Approximation Methods in Reinforcement Learning

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1 2018 CS420, Machine Learning, Lecture 12 Approximation Methods in Reinforcement Learning Weinan Zhang Shanghai Jiao Tong University

Reinforcement Learning Materials Our course on RL is mainly based on the materials from these masters. Prof.

html Reinforcement Learning: An Introduction (2 nd edition) http://incompleteideas.

uk/staff/d.silver/web/home.html UCL Reinforcement Learning Course http://www0.cs.ucl.ac.uk/staff/d.silver/web/teaching.

2 Reinforcement Learning Materials Our course on RL is mainly based on the materials from these masters. Prof. Richard Sutton University of Alberta, Canada Reinforcement Learning: An Introduction (2 nd edition) Dr. David Silver Google DeepMind and UCL, UK UCL Reinforcement Learning Course Prof. Andrew Ng Stanford University, US Machine Learning (CS229) Lecture Notes 12: RL

3 Last Lecture Model-based dynamic programming Value iteration Policy iteration V (s) =R(s)+max X P sa (s 0 )V (s 0 ) a2a X s 0 2S ¼(s) =argmax P sa (s 0 )V (s 0 ) a2a s 0 2S Model-free reinforcement learning On-policy MC V (s t ) Ã V (s t )+ (G t V (s t )) On-policy TD V (s t ) Ã V (s t )+ (r t+1 + V (s t+1 ) V (s t )) On-policy TD SARSA Q(s t ;a t ) Ã Q(s t ;a t )+ (r t+1 + Q(s t+1 ;a t+1 ) Q(s t ;a t )) Off-policy TD Q-learning Q(s t ;a t ) Ã Q(s t ;a t )+ (r t+1 + max Q(s t+1 ;a t+1 ) Q(s t ;a t )) a 0

4 Key Problem to Solve in This Lecture In all previous models, we have created a lookup table to maintain a variable V(s) for each state or Q(s,a) for each state-action What if we have a large MDP, i.e. the state or state-action space is too large or the state or action space is continuous to maintain V(s) for each state or Q(s,a) for each state-action? For example Game of Go ( states) Helicopter, autonomous car (continuous state space)

5 Content Solutions for large MDPs Discretize or bucketize states/actions Build parametric value function approximation Policy gradient Deep reinforcement learning and multi-agent RL

6 Content Solutions for large MDPs Discretize or bucketize states/actions Build parametric value function approximation Policy gradient Deep reinforcement learning and multi-agent RL

7 Discretization Continuous MDP For a continuous-state MDP, we can discretize the state space For example, if we have 2D states (s 1, s 2 ), we can use a grid to discretize the state space The discrete state ¹s¹s The discretized MDP: ( S;A;fP ¹ ¹sa g; ;R) S 2 ¹s¹s Then solve this MDP with any previous solutions S 1

8 Bucketize Large Discrete MDP For a large discrete-state MDP, we can bucketize the states to down sample the states To use domain knowledge to merge similar discrete states For example, clustering using state features extracted from domain knowledge

9 Discretization/Bucketization Pros Straightforward and off-theshelf Efficient Can work well for many problems Cons A fairly naïve representation for V Assumes a constant value over each discretized cell Curse of dimensionality S = R n ) S ¹ = f1;:::;kg n S 2 ¹s¹s S 1

10 Parametric Value Function Approximation Create parametric (thus learnable) functions to approximate the value function V μ (s) ' V ¼ (s) Q μ (s; a) ' Q ¼ (s; a) θ is the parameters of the approximation function, which can be updated by reinforcement learning Generalize from seen states to unseen states

11 Main Types of Value Function Approx. V μ (s) Q μ (s; a) Many function approximations (Generalized) linear model Neural network Decision tree Nearest neighbor Fourier / wavelet bases μ μ Differentiable functions (Generalized) linear model Neural network s s a We assume the model is suitable to be trained for nonstationary, non-iid data

12 Value Function Approx. by SGD Goal: find parameter vector θ minimizing mean-squared error between approximate value function V θ (s) and true value V π (s) J(μ) =E ¼ h 1 2 (V ¼ (s) V μ (s)) 2i J(μ) =E ¼ h 1 2 (V ¼ (s) V μ (s)) 2i Gradient to minimize the h = E ¼ (V ¼ (s) V μ i μ(s) Stochastic gradient descent on one sample μ Ã = μ + (V ¼ (s) V μ μ(s)

Featurize the State Represent state by a feature vector x(s) = 2 3 x 1 (s) 6 7 4.

13 Featurize the State Represent state by a feature vector x(s) = 2 3 x 1 (s) x k (s) For example of a helicopter 3D location 3D speed (differentiation of location) 3D acceleration (differentiation of speed)

14 Linear Value Function Approximation Represent value function by a linear combination of features V μ (s) =μ > x(s) Objective function is quadratic in parameters θ h 1 J(μ) =E ¼ 2 (V ¼ (s) μ > x(s)) 2i Thus stochastic gradient descent converges on global optimum μ Ã = μ + (V ¼ (s) V μ (s))x(s) Step size Prediction error Feature value

15 Monte-Carlo with Value Function Approx. μ Ã μ + (V ¼ (s) V μ (s))x(s) Now we specify the target value function V π (s) We can apply supervised learning to training data hs 1 ;G 1 i; hs 2 ;G 2 i;:::;hs T ;G T i For each data instance <s t, G t > μ Ã μ + (G t V μ (s))x(s t ) MC evaluation at least converges to a local optimum In linear case it converges to a global optimum

16 TD Learning with Value Function Approx. μ Ã μ + (V ¼ (s) V μ (s))x(s) TD target r t+1 + V μ (s t+1 ) is a biased sample of true target value V ¼ (s t ) Supervised learning from training data hs 1 ;r 2 + V μ (s 2 )i; hs 2 ;r 3 + V μ (s 3 )i;:::;hs T ;r T i For each data instance hs t ;r t+1 + V μ (s t+1 )i μ Ã μ + (r t+1 + V μ (s t+1 ) V μ (s))x(s t ) Linear TD converges (close) to global optimum

17 Action-Value Function Approximation Approximate the action-value function Minimize mean squared error Q μ (s; a) ' Q ¼ (s; a) h 1 J(μ) =E ¼ 2 (Q¼ (s; a) Q μ (s; a)) 2i Stochastic gradient descent on one sample μ Ã = μ + (Q ¼ (s; a) Q μ (s; μ(s; a)

18 Linear Action-Value Function Approx. Represent state-action pair by a feature vector x(s; a) = 2 3 x 1 (s; a) x k (s; a) Parametric Q function, e.g., the linear case Q μ (s; a) =μ > x(s; a) Stochastic gradient descent update μ Ã = μ + (Q ¼ (s; a) μ > x(s; a))x(s; a)

19 TD Learning with Value Function Approx. μ Ã μ + (Q ¼ (s; a) Q μ (s; μ(s; a) For MC, the target is the return G t μ Ã μ + (G t Q μ (s; μ(s; a) For TD, the target is r t+1 + Q μ (s t+1 ;a t+1 ) μ Ã μ + (r t+1 + Q μ (s t+1 ;a t+1 ) Q μ (s; μ(s; a)

20 Control with Value Function Approx. Q μ = Q ¼ μ Q μ ' Q ¼ Q ;¼ ¼ = ²-greedy(Q μ ) Policy evaluation: approximately policy evaluation Policy improvement: ɛ-greedy policy improvement Q μ ' Q ¼

21 NOTE of TD Update For TD(0), the TD target is State value Action value μ Ã μ + (V ¼ (s t ) V μ (s t μ(s t ) = μ + (r t+1 + V μ (s t+1 ) V μ μ(s t ) μ Ã μ + (Q ¼ (s; a) Q μ (s; μ(s; a) = μ + (r t+1 + Q μ (s t+1 ;a t+1 ) Q μ (s; μ(s; a) Although θ is in the TD target, we don t calculate gradient from the target. Think about why.

22 Case Study: Mountain Car The gravity is stronger than the car s engine Cost-to-go function

23 Case Study: Mountain Car

24 Deep Q-Network (DQN) Volodymyr Mnih, Koray Kavukcuoglu, David Silver et al. Playing Atari with Deep Reinforcement Learning. NIPS 2013 workshop. Volodymyr Mnih, Koray Kavukcuoglu, David Silver et al. Human-level control through deep reinforcement learning. Nature 2015.

25 Deep Q-Network (DQN) Implement Q function with deep neural network Volodymyr Mnih, Koray Kavukcuoglu, David Silver et al. Playing Atari with Deep Reinforcement Learning. NIPS 2013 workshop. Volodymyr Mnih, Koray Kavukcuoglu, David Silver et al. Human-level control through deep reinforcement learning. Nature 2015.

26 Deep Q-Network (DQN) The loss function of Q-learning update at iteration i r L i (μ i )=E (s;a;r;s )»U(D)h 0 + max Q(s 0 ;a 0 ; μ i ) Q(s; a; μ i) i 2 a 0 target Q value estimated Q value θ i are the network parameters to be updated at iteration i Updated with standard back-propagation algorithms θ i- are the target network parameters Only updated with θ i for every C steps (s,a,r,s )~U(D): the samples are uniformly drawn from the experience pool D Thus to avoid the overfitting to the recent experiences Volodymyr Mnih, Koray Kavukcuoglu, David Silver et al. Playing Atari with Deep Reinforcement Learning. NIPS 2013 workshop. Volodymyr Mnih, Koray Kavukcuoglu, David Silver et al. Human-level control through deep reinforcement learning. Nature 2015.

27 Content Solutions for large MDPs Discretize or bucketize states/actions Build parametric value function approximation Policy gradient Deep reinforcement learning and multi-agent RL

28 Parametric Policy We can parametrize the policy ¼ μ (ajs) which could be deterministic a = ¼ μ (s) or stochastic ¼ μ (ajs) =P (ajs; μ) θ is the parameters of the policy Generalize from seen states to unseen states We focus on model-free reinforcement learning

29 Policy-based RL Advantages Better convergence properties Effective in high-dimensional or continuous action spaces No.1 reason: for value function, you have to take a max operation Can learn stochastic polices Disadvantages Typically converge to a local rather than global optimum Evaluating a policy is typically inefficient and of high variance

30 Policy Gradient For stochastic policy ¼ μ (ajs) =P (ajs; μ) Intuition lower the probability of the action that leads to low value/reward higher the probability of the action that leads to high value/reward A 5-action example 1. Initialize θ 3. Update θ by policy gradient 5. Update θ by policy gradient Action Probability Action Probability Action Probability A1 A2 A3 A4 A A1 A2 A3 A4 A A1 A2 A3 A4 A5 2. Take action A2 Observe positive reward 4. Take action A3 Observe negative reward

31 Policy Gradient in One-Step MDPs Consider a simple class of one-step MDPs Starting in state s» d(s) Terminating after one time-step with reward r sa Policy expected value J(μ) =E ¼μ [r] = X s2s d(s) X ¼ μ (ajs)r sa = X s2s d(s) X μ (ajs) r sa

32 Likelihood Ratio Likelihood ratios exploit the following μ (ajs) μ (ajs) = ¼ μ (ajs) ¼ μ (ajs) = ¼ μ log ¼ μ(ajs) Thus the policy s expected value J(μ) =E ¼μ [r] = X s2s d(s) X ¼ μ (ajs)r sa = X s2s d(s) X μ (ajs) r sa = X d(s) X ¼ μ log ¼ μ(ajs) s2s a2a log ¼μ (ajs) i = E ¼μ r sa r sa This can be approximated by sampling state s from d(s) and action a from π θ

33 Policy Gradient Theorem The policy gradient theorem generalizes the likelihood ratio approach to multi-step MDPs Replaces instantaneous reward r sa with long-term value Policy gradient theorem applies to start state objective J 1, average reward objective J avr, and average value objective J avv Theorem Q ¼ μ (s; a) For any differentiable policy ¼ μ (ajs), for any of policy objective function J = J 1, J avr, J avv, the policy gradient log ¼μ (ajs) i = E ¼μ Q ¼ μ (s; a) Please refer to appendix of the slides for detailed proofs

34 Monte-Carlo Policy Gradient (REINFORCE) Update parameters by stochastic gradient ascent Using policy gradient theorem Using return G t as an unbiased sample of Q ¼ μ (s; a) μ t log ¼ μ(a t js t ) G t REINFORCE Algorithm Initialize θ arbitrarily for each episode fs 1 ;a 1 ;r 2 ;:::;s T 1 ;a T 1 ;r T g»¼ μ do for t=1 to T-1 do end for return θ μ Ã μ log ¼ μ(a t js t )G t end for

35 Puck World Example Continuous actions exert small force on puck Puck is rewarded for getting close to target Target location is reset every 30 seconds Policy is trained using variant of MC policy gradient

36 Softmax Stochastic Policy Softmax policy is a very commonly used stochastic policy ¼ μ (ajs) = P ef μ(s;a) a 0 e f μ(s;a 0 ) where f θ (s,a) is the score function of a state-action pair parametrized by θ, which can be defined with domain knowledge The gradient of its log ¼ μ (ajs) μ(s; a) μ(s; a) 1 X P a 0 e f e μ(s; a00 ) μ(s;a 0 ) a 00 (s; a 0 ) i E a 0»¼ μ (a 0 js)

37 Softmax Stochastic Policy Softmax policy is a very commonly used stochastic policy ¼ μ (ajs) = P ef μ(s;a) a 0 e f μ(s;a 0 ) where f θ (s,a) is the score function of a state-action pair parametrized by θ, which can be defined with domain knowledge For example, we define the linear score function f μ (s; a) =μ > x(s; log ¼ μ (ajs) μ(s; a) (s; a 0 ) i E a 0»¼ μ (a 0 js) = x(s; a) E a 0»¼ μ (a 0 js)[x(s; a 0 )]

38 Sequence Generation Example Generator is a reinforcement learning policy G μ (y t jy 1:t 1 ) of generating a sequence Decide the next word (discrete action) to generate given the previous ones, implemented by softmax policy Discriminator provides the reward (i.e. the probability of being true data) for the whole sequence G is trained by MC policy gradient (REINFORCE) [Lantao Yu, Weinan Zhang, Jun Wang, Yong Yu. SeqGAN: Sequence Generative Adversarial Nets with Policy Gradient. AAAI 2017.]

39 Experiments on Synthetic Data Evaluation measure with Oracle h X T i NLL oracle = E Y1:T»G μ log G oracle (y t jy 1:t 1 ) t=1 [Lantao Yu, Weinan Zhang, Jun Wang, Yong Yu. SeqGAN: Sequence Generative Adversarial Nets with Policy Gradient. AAAI 2017.]

40 Content Solutions for large MDPs Discretize or bucketize states/actions Build parametric value function approximation Policy gradient Deep reinforcement learning and multi-agent RL By our invited speakers Yuhuai Wu, Shixiang Gu and Ying Wen

41 APPENDIX Policy Gradient Theorem: Average Reward Setting Average reward objective 1 h i J(¼) = lim n!1 n E r 1 + r r n j¼ = X s 1X h i Q ¼ (s; a) = E r t J(¼)js 0 = s; a 0 = a; ¼ (s) t=1 def X = X a = X a = X a = X a a ¼(s; a)q ¼ (s; a); 8s a) a) Q ¼ (s; a)+¼(s; a) a) Q ¼ (s; a)+¼(s; i Q¼ (s; a) ³ Q ¼ (s; a)+¼(s; d ¼ (s) X a Q ¼ (s; a)+¼(s; a) X s 0 P a ss ¼ (s 0 ) ¼(s; a)r(s; a) ³ r(s; a) J(¼)+ X i Pss a 0V ¼ (s 0 ) s 0 X i Pss a 0V ¼ (s 0 ) s i ¼ (s) Please refer to Chapter 13 of Rich Sutton s Reinforcement Learning: An Introduction (2 nd Edition)

42 APPENDIX Policy Gradient Theorem: Average Reward Setting Average reward = X a) Q ¼ (s; a)+¼(s; a) X P ¼ (s 0 ) i ¼ (s) 0 a s X 0 d ¼ = X d ¼ (s) a) Q ¼ (s; a)+ X d ¼ (s) X ¼(s; a) X P ¼ (s 0 ) ss X 0 s s a s a s 0 s X d ¼ (s) X ¼(s; a) X P ¼ (s 0 ) ss = X X X d ¼ (s)¼(s; a)p ¼ (s0 ) 0 ss 0 s a s 0 s a s 0 = X X ³ X d ¼ (s) ¼(s; a)p a ¼ (s 0 ) ss = X X d ¼ (s 0 ) (s)p ss 0 s s 0 a s s 0 = X ³ d ¼ ¼ (s 0 ) (s)p ss 0 = X d ¼ (s 0 ¼ (s 0 ) s 0 s s 0 ) X d ¼ = X d ¼ (s) a) Q ¼ (s; a)+ X d ¼ (s 0 ¼ (s 0 ) X s s a s 0 s = X d ¼ (s) a) Q ¼ (s; a) s a d ¼ ¼ (s) d ¼ ¼ (s) Please refer to Chapter 13 of Rich Sutton s Reinforcement Learning: An Introduction (2 nd Edition)

43 APPENDIX Policy gradient theorem: Start Value Setting Start state value objective h X 1 i J(¼) =E t 1 r t s0 ;¼ t=1 h X 1 Q ¼ (s; a) ¼ (s) def = X a = X a = X a k=1 i k 1 r t+k st = s; a t = a; ¼ X ¼(s; a)q ¼ (s; a); a 8s Q ¼ (s; a)+¼(s; i Q¼ (s; a) a) a) Q ¼ (s; a) Q ¼ (s; a)+ X a ³ r(s; a)+ X i Pss a 0V ¼ (s 0 ) s 0 ¼(s; a) X s 0 P a ss ¼ (s 0 ) Please refer to Chapter 13 of Rich Sutton s Reinforcement Learning: An Introduction (2 nd Edition)

44 APPENDIX Policy gradient theorem: Start Value Setting Start state value objective X ¼ (s) = X a X a) Q ¼ (s; a)+ X a) Q ¼ (s; a) = 0 Pr(s! s; 0;¼) X a ¼(s; a) X s 1 P a ss ¼ (s 1 ¼ (s 1 ) = X a = X s 1 X a = X s 1 P ¼ (s 1 ) ¼(s; a) X P ¼ (s 1 ) ss 1 s a) Q ¼ (s; a) ¼(s; a) P a ss ¼ (s 1 ) = 1 X s 1 Pr(s! s 1 ; ¼ (s 1 a) Q ¼ (s; a)+ X 1 Pr(s 1! s 2 ; ¼ (s 2 ) s 2 Please refer to Chapter 13 of Rich Sutton s Reinforcement Learning: An Introduction (2 nd Edition)

45 APPENDIX Policy gradient theorem: Start Value Setting Start state value ¼ (s) = 0 Pr(s! s; 0;¼) X a) Q ¼ (s; a)+ X 1 s 1 Pr(s! s 1 ; 1;¼) X a + 2 X s 1 Pr(s! s 1 ; 1;¼) X s 2 Pr(s 1! s 2 ; ¼ (s 2 ) = 0 Pr(s! s; 0;¼) a) Q ¼ (s; a)+ X 1 a s 1 + X 2 Pr(s! s 2 ; ¼ (s 2 ) s 2 1X X = k Pr(s! x; k; ¼) X k=0 x a ¼ (s 0 ) = X 1X k Pr(s 0! s; k; ¼) X s k=0 a) Q ¼ (x; a) = X x Pr(s! s 1 ; 1;¼) X 1 ;a) Q ¼ (s 1 1 ;a) Q ¼ (s 1 ;a) 1X k Pr(s! x; k; ¼) X a a) Q ¼ (s; a) = X s d ¼ (s) X a) Q ¼ (x; a) Q ¼ (s; a) Please refer to Chapter 13 of Rich Sutton s Reinforcement Learning: An Introduction (2 nd Edition)

Lecture 8: Policy Gradient

Lecture 8: Policy Gradient Hado van Hasselt Outline 1 Introduction 2 Finite Difference Policy Gradient 3 Monte-Carlo Policy Gradient 4 Actor-Critic Policy Gradient Introduction Vapnik s rule Never solve