from state s i to state s j and a reward r, we update using the following rule: J ( si ) = (1 #") J ( si ) + "( r +! j ( s j )) parameter to determine how much weight we place on current observation We have seen similar update rule before, as always, choosing α is an issue
21 Convergence TD learning is guaranteed to converge if: All states are visited often And: "# t t " t # 2 t =! <! For example, α t =C/t for some constant C would satisfy both requirements
22 TD: Complexity and space Time to update: O(1) Space: O(n) Method Time Space Supervised learning O(n) O(n) CE learning O(n 2 *#it) O(n 2 ) One backup CE O(1) O(n 2 )
23 RL No actions With actions
24 Policy learning So far we assumed that we cannot effect the environment. I real world situations we often have a choice of actions we take (as we discussed for MDPs). How can we learn the b policy for such cases?
25 Policy learning using CE We can easily update CE by setting: & # J ( sk ) = r ( sk ) + max$ ( ' p ( s j sk, a) J ( s j )! a % j " We revise our transition model to include actions But which action should we chose next?
26 Policy learning for TD TD is model free We can adjust TD to learn policies by defining the Q function: Q*(s i,a) = expected sum of future (discounted) rewards if we start at state s i and take action a When we take a specific action a in state s i and then transition to state s j we can update the Q function directly by setting: Q ( S i, a) = (1 #") Q ( S, a) + "( r +! maxq i i a' ( S j, a')) Instead of the J vector we maintain the Q matrix, which is a rather sparse n by m matrix (n states and m actions)
27 Choosing the next action We can select the action that results in the high expected sum of future rewards But that may not be the b action. Remember, we are only sampling from the distribution of possible outcomes. We do not want to avoid potentially beneficial actions. Instead, we can take a more probabilistic approach: p( a) = 1 Z Q exp(! ( si, a) ) f ( t) The probability we will use action a Normalizing constant Decreases as time goes by and we are more confident in the model we learned
28 Choosing the next action Instead, we can take a more probabilistic approach: p( a) " exp(! Q ( si, a) ) f ( t) We can initialize Q values to be high to increase the likelihood that we will explore more options It can be shown that Q learning converges to optimal policy
29 Demo
30 What you should know Differences between MDP and RL Strategies for computing with expected rewards Strategies for computing rewards and actions Q learning
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