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1 Introduction to Game Theoretic Multi- Agent Learning Game Theory University of Tehran Spring 2009

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2 Introduction What is MAL? Why Game Theory? Multi Disciplinary Area of Research Economics Control Eng. Computer Science * Neurosciences Reinforcement Learning

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3 Introduction to Reinforcement Learning Gerry Tesauro IBM T.J.Watson Research Center

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4 Outline Statement of the problem: What RL is all about How it’s different from supervised learning Mathematical Foundations Markov Decision Problem (MDP) framework Dynamic Programming: Value Iteration,... Temporal Difference (TD) and Q Learning Applications: Combining RL and function approximation

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5 Outline Statement of the problem: What RL is all about How it’s different from supervised learning Mathematical Foundations Markov Decision Problem (MDP) framework Dynamic Programming: Value Iteration,... Temporal Difference (TD) and Q Learning Applications: Combining RL and function approximation

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6 Acknowledgement Lecture material shamelessly adapted from: R. S. Sutton and A. G. Barto, “Reinforcement Learning” –Book published by MIT Press, 1998 –Available on the web at: RichSutton.com –Many slides shamelessly stolen from web site

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7 Basic RL Framework 1. Learning with evaluative feedback –Learner’s output is “scored” by a scalar signal (“Reward” or “Payoff” function) saying how well it did –Supervised learning: Learner is told the correct answer! –May need to try different outputs just to see how well they score (exploration …)

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10 Basic RL Framework 2. Learning to Act: Learning to manipulate the environment –Supervised learning is passive: Learner doesn’t affect the distribution of exemplars or the class labels –Model of Environment and RL

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12 Basic RL Framework –Learner has to figure out which action is best, and which actions lead to which states. Might have to try all actions! –Exploration vs. Exploitation: when to try a “wrong” action vs. sticking to the “best” action

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13 Basic RL Framework 3. Learning Through Time: –Reward is delayed (Act now, reap the reward later) –Agent may take long sequence of actions before receiving reward –“Temporal Credit Assignment” Problem: Given sequence of actions and rewards, how to assign credit/blame for each action?

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17 Agent’s objective is to maximize expected value of “return” R: sum of future rewards: – is a “discount parameter” (0 1) –Example: Cart-Pole Balancing Problem: – reward = -1 at failure, else 0 – expected return = - k for k steps to failure reward maximized by making k

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18 We consider non-deterministic environments: –Action a t in state s t Probability distribution of rewards r t+1 Probability distribution of new states s t+1 Some environments have nice property: distributions are history-independent and stationary. These are called Markov environments and the agent’s task is a Markov Decision Problem (MDP)

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19 An MDP specification consists of: –list of states s S –list of legal action set A(s) for every s –set of transition probabilities for every s,a,s’: –set of expected rewards for every s,a,s’:

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20 Given an MDP specification: –Agent learns a policy : deterministic policy (s) = action to take in state s non-deterministic policy (s,a) = probability of choosing action a in state s –Agent’s objective is to learn the policy that maximizes expected value of return R t –“Value Function” associated with a policy tells us how good the policy is. Two types of value functions...

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21 State-Value Function V (s) = Expected return starting in state s and following policy : Action-Value Function Q (s,a) = Expected return starting from action a in state s, and then following policy :

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22 Bellman Equation for a Policy The basic idea: Apply expectation for state s under policy : A linear system of equations for V ; unique solution

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25 Why V*, Q* are useful Any policy that is greedy w.r.t. V* or Q* is an optimal policy *. One-step lookahead using V*: Zero-step lookahead using Q*:

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26 Two methods to solve for V*, Q* Policy improvement: given a policy , find a better policy ’. –Policy Iteration: Keep repeating above and ultimately you will get to *. Value Iteration: Directly solve Bellman’s optimality equation, without explicitly writing down the policy.

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27 Policy Improvement Evaluate the policy: given , compute V (s) and Q (s,a) (from linear Bellman equations). For every state s, construct new policy: do the best initial action, and then follow policy thereafter. The new policy is greedy w.r.t. Q (s,a) and V (s) V ’ (s) V (s) ’ in our partial ordering.

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28 Policy Improvement, contd. What if the new policy has the same value as the old policy? ( V ’ (s) = V (s) for all s) But this is the Bellman Optimality equation: if V solves it, then it must be the optimal value function V*.

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30 Value Iteration Use the Bellman Optimality equation to define an iterative “ bootstrap ” calculation: This is guaranteed to converge to a unique V* (backup is a contraction mapping)

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31 Summary of DP methods Guaranteed to converge to * in polynomial time (in size of state space); in practice often faster than linear The method of choice if you can do it. Why it might not be doable: –your problem is not an MDP –the transition probs and rewards are unknown or too hard to specify –Bellman’s “curse of dimensionality:” the state space is too big (>> O(10 6 ) states) RL may be useful in these cases

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32 Monte Carlo Methods Estimate V (s) by sampling –perform a trial: run the policy starting from s until termination state reached; measure actual return R t –N trials: average R t accurate to ~ 1/sqrt(N) –no “bootstrapping:” not using V(s’) to estimate V(s) Two important advantages of Monte Carlo: –Can learn online without a model of the environment –Can learn in a simulated environment

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34 Temporal Difference Learning Error signal: difference between current estimate and improved estimate; drives change of current estimate –Supervised learning error: error(x) = target_output(x) - learner_output(x) –Bellman error (DP): “1-step full-width lookahead” - “0-step lookahead” –Monte Carlo error: error(s) = - V(s) “many-step sample lookahead” - “0-step lookahead”

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35 TD error signal Temporal Difference Error Signal: take one step using current policy, observe r and s’, then: “1-step sample lookahead” - “0-step lookahead” –In particular, for undiscounted sequences with no intermediate rewards, we have simply: –Self-consistent prediction goal: predicted returns should be self-consistent from one time step to the next (true of both TD and DP)

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36 Learning using the Error Signal: we could just do a reassignment: But it’s often a good idea to learn incrementally: where is a small “learning rate” parameter (either constant, or decreases with time) the above algorithm is known as “TD(0)” ; convergence to be discussed later...

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37 Advantages of TD Learning Combines the “bootstrapping” (1-step self-consistency) idea of DP with the “sampling” idea of MC; maybe the best of both worlds Like MC, doesn’t need a model of the environment, only experience TD, but not MC, can be fully incremental –you can learn before knowing the final outcome –you can learn without the final outcome (from incomplete sequences) Bootstrapping TD has reduced variance compared to Monte Carlo, but possibly greater bias

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42 The point of the parameter (My view): in TD( ) is a knob to twiddle: provides a smooth interpolation between =0 (pure TD) and =1 (pure MC) For many toy grid-world type problems, can show that intermediate values of work best. For real-world problems, best will be highly problem-dependent.

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43 Convergence of TD ( ) TD( ) converges to the correct value function V (s) with probability 1 for all . Requires: – lookup table representation (V(s) is a table), – must visit all states an infinite # of times, – a certain schedule for decreasing (t). (Usually (t) ~ 1/t) BUT: TD( ) converges only for a fixed policy. What if we want to learn as well as V? We still have more work to do...

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44 Q-Learning: TD Idea to Learn * Q-Learning (Watkins, 1989): one-step sample backup to learn action-value function Q(s,a). The most important RL algorithm in use today. Uses one-step error: to define an incremental learning algorithm: where (t) follows same schedule as in TD algorithm.

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Q-Learning Algorithmic Components Learning update (to Q-Table): Q(s, a) (1-α)Q(s, a) + α[r + γ Q(s’, a’)] or Q(s, a) Q(s, a) + α[r + γ Q(s’, a’) - Q(s, a)] Action selection (from Q-Table): a = f(Q(s, a))

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Q-Learning Algorithm 46

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47 Nice properties of Q-learning Q guaranteed to converge to Q* w/probability 1. Greedy guaranteed to converge to *. But (amazingly), don’t need to follow a fixed policy, or the greedy policy, during learning! Virtually any policy will do, as long as all (s,a) pairs visited infinitely often. As with TD, don’t need a model, can learn online, both bootstraps and samples.

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48 RL and Function Approximation DP infeasible for many real applications due to curse of dimensionality: |S| too big. FA may provide a way to “lift the curse:” –complexity D of FA needed to capture regularity in environment may be << |S|. –no need to sweep thru entire state space: train on N “plausible” samples and then generalize to similar samples drawn from the same distribution. –PAC learning tells us generalization error ~D/N; N need only scale linearly with D.

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49 RL + Gradient Parameter Training Recall incremental training of lookup tables: If instead V(s) = V (s), adjust to reduce MSE (R-V(s)) 2 by gradient descent:

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50 Example: TD( ) training of neural networks (episodic; =1 and intermediate r = 0):

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51 Case-Study Applications Several commonalities: – Problems are more-or-less MDPs – |S| is enormous can’t do DP – State-space representation critical: use of “features” based on domain knowledge – FA is reasonably simple (linear or NN) – Train in a simulator! Need lots of experience, but still << |S| – Only visit plausible states; only generalize to plausible states

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53 Learning backgammon using TD( ) Neural net observes a sequence of input patterns x 1, x 2, x 3, …, x f : sequence of board positions occurring during a game Representation: Raw board description (# of White or Black checkers at each location) using simple truncated unary encoding. (“hand-crafted features” added in later versions) At final position x f, reward signal z given: –z = 1 if White wins; –z = 0 if Black wins Train neural net using gradient version of TD( ) Trained NN output V t = V (x t, w) should estimate prob (White wins | x t )

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55 Q: Who makes the moves?? A: Let neural net make the moves itself, using its current evaluator: score all legal moves, and pick max V t for White, or min V t for Black. Hopelessly non-theoretical and crazy: –Training V using non-stationary (no convergence proof) –Training V using nonlinear func. approx. (no cvg. proof) –Random initial weights Random initial play! Extremely long sequence of random moves and random outcome Learning seems hopeless to a human observer But what the heck, let’s just try and see what happens...

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56 TD-Gammon can teach itself by playing games against itself and learning from the outcome –Works even starting from random initial play and zero initial expert knowledge (surprising!) achieves strong intermediate play –add hand-crafted features: advanced level of play (1991) –2-ply search: strong master play (1993) –3-ply search: superhuman play (1998) “TD-Leaf:” n-step TD backups in 2-player games (Beal; Baxter et al.): great results for checkers and chess

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57 RL Success Stories/Videos U. Michigan RL wiki page: – “keep-away” in Robocup simulator – Aibo fast walk gate; ball acquisition – Humanoid robot Air hockey – Helicopter aerobatics (Ng et al.) Human flies helicopter for mins Perform System Identification: learn model of helicopter dynamics Using model, train RL policy in simulator

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58 Cell-phone channel allocation S. Singh and D. Bertsekas, NIPS-96 Dynamic resource allocation: assign channels to calls in a cell; can’t interfere with neighboring cell Problem is a real-time discrete-event MDP with huge state space ~ states Objective: maximize:

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59 Modified Bellman optimality equation Modify equation to handle continuous time, discrete events: where: s = configuration, e=random event (arrival, handoff, departure) a=action, t=random time to next event, c(s,a, t) = effective immediate payoff

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60 represent s x using 2 features for each cell: –Availability: # of free channels in a cell –Cell-channel packing: # of times channel is used in 4-cell radius represent V using linear FA: V = x train in simulator using gradient version of TD(0) 54

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71 RL for Spoken Dialogue Systems Singh, Litman, Kearns, Walker (JAIR 2002) Sequence of human-computer speech interactions Use in DB-query system “NJFun:” database of leisure activities in NJ, organized by (type, location, time) Humans aren’t MDPs, but pretend they are: devise MDP representation of system-human interaction:

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72 Severely restrict state space: 7 state variables and 42 “choice-state” combinations

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73 Severely restrict the policy: 2 actions possible in each choice-state: 2 42 possible policies; train using random exploration Actions are spoken requests to the user, classified as: –system initiative: “Please state the type of activity you are interested in” –user initative: “How may I help you?” –mixed initiative: “Please say the location you are interested in. You can also tell me the time.” –confirmation of an attribute: “Did you say you are interested in going to a museum?” Train on a corpus of 311 dialogues (using AT&T volunteers); test trained system on 124 test dialogues. “Reward” after each dialogue is both objective (was the specific task completed exactly or partially) as well as subjective (“good,” “bad,” or “so-so” performance) from the human Small MDP but don’t have a model! Do Q-Learning using sample trajectories with the above random-exploration policy

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74 Results: Learned policy much better than random exploration

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75 Results: Learned policy much better than standard policies

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77 RL Mashups RL + semi-supervised learning RL + active learning RL + metric learning RL + dimensionality reduction Bayesian RL RL + SVMs/kernel methods RL + semi-definite programming RL + Gaussian process models etc. NIPS 2006 workshop “Towards A New Reinforcement Learning:”

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78 Final remarks on RL Can solve MDPs on-line, in real environment, without knowing underlying MDP Function Approximators can avoid the “curse of dimensionality” Beyond MDPs: active research in RL for: –high-level planning, –structured (e.g. factored, hierarchical) MDPs, –partially observable MDPs (POMDPs), –history dependent problems, –non-stationary problems, –multi-agent problems For more info, go to: RichSutton.com

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79 Game Theory and Multi-Agent Learning

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80 Outline Description of the problem Tools and concepts from RL & game theory “Naïve” approaches to multi-agent learning ordinary single-agent RL evolutionary game theory “Sophisticated” approaches minimax-Q, FriendOrFoe-Q (Littman), tinkering with learning rates: WoLF (Bowling), “strategic teaching” (Camerer) Challenges and Opportunities

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81 Normal single-agent learning Assume that environment has observable states, characterizable expected rewards and state transitions, and all of the above is stationary (MDP-ish) Non-learning, theoretical solution to fully specified problem: DP formalism Learning: solve by trial and error without a full specification: RL + exploration, Monte Carlo,...

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82 Multi-Agent Learning Problem: Agent tries to solve its learning problem, while other agents in the environment also are trying to solve their own learning problems. challenging non-stationarity. Main scenarios: (1) cooperative; (2) self-interest (many deep issues swept under the rug) Agent may know very little about other agents: –payoffs may be unknown –learning algorithms unknown Traditional method of solution: game theory (uses several questionable assumptions)

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83 MAL needs foundational principles! A precise problem formulation is still lacking! See: “If Multi-Agent Learning is the Answer, What is the Question?” Shoham et al, 2006 Some (debatable) MAL objectives: –Learning should converge to a stationary strategy –In “self-play” learning (all agents use same learning algorithm), learners should jointly converge to an equilibrium strategy –Learning should achieve payoffs as good as a best-response to other agents’ strategies –(Worst case bound) Learning should guarantee a minimum payoff (“security payment,” “no-regret” property)

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84 Game Theory Provides essential theoretical/conceptual background for tackling multi-agent learning Wikipedia definition: Game theory is most often described as a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns. The essential feature, however, is that it provides a formal modelling approach to social situations in which decision makers interact with other minds.applied mathematicseconomics Today, widely used in politics, business, economics, biology, psychology, computer science etc.

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85 Fundamental Postulate of Game Theory: “Rationality” A rational player/agent will make decisions that maximize her individual expected utility (= expected payoff for simplicity) given her understanding/beliefs about the problem. Also, perfectly indifferent to payoffs received by other players.

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86 Basics of game theory A game is specified by: players (1…N), actions, and (expected) payoff matrices (functions of joint actions) B’s action A’s action A’s payoff B’s payoff If payoff matrices are identical, A and B are cooperative, else non-cooperative (zero-sum = purely competitive)

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87 Basic lingo…(2) Games with no states: (bi)-matrix games Games with states: stochastic games, Markov games; (state transitions are functions of joint actions) Games with simultaneous moves: normal form Games with alternating turns: extensive form No. of rounds = 1: one-shot game No. of rounds > 1: repeated game deterministic action policy: pure strategy non-deterministic action policy: mixed strategy e.g. Prob(R,P,S) = (½,¼,¼)

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88 Stochastic vs. Matrix Games A stochastic game (a.k.a. “Markov game” ) generalizes MDPs to multiple agents –finite state space S –joint action set –stationary reward distribution –stationary transition probabilities A matrix game has no state information, only joint actions and payoffs (|S| = 1)

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89 Basic Analysis Agent i’s mixed strategy x i is a best-response to others’ x -i if it maximizes payoff given x -i x i is a dominant strategy if it maximizes payoff regardless of what others do A joint strategy x is an equilibrium if each agent’s strategy is simultaneously a best-response to everyone else’s strategy, i.e. no incentive to deviate. Nash equilibrium is the main one, but there are others (e.g. correlated equilibrium) A Nash equilibrium always exists, but may be exponentially many of them, and very hard to compute tequilibrium coordination (players agree on which eqm to choose) is a big problem

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90 What about imperfect information games? Nash eqm. requires full observability of all game info. For imperfect info. games (e.g. each player has private info), corresponding concept is Bayes-Nash equilibrium (Nash plus Bayesian inference over hidden information). Even more intractable than regular Nash.

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91 Pros and Cons of game theory Game theory provides a basic conceptual/theoretical framework for thinking about multi-agent learning. Game theory is appropriate provided that: –Game is stationary and fully specified; X –Enough computer power to compute equilibrium; X –Can assume other agents are also game theorists; X –Can solve equilibrium coordination problem. X Above conditions rarely hold in real applications Multi-agent learning is not only a fascinating problem, it may be the only viable option.

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92 Real-Life vs. Game Theory games NFL playoffs World Series of Poker World of Warcraft Buying a house Salary negotiations Competitive pricing: –Best Buy vs. Circuit City –Airline fare wars OPEC production cuts NASDAQ, NYSE, … FCC spectrum auctions Matching Pennies Rock-Paper-Scissors Prisoners’ Dilemma Battle-of-the-Sexes Chicken Ultimatum

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93 Assumptions in Normal-Form Games Game specification is fully known; actions and payoffs are fully observable by all players Players act “simultaneously”, i.e. without observing actions of others (not scalable!) Assume no communication between players, or it doesn’t affect play (communication is “cheap talk”) Basic analysis assumes the game is only played once (called one-shot)

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94 Presentation of Rock Paper Scissors Payoffs in a Bimatrix This is a zero-sum game since for each pair of joint actions, the players’ payoffs add up to zero. This is a symmetric game: invariant under swapping of player labels This game has a unique mixed strategy Nash equilibrium: both players play uniform random strategies: prob(R,P,S)=(1/3,1/3,1/3) Column player R P S R Row player P S

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95 Prisoners’ Dilemma Game Prisoner 2 Confess (Defect) Hold out (Cooperate) Prisoner 1 Confess (Defect) Hold out (Cooperate) -10 0

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96 Prisoners’ Dilemma Game Prisoner 2 Confess (Defect) Hold out (Cooperate) Prisoner 1 Confess (Defect) Hold out (Cooperate) Whatever Prisoner 2 does, the best that Prisoner 1 can do is Confess

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97 Prisoners’ Dilemma Game Prisoner 2 Confess (Defect) Hold out (Cooperate) Prisoner 1 Confess (Defect) Hold out (Cooperate) Whatever Prisoner 1 does, the best that Prisoner 2 can do is Confess.

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98 Prisoners’ Dilemma Game Prisoner 2 Confess (Defect) Hold out (Cooperate) Prisoner 1 Confess (Defect) Hold out (Cooperate) A strategy is a dominant strategy if it is a player’s strictly best response to any strategies the other players might pick. A dominant strategy equilibrium is a strategy combination consisting of each players dominant strategy. Each player has a dominant strategy to Confess. The dominant strategy equilibrium is (Confess,Confess)

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99 Prisoners’ Dilemma Game Prisoner 2 Confess (Defect) Hold out (Cooperate) Prisoner 1 Confess (Defect) Hold out (Cooperate) The payoff in the dominant strategy equilibrium (-8,-8) is worse for both players than (-1,-1), the payoff in the case that both players hold out. Thus, the Prisoners’ Dilemma Game is a game of social conflict. Opportunity for multi-agent learning: by learning during repeated play, the Pareto optimal solution (-1,-1) can emerge as a result of learning (also can arise in evolutionary game theory).

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100 Battle of the Sexes Bob Prize FightBallet Alice Prize fight2121 Ballet

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101 Battle of the Sexes Bob Prize FightBallet Alice Prize fight2121 Ballet This game has no (iterated) dominant strategy equilibrium

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102 Battle of the Sexes Bob Prize FightBallet Alice Prize fight2121 Ballet This game has no (iterated) dominant strategy equilibrium

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103 Battle of the Sexes Bob Prize FightBallet Alice Prize fight2121 Ballet This game has no (iterated) dominant strategy equilibrium two Nash equilibria (Prize Fight, Prize Fight) and (Ballet, Ballet)

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104 Battle of the Sexes Bob Prize FightBallet Alice Prize fight2121 Ballet This game has two Nash equilibria How can these two players coordinate ?

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105 Multiagent Q-learning desiderata “ performs well ” vs. arbitrarily adapting other agents best-response probably impossible Doesn ’ t need correct model of other agents ’ learning algorithms But modeling is fair game Doesn ’ t need to know other agents ’ payoffs Estimate other agents ’ strategies from observation do not assume game-theoretic play No assumption of stationary outcome: population may never reach eqm, agents may never stop adapting Self-play: convergence to repeated Nash would be nice but not necessary. (unreasonable to seek convergence to a one-shot Nash)

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106 Naïve Approaches to Multi-Agent Learning Basic idea: agent adapts, ignoring non-stationarity of other agents’ strategies 1. Evolutionary game theory: “Replicator Dynamics” models: large population of agents using different strategies, fittest agents breed more copies. –Let x= population strategy vector, and x k = fraction of population playing strategy k. Growth rate then: –Above eqn also derived from an “imitation” model –NE are fixed points of above equation, but not necessarily attractors (unstable or neutral stable)

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107 Many possible dynamic behaviors... limit cycles attractors unstable f.p. Also saddle points, chaotic orbits,...

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108 Replicator dynamics: auction bidding strategies

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109 More Naïve Approaches… 2. Iterated Gradient Ascent: (Singh, Kearns and Mansour): Again does a myopic adaptation to other players’ current strategy. –Coupled system of linear equations: u is linear in x i and x -i –Analysis for two-player, two-action games: either converges to a Nash fixed point on the boundary (at least one pure strategy), or get limit cycles

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110 Further Naïve Approaches… 3. Dumb Single-Agent Learning: Use a single-agent algorithm in a multi-agent problem & hope that it works –No-regret learning by pricebots (Greenwald & Kephart) –Simultaneous Q-learning by pricebots (Tesauro & Kephart) –In many cases, this actually works: learners converge either exactly or approximately to self-consistent optimal strategies Naïve approaches are “rational” i.e. they converge to a best response against a stationary opponent –but they generally don’t converge to Nash equilibrium

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111 A Fancier Approach 4. No-regret learning: (Hart & Mas-Colell, Freund & Schapire, many others): Define regret for playing a sequence s i instead of constant action a j for t time steps: –Then choose next action with probability proportional to: prob (action j) ~ –This has a worst-case guarantee that asymptotic regret per time step 0, i.e., will be as good as best (constant) action choice

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112 “Sophisticated” approaches Takes into account the possibility that other agents’ strategies might change. 4. Equilibrium Q-learners: –Minimax-Q (Littman): converges to Nash equilibrium for two-player zero-sum stochastic games –FriendOrFoe-Q (Littman): convergent algorithm for games where every other player can be identified as “friend” (same payoffs as me) or “foe” (payoffs are zero-sum) These algorithms converge to Nash equilibrium but aren’t “rational” since they don’t best-respond to a fixed opponent

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113 More sophisticated approaches Varying learning rates –WoLF: “Win or Learn Fast” (Bowling): agent reduces its learning rate when performing well, and increases when doing badly. Improves convergence of IGA and policy hill-climbing –GIGA-WoLF (Bowling): Combines the IGA algorithm with WoLF idea. Provably convergent + no-regret.

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114 More sophisticated approaches “Strategic Teaching:” recognizes that other players’ strategy are adaptive –“A strategic teacher may play a strategy which is not myopically optimal (such as cooperating in Prisoner’s Dilemma) in the hope that it induces adaptive players to expect that strategy in the future, which triggers a best-response that benefits the teacher.” (Camerer, Ho and Chong)

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115 Theoretical Research Challenges Proper theoretical formulation? –“No short-cut” hypothesis: Massive on-line search a la Deep Blue to maximize expected long-term reward –(Bayesian) Model and predict behavior of other players, including how they learn based on my actions (beware of infinite model recursion) –trial-and-error exploration –continual Bayesian inference using all evidence over all uncertainties (Boutilier: Bayesian exploration) When can you get away with simpler methods?

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116 Real-World Opportunities Multi-agent systems where you can’t do game theory (covers everything :-)) –Electronic marketplaces –Mobile networks –Self-managing computer systems –Teams of robots –Video games –Military/counter-terrorism applications

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