Reinforcement Learning. Machine Learning, Fall 2010
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1 Reinforcement Learning Machine Learning, Fall
2 Administrativia This week: finish RL, most likely start graphical models LA2: due on Thursday LA3: comes out on Thursday TA Office hours: Today 1:30-2:30 2
3 Learning Setting A learning agent L interacts with an environment L can observe the current state S of the environment, e.g., as a vector of attribute-value pairs L can take actions (from a set A) in the environment to change its state L receives an immediate reward r after it takes an action L s goal is to learn a policy in order to maximize its long-term reward 3
4 Exploration vs Exploitation From Sutton & Barto Book 4
5 What makes this interesting is that δ and R are unknown MDPs Markov Decision Process Fancy term for an RL task that satisfies the Markov property MDPs are defined by providing two functions: Transition function: given a state/action pair, computes the next state s t+1 = δ(s t,a t ) Reward function: given a state/action pair, computes the reward r t+1 = R(s t,a t ) 5
6 Example MDP: Recycling Robot Robot with rechargeable battery roams around building looking for empty cans Actions: Search for cans Head for recharging station Wait for someone to bring can Rewards: Positive when can is obtained Strong negative when power is lost 6
7 MDP for recycling robot From Sutton & Barto Book 7
8 Q Function Q(s, a) r(s, a) + γv*(δ(s, a)) Sum of taking a in s and following the optimal policy after that If agent learns Q instead of V*, it can select optimal action even if it doesn t know the reward and transition functions of the MDP 8
9 Q Learning Set-up: Agent maintains a function Q^, which is its current estimate of Q Q^ is represented as a simple table, with one entry for each <s,a> pair (initialize arbitrarily, e.g. to 0) 9
10 Q Learning s: current state Iterate: Take action a Observe immediate reward r, next state s Update Q^(s, a) ˆQ(s, a) r + γ max ˆQ(s,a ) s s a 10
11 Q-Learning Properties If the RL task is: deterministic MDP has bounded immediate rewards If actions are visited infinitely often Then, Q^ converges to correct Q 11
12 Q Learning in nondeterministic MDPs s: current state Iterate: Take action a Observe immediate reward r, next state s Update Q^(s, a) ˆQ(s, a) (1 α n ) ˆQ(s, a)+α n (r + γ max a ˆQ(s,a )) s s Note: Typo in the textbook 12
13 Non-deterministic Q- Learning Properties If: Bounded rewards State/action pairs are visited infinitely often α n(i,s,a) = and i=1 (α n(i,s,a) ) 2 < i=1 Then: Q^ converges to Q with probability 1 as the number of steps goes to infinity 13
14 On-Policy vs Off-Policy Updates Q-learning is an off-policy method: ˆQ(s, a) (1 α n ) ˆQ(s, a)+α n (r + γ max a ˆQ(s,a )) The update does not depend on the action agent actually took in s An on-policy method would make its update based on the actions taken Learns Q π, for policy π being followed, not Q* 14
15 SARSA: On-Policy Learning s current state Choose action a in s, according to policy derived from Q^ Iterate: Take a Observe immediate reward r, next state s Choose action a to take in s according to policy derived from Q^ Update Q^ ˆQ(s, a) (1 α n ) ˆQ(s, a)+α n (r + γ ˆQ(s,a )) s s, a a 15
16 Example from Sutton and Barto Learner follows ε-greedy strategy 16
17 Very Large/Infinite State Spaces So far, assumed Q(s, a) is represented as a simple look-up table In a large/infinite state space agent may never visit the exact same state more than once So we need a way of generalizing across states... and we are still assuming a finite number of possible actions 17
18 Can we cast this as a supervised learning task? For each action a, can we learn a function F a ( θ 1, θ 2,..., θ n ) Q(s, a), where state s is represented as a vector of n attribute values θ 1, θ 2,..., θ n Example: Linear F 18
19 We can, if: We have training data of the form X = θ 1, θ 2,..., θ n y = Q(s, a)... but we don t actually have that kind of training data: no correct values for y! 19
20 ... but we have a best estimate for Q(s,a) ˆQ(s, a) (1 α n ) ˆQ(s, a)+α n (r + γ max a ˆQ(s,a )) So, our training data will look like this: For Q-learning: X = θ 1, θ 2,..., θ n y = r + γ max ˆQ(s,a) a For Sarsa: X = θ 1, θ 2,..., θ n y = r + γ ˆQ(s,a ) 20
21 Learning from such data Can cast learning task as aiming to minimize the mean squared error with respect to the parameters W The correct value MSE(W )= [y F a (s)] 2 s,a Our estimate of it What methods do we know that will allow us to learn from such data? 21
22 Stochastic Gradient Descent (Reminder) Initialize W arbitrarily Iterate over state/action pairs, Compute the gradient of the MSE wrt each parameter take small step in direction opposite to gradient w t+1 i w t i 1 2 α [y F t a(s)] 2 22
23 Where do the Theta-s Come From? State s can be described as a set of Boolean attributes derived from a coarse coding scheme: From Sutton and Barto book. Each circle corresponds to an attribute 23
24 Tile Coding Circles are not so easy to work with. Instead, use a set of super-imposed grids, called tilings From Sutton and Barto Book 24
25 Kanerva Coding As the number of dimensions (number of variables describing a state) increase, coarse and tile coding methods become intractable Instead, pick a set of n prototypical states Each attribute θ i corresponds to one of the prototypical states A given s, is described by how similar it is to each of the prototypical states 25
26 Curse of Dimensionality Images from Bishop Pattern Recognition book 26
27 Curse of Dimensionality Images from Bishop Pattern Recognition book 27
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