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Reinforcement Learning Presented by: Bibhas Chakraborty and Lacey Gunter 5/22/2017 1 What is Machine Learning? A method to learn about some phenomenon from data, when there is little scientific theory (e.g., physical or biological laws) relative to the size of the feature space. The goal is to make an “intelligent” machine, so that it can make decisions (or, predictions) in an unknown situation. The science of learning plays a key role in areas like statistics, data mining and artificial intelligence. It also arises in engineering, medicine, psychology and finance. 5/22/2017 2 Types of Learning Supervised Learning - Training data: (X,Y). (features, label) - Predict Y, minimizing some loss. - Regression, Classification. Unsupervised Learning - Training data: X. (features only) - Find “similar” points in high-dim X-space. - Clustering. 5/22/2017 3 Types of Learning (Cont’d) Reinforcement Learning - Training data: (S, A, R). (State-Action-Reward) - Develop an optimal policy (sequence of decision rules) for the learner so as to maximize its long-term reward. - Robotics, Board game playing programs. 5/22/2017 4 Example of Supervised Learning Predict the price of a stock in 6 months from now, based on economic data. (Regression) Predict whether a patient, hospitalized due to a heart attack, will have a second heart attack. The prediction is to be based on demographic, diet and clinical measurements for that patient. (Logistic Regression) 5/22/2017 5 Example of Supervised Learning Identify the numbers in a handwritten ZIP code, from a digitized image (pixels). (Classification) 5/22/2017 6 Example of Unsupervised Learning From the DNA micro-array data, determine which genes are most “similar” in terms of their expression profiles. (Clustering) 5/22/2017 7 Examples of Reinforcement Learning How should a robot behave so as to optimize its “performance”? (Robotics) How to automate the motion of a helicopter? (Control Theory) How to make a good chess-playing program? (Artificial Intelligence) 5/22/2017 8 History of Reinforcement Learning Roots in the psychology of animal learning (Thorndike,1911). Another independent thread was the problem of optimal control, and its solution using dynamic programming (Bellman, 1957). Idea of temporal difference learning (on-line method), e.g., playing board games (Samuel, 1959). A major breakthrough was the discovery of Qlearning (Watkins, 1989). 5/22/2017 9 What is special about RL? RL is learning how to map states to actions, so as to maximize a numerical reward over time. Unlike other forms of learning, it is a multistage decision-making process (often Markovian). An RL agent must learn by trial-and-error. (Not entirely supervised, but interactive) Actions may affect not only the immediate reward but also subsequent rewards (Delayed effect). 5/22/2017 10 Elements of RL A policy - A map from state space to action space. - May be stochastic. A reward function - It maps each state (or, state-action pair) to a real number, called reward. A value function - Value of a state (or, state-action pair) is the total expected reward, starting from that state (or, state-action pair). 5/22/2017 11 The Precise Goal To find a policy that maximizes the Value function. There are different approaches to achieve this goal in various situations. Q-learning and A-learning are just two different approaches to this problem. But essentially both are temporal-difference methods. 5/22/2017 12 The Basic Setting Training data: n finite horizon trajectories, of the form {s0 , a0 , r0 ,..., sT , aT , rT , sT 1}. Deterministic policy: A sequence of decision rules { 0 , 1 ,..., T }. Each π maps from the observable history (states and actions) to the action space at that time point. 5/22/2017 13 Value and Advantage Time t state value function, for history (s t , a t 1 ) is T Vt (st , at 1 ) E rj (S j , A j , S j 1 ) | St st , At 1 at 1 . j t Time t state-action value function, Q-function, is Qt (s t , a t ) E rt (S t , A t , St 1 ) Vt 1 (S t 1 , A t ) | S t s t , A t a t . Time t advantage, A-function, is t (s t , at ) Qt (s t , a t ) Vt (s t , a t 1 ). 5/22/2017 14 Optimal Value and Advantage Optimal time t value function for history (s t , a t 1 ) is T * Vt (st , at 1 ) max E rj (S j , A j , S j 1 ) | St st , At 1 at 1 . j t Optimal time t Q-function is Qt* (s t , a t ) E rt (S t , A t , St 1 ) Vt*1 (S t 1 , A t ) | S t s t , A t a t . Optimal time t A-function is t* (s t , at ) Qt* (st , at ) Vt* (st , at 1 ). 5/22/2017 15 Return (sum of the rewards) The conditional expectation of the return is T 1 T T E rt ST , AT t (S t , A t ) t 1 (S t 1 , At ) V0 ( S0 ) t 0 t 0 t 0 where the advantages μ are t (St , At ) Qt (St , At ) Vt (St , At 1 ) and the , called temporal difference errors, are t (St , At 1 ) rt 1 (St , At 1 ) Vt (St , At 1 ) Qt 1 (St 1 , At 1 ) 5/22/2017 16 Return (continued) Conditional expectation of the return is a telescoping sum T 1 T T E rt ST , AT t t 1 V0 ( S0 ) t 0 t 0 t 0 T T 1 t 0 t 0 [Qt Vt ] [rt Vt 1 Qt ] V0 ( S 0 ) Temporal difference errors have conditional mean zero E[rt 1 (St , At 1 ) Vt (St , At 1 ) | St 1 , At 1 ] Qt 1 (St 1 , At 1 ) 5/22/2017 17 Q-Learning Watkins,1989 Estimate the Q-function using some approximator (for example, linear regression or neural networks or decision trees etc.). Derive the estimated policy as an argument of the maximum of the estimated Q-function. Allow different parameter vectors at different time points. Let us illustrate the algorithm with linear regression as the approximator, and of course, squared error as the appropriate loss function. 5/22/2017 18 Q-Learning Algorithm QT 1 0. For t T , T 1,...,0, Set Yt rt max Qt 1 (S t 1 , A t , at 1;ˆt 1 ). at 1 2 1 ˆ t arg min Yt ,i Qt (S t ,i , A t ,i ; ) . n i 1 n The estimated policy satisfies ˆ Q ,t (s t , at 1 ) arg max Qt (s t , at ;ˆt ), t. at 5/22/2017 19 What is the intuition? Bellman equation gives Qt* ( St , At ) E rt max Qt*1 (S t 1 , A t , at 1 ) | S t , A t . at 1 * Q Q If t 1 t 1 and the training set were infinite, then Q-learning minimizes 2 E rt max Qt*1 (S t 1 , A t , at 1 ) Qt (S t , A t ; ) . at 1 which is equivalent to minimizing E[Qt* Qt (S t , A t ; )]2 . 5/22/2017 20 A Success Story TD Gammon (Tesauro, G., 1992) - A Backgammon playing program. - Application of temporal difference learning. - The basic learner is a neural network. - It trained itself to the world class level by playing against itself and learning from the outcome. So smart!! - More information: http://www.research.ibm.com/massive/tdl.html 5/22/2017 21 A-Learning Murphy, 2003 and Robins, 2004 Estimate the A-function (advantages) using some approximator, as in Q-learning. Derive the estimated policy as an argument of the maximum of the estimated A-function. Allow different parameter vectors at different time points. Let us illustrate the algorithm with linear regression as the approximator, and of course, squared error as the appropriate loss function. 5/22/2017 22 A-Learning Algorithm (Inefficient Version) For t T , T 1,...,0, T Yt rj j t T (S , A ;ˆ ). j t 1 j j j j 2 n 1 ˆt arg min Yt ,i t (S t ,i , A t ,i ; ) E ( t | S t ,i , A t 1,i ) . n i 1 t (S t , A t ;ˆt ) t (S t , A t ;ˆt ) max t (S t , , A t 1 , at ;ˆt ) at The estimated policy satisfies ˆ A,t (s t , a t 1 ) arg max t (s t , a t ;ˆt ), t. at 5/22/2017 23 Differences between Q and A-learning Q-learning At time t we model the main effects of the history, (St,,At-1) and the action At and their interaction Our Yt-1 is affected by how we modeled the main effect of the history in time t, (St,,At-1) A-learning At time t we only model the effects of At and its interaction with (St,,At-1) Our Yt-1 does not depend on a model of the main effect of the history in time t, (St,,At-1) 5/22/2017 24 Q-Learning Vs. A-Learning Relative merits and demerits are not completely known till now. Q-learning has low variance but high bias. A-learning has high variance but low bias. Comparison of Q-learning with A-learning involves a bias-variance trade-off. 5/22/2017 25 References Sutton, R.S. and Barto, A.G. (1998). Reinforcement Learning- An Introduction. Hastie, T., Tibshirani, R. and Friedman, J. (2001). The Elements of Statistical Learning-Data Mining, Inference and Prediction. Murphy, S.A. (2003). Optimal Dynamic Treatment Regimes. JRSS-B. Blatt, D., Murphy, S.A. and Zhu, J. (2004). A-Learning for Approximate Planning. Murphy, S.A. (2004). A Generalization Error for Q-Learning. 5/22/2017 26