Deep Reinforcement Learning
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1 Martin Matyášek Artificial Intelligence Center Czech Technical University in Prague October 27, 2016 Martin Matyášek VPD, / 50
2 Reinforcement Learning in a picture R. S. Sutton and A. G. Barto 2015 learning what to do to maximize future reward general-purpose framework extending sequential decision making when the model of the environment is unknown Martin Matyášek VPD, / 50
3 RL background let s assume MDP S, A, P, R, s 0 RL deals with situation where the environment model P and R is unknown can be generalized to Stochastic Games S, N, A, P, R RL agent includes: policy a = π(s) (deterministic), π(a s) = P(a s) (stochastic) value function Q π (a s) = E[ t=0 γt r t s, a] Bellman Eq.: Q π (a s) = E s,a [r + γqπ (a s ) s, a] opt. value funcions: Q (s, a) = E s [r + γ max a Q (s, a ) s, a] opt. policy: π (s) = argmax a Q (s, a) model - learned proxy for environment Martin Matyášek VPD, / 50
4 RL Types 1. value-based RL estimate the opt. value function Q (s, a) max. value achievable under any policy 2. policy-based RL search directly for the opt. policy π i.e. policy yielding max. future reward 3. model-based RL build a model of the environment plan using this model Martin Matyášek VPD, / 50
5 Q-learning Algorithm 1 Q-learning 1: initialize the Q-function and V values (arbitrarily) 2: repeat 3: observe the current state s t 4: select action a t and take it 5: observe the reward R(s t, a t, s t+1 ) 6: Q t+1 (s t, a t ) (1 α t )Q t (s t, a t ) + α t (R(s t, a t, s t+1 ) + γv t (s t+1 )) 7: V t+1 (s) max a Q t (s, a) 8: until convergence Martin Matyášek VPD, / 50
6 Q-learning model-free method temporal-difference version: Q(s, a) Q(s, a) + α(r + γ max Q(s, a ) Q(s, a)) a }{{} value based on next state converges to Q, V iff 0 α t <, t=0 α t = and t=0 α2 t < zero-sum Stochastic Games: cannot simply use Q π i : S A i R but rather Q π i : S A R minimax-q converges to NE R-max: converge to ɛ Nash with prob. (1 δ) in poly. # steps (PAC learn) Martin Matyášek VPD, / 50
7 Q (s, a) Q(s, a, w) Q-Networks treat right hand side r + γ max a Q(s, a, w) of Bellman s Eq. as target minimize MSE loss l = (r + max a Q(s, a, w) Q(s, a, w)) 2 by stochastic gradient descent David Silver, Google DeepMind Martin Matyášek VPD, / 50
8 Q-learning summary + converges to Q using table lookup representation diverges using NN: correlations between samples non-stationary targets! go deep Martin Matyášek VPD, / 50
9 Deep Q-Networks (DQN) basic approach is called experience replay idea: remove correlations by building data-set from agent s experience e = (s, a, r, s ) sample experiences from D t = {e 1, e 2,... e t } and apply update deal with non-stationarity by fixing w in l = (r + γ max a Q(s, a, w ) Q(s, a, w)) 2 Martin Matyášek VPD, / 50
10 Algorithm 2 Deep Q-learning algorithm 1: init. replay memory D, init. Q with random weights 2: observe initial state s 3: repeat 4: with prob. ɛ select random a, select a = argmax a Q(s, a ) 5: carry out a, observe (r, s ) and store (s, a, r, s ) in D 6: sample random transition (ss, aa, rr, ss ) from D 7: calculate target for each minibatch transition: 8: if ss is terminal state then 9: tt rr 10: else 11: tt rr + γ max a Q(ss, aa ) 12: end if 13: train the Q-network using (tt Q(ss, aa)) 2 as loss 14: s s 15: until convergence Martin Matyášek VPD, / 50
11 DQN in Atari David Silver, Google DeepMind Martin Matyášek VPD, / 50
12 DQN in Atari - setting state stack of raw pixels from last 4 frames actions 18 joystick/button positions reward delta in score learn Q(s, a) David Silver, Google DeepMind Martin Matyášek VPD, / 50
13 DQN in Atari - results Martin Matyášek VPD, / 50
14 DQN improvements Double DQN removes bias caused by max a Q( ) current QN - w used to select actions old QN - w used to evaluate actions Prioritized Replay weight experience according to DQN error (stored in PQ) Duelling Network split Q-Network into: action-independent value function action-dependent advantage function Martin Matyášek VPD, / 50
15 General Reinforcement Learning Architecture (GORILA) Martin Matyášek VPD, / 50
16 Deep Policy Network parametrize the policy π by a DNN and use SGD to optimize weights u π(a s, u) or π(s, u) max u L(u) = E[ t=0 γt r t π(, u)] policy gradients: L(u) log π(a s,u) u = E[ u Q π (s, a)] for stochastic policy π(a s, u) L(u) u = E[ Qπ (s,a) a a u ] for deterministic policy a = π(s) where a is cont. and Q diff. variations: Actor-Critic alg., A3C, Fictitious Self-Play Martin Matyášek VPD, / 50
17 Game Theory 101 normal-form game G = (N, A, u) N = {1, 2,..., n} - players A = i N A i - pure strategies (actions) Π = i Π i = i (A i ) - mixed stratedies u = (u 1,..., u n ), u i : A R - utilities best response Nash equlibrium BR i (π i ) = {π i Π i π i π i : u i (π i, π i ) u i (π i, π i)} NE(G) = {π Π i N : π i BR i (π i )} Martin Matyášek VPD, / 50
18 Fictitious Self Play (FSP) fictitious play (FP) 1. initialize beliefs about the opponent s strategy 2. play a best response to the assessed strategy of the opponent 3. observe the opponent s actual play and update beliefs accordingly, goto 2 fictitious self play (FSP) DQN with experience replay learns BR to opponent policies policy network learns an average of BRs l w = log π w(a s) w actions a sample mix of policy network and best response Martin Matyášek VPD, / 50
19 Deep RL Summary RL steps in when the model of the environment is unknown NN employed when the state space is too large RL sets the learning objective, NN then approximates: value function V V w, Q Q w policy π π u use DNN to deal with convergence and correlations What about some applications? Martin Matyášek VPD, / 50
20 MvM History Man vs Machine: 1992 backgammon Tesauro very close to top human experts 1996 chess Kasparov loses vs Deep Blue 1997 othello Logistello vs Murakami checkers solved 2008 poker Polaris wins vs poker pros Heads-up limit Texas hold em solved Martin Matyášek VPD, / 50
21 Go Martin Matyášek VPD, / 50
22 The Rules of Go (a) capture (b) territory start empty board move place one stone (of your color) goal surround win control more than half of the board Martin Matyášek VPD, / 50
23 Go in Theory 2-players (just me and the opponent) zero-sum game (win for me = loss for opponent) perfect information finite (the game rules ensure this) NE exists and we can search for it: minimax alpha-beta negascout Martin Matyášek VPD, / 50
24 Go in Reality there is O(b d ) game states where b 250 and d 150 Martin Matyášek VPD, / 50
25 Alternative to Exhaustive Search Monte Carlo Tree Search (MCTS): popular heuristic search algorithm for game play Monte Carlo rollouts to estimate v(s) v (s) (reduces d) sampling actions from p(a s) reduces b MC rollouts search to max. depth without branching at all Martin Matyášek VPD, / 50
26 MCTS Martin Matyášek VPD, / 50
27 t 2015 the strongest Go programs are based on MCTS enhanced by policies that are trained to predict human expert moves early rules hand-made later ML based on simple features (lin. comb. of inputs) knowledge learned: (i) fast (simple) knowledge used for move selection in simulation (rollout policy) (ii) slower (better) knowledge used for move ordering in tree search (SL policy) Martin Matyášek VPD, / 50
28 2016: Lee Sedol vs. AlphaGo Martin Matyášek VPD, / 50
29 March 9-15, 2016 name Lee Sedol AlphaGo age 33 2 rank 9 dan prof. none titles 18 0 power 1 brain 1200 CPU and 200 GPU results loss, loss, loss, win, loss win, win, win, loss, win experience c 1k games c 1M self-play games Martin Matyášek VPD, / 50
30 How? Martin Matyášek VPD, / 50
31 Science Martin Matyášek VPD, / 50
32 AlphaGo Design normal search - MCTS simulation (rollout) policy - relatively normal supervised learning (SL) policy from master games - improved in details, more data RL from self-play for value network RL from self-play for policy network Martin Matyášek VPD, / 50
33 Convolutional Neural Network Martin Matyášek VPD, / 50
34 Value Network Martin Matyášek VPD, / 50
35 Policy Network Martin Matyášek VPD, / 50
36 Reducing d with Value Network Martin Matyášek VPD, / 50
37 Reducing d with Value Network Martin Matyášek VPD, / 50
38 Reducing b with Policy Network Martin Matyášek VPD, / 50
39 Deep RL in AlphaGo Martin Matyášek VPD, / 50
40 SL of Policy Networks network 12 layer convolutional NN training data 30M position from human experts (KGS 5+ dan) training alg. max. likelihood by SGD σ log p σ(a s) σ training time 4 weeks on 50 GPUs (Google Cloud) results 57% accuracy on held out test data state-of-the art was 44% Martin Matyášek VPD, / 50
41 RL of Policy Networks network 12 layer convolutional NN training data games of self-play between policy networks training alg. max. wins z by policy gradient RL ρ log p ρ(a s) z ρ training time 1 week on 50 GPUs (Google Cloud) results 80% vs. SL network p σ raw network 3 amateur dan Martin Matyášek VPD, / 50
![RL of Value Networks network 12 layer convolutional NN training data 30M games of self-play idea v p (s) = E[z t s t = s, a t,...,t p] training alg. min.](/docs-images/91/107015914/images/42-1.jpg "MSE by SGD v θ (s) v pρ (s) v (s) θ v θ(s) (z v θ (s)) θ training time 1 week on 50 GPUs (Google Cloud) results first strong position eval.")
42 RL of Value Networks network 12 layer convolutional NN training data 30M games of self-play idea v p (s) = E[z t s t = s, a t,...,t p] training alg. min. MSE by SGD v θ (s) v pρ (s) v (s) θ v θ(s) (z v θ (s)) θ training time 1 week on 50 GPUs (Google Cloud) results first strong position eval. function Martin Matyášek VPD, / 50
43 MCTS in AlphaGo each edge stores: action value Q(s, a) visit count N(s, a) prior prob. P(s, a) (initialized to P(s, a) = p σ (a s)) at each step t we select in state s t : a t = argmax a (Q(s t, a) + u(s t, a)) u(s, a) P(s,a) 1+N(s,a) decays with repeated visits (encourages exploration) leaf node s L is evaluated in two different ways: V (s L ) = (1 λ)v θ (s L ) + λz L 1. by value network v θ (s L ) 2. by outcome z L p π Martin Matyášek VPD, / 50
44 MCTS in AlphaGo: selection Martin Matyášek VPD, / 50
45 MCTS in AlphaGo: expansion Martin Matyášek VPD, / 50
46 MCTS in AlphaGo: evaluation Martin Matyášek VPD, / 50
47 MCTS in AlphaGo: rollout Martin Matyášek VPD, / 50
48 MCTS in AlphaGo: backup Martin Matyášek VPD, / 50
49 AlphaGo: results Martin Matyášek VPD, / 50
50 References Sutton, Richard S and Barto, Andrew G. Reinforcement learning: An introduction. MIT press Cambridge, Silver, David and Huang, Aja and Maddison, Chris J and others. Mastering the game of Go with deep neural networks and tree search. Nature Publishing Group, Shoham, Yoav and Leyton-Brown, Kevin. Multiagent systems: Algorithmic, game-theoretic, and logical foundations. Cambridge University Press, Silver, David. Tutorial: Deep Reinforcement Learning. Silver, David. AlphaGo slides. Resources_files/AlphaGo_IJCAI.pdf Martin Matyášek VPD, / 50
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