Monte-Carlo Planning: Policy Improvement

Monte-Carlo Planning: Policy Improvement Alan Fern

Monte-Carlo Planning • Often a simulator of a planning domain is availableor can be learned from data Conservation Planning Fire & Emergency Response 2

Large Worlds: Monte-Carlo Approach • Often a simulator of a planning domain is availableor can be learned from data • Monte-Carlo Planning: compute a good policy for an MDP by interacting with an MDP simulator action World Simulator RealWorld State + reward 3

MDP: Simulation-Based Representation • A simulation-based representation gives: S, A, R, T, I: • finite state set S (|S|=n and is generally very large) • finite action set A (|A|=m and will assume is of reasonable size) • Stochastic, real-valued, bounded reward function R(s,a) = r • Stochastically returns a reward r given input s and a • Stochastic transition function T(s,a) = s’ (i.e. a simulator) • Stochastically returns a state s’ given input s and a • Probability of returning s’ is dictated by Pr(s’ | s,a) of MDP • Stochastic initial state function I. • Stochastically returns a state according to an initial state distribution These stochastic functions can be implemented in any language!

Outline • You already learned how to evaluate a policy given a simulator • Just run the policy multiple times for a finite horizon and average the rewards • In next two lectures we’ll learn how to select good actions

Monte-Carlo Planning Outline • Single State Case (multi-armed bandits) • A basic tool for other algorithms • Monte-Carlo Policy Improvement • Policy rollout • Policy Switching • Monte-Carlo Tree Search • Sparse Sampling • UCT and variants Today

Single State Monte-Carlo Planning • Suppose MDP has a single state and k actions • Can sample rewards of actions using calls to simulator • Sampling action a is like pulling slot machine arm with random payoff function R(s,a) s ak a1 a2 … … R(s,ak) R(s,a2) R(s,a1) Multi-Armed Bandit Problem

Multi-Armed Bandits • We will use bandit algorithms as components for multi-state Monte-Carlo planning • But they are useful in their own right • Pure bandit problems arise in many applications • Applicable whenever: • We have a set of independent options with unknown utilities • There is a cost for sampling options or a limit on total samples • Want to find the best option or maximize utility of our samples

Multi-Armed Bandits: Examples • Clinical Trials • Arms = possible treatments • Arm Pulls = application of treatment to inidividual • Rewards = outcome of treatment • Objective = Find best treatment quickly (debatable) • Online Advertising • Arms = different ads/ad-types for a web page • Arm Pulls = displaying an ad upon a page access • Rewards = click through • Objective = find best add quickly (the maximize clicks)

Simple Regret Objective • Different applications suggest different types of bandit objectives. • Today minimizing simple regret will be the objective • Simple Regret: quickly identify arm with high reward (in expectation) s ak a1 a2 … … R(s,ak) R(s,a2) R(s,a1) Multi-Armed Bandit Problem

Simple Regret Objective: Formal Definition • Protocol: At time step n the algorithm picks an “exploration” arm to pull and observes reward and also picks an arm index it thinks is best ( are random variables). • If interrupted at time n the algorithm returns . • Let denote the expected reward of best arm • Expected Simple Regret (: difference between and expected reward of arm selected by our strategy at time n

UniformBanditAlgorith(or Round Robin) Bubeck, S., Munos, R., & Stoltz, G. (2011). Pure exploration in finitely-armed and continuous-armed bandits. Theoretical Computer Science, 412(19), 1832-1852 UniformBandit Algorithm: • At round n pull arm with index (k mod n) + 1 • At round n return arm (if asked) with largest average reward • I.e. is the index of arm with best average so far • This bound is exponentially decreasing in n! • So even this simple algorithm has a provably small simple regret. Theorem: The expected simple regret of Uniform after n arm pulls is upper bounded by Ofor a constant c.

Can we do better? Tolpin, D. & Shimony, S, E. (2012). MCTS Based on Simple Regret. AAAI Conference on Artificial Intelligence. Algorithm -GreedyBandit: (parameter ) • At round n, with probability pull arm with best average reward so far, otherwise pull one of the other arms at random. • At round n return arm (if asked) with largest average reward Theorem: The expected simple regret of -Greedy for after n arm pulls is upper bounded by Ofor a constant c that is larger than the constant for Uniform(this holds for “large enough” n). Often is seen to be more effective in practice as well.

Monte-Carlo Planning Outline • Single State Case (multi-armed bandits) • A basic tool for other algorithms • Monte-Carlo Policy Improvement • Policy rollout • Policy Switching • Monte-Carlo Tree Search • Sparse Sampling • UCT and variants Today

Policy Improvement via Monte-Carlo • Now consider a very large multi-state MDP. • Suppose we have a simulator and a non-optimal policy • E.g. policy could be a standard heuristic or based on intuition • Can we somehow compute an improved policy? World Simulator + Base Policy action RealWorld State + reward 15

Policy Improvement Theorem • Definition: The Q-value function gives the expected future reward of starting in state s, taking action a, and then following policy until the horizon h. • Define: • Theorem [Howard, 1960]: For any non-optimal policy the policy is strictly better than . • So if we can compute at any state we encounter, then we can execute an improved policy • Can we use bandit algorithms to compute

Policy Improvement via Bandits s ak a1 a2 … SimQ(s,a1,π,h) SimQ(s,a2,π,h) SimQ(s,ak,π,h) • Idea: define a stochastic function SimQ(s,a,π,h)that we can implement and whose expected value is Qπ(s,a,h) • Then use Bandit algorithm to select (approximately) the action with best Q-value How to implement SimQ?

Policy Improvement via Bandits • SimQ(s,a,π,h) • r = R(s,a) simulate a in s • s = T(s,a) • for i = 1 to h-1 r = r + R(s, π(s)) simulate h-1 steps • s = T(s, π(s)) of policy • Return r • Simply simulate taking a in s and following policy for h-1 steps, returning discounted sum of rewards • Expected value of SimQ(s,a,π,h) is Qπ(s,a,h) which can be made arbitrarily close to Qπ(s,a) by increasing h

… … … Policy Improvement via Bandits • SimQ(s,a,π,h) • r = R(s,a) simulate a in s • s = T(s,a) • for i = 1 to h-1 r = r + R(s, π(s)) simulate h-1 steps • s = T(s, π(s)) of policy • Return r Trajectory under p Sum of rewards = SimQ(s,a1,π,h) a1 Sum of rewards = SimQ(s,a2,π,h) s a2 … ak Sum of rewards = SimQ(s,ak,π,h)

Policy Improvement via Bandits s ak a1 a2 … SimQ(s,a1,π,h) SimQ(s,a2,π,h) SimQ(s,ak,π,h) • Now apply your favorite bandit algorithm for simple regret • UniformRollout : use UniformBandit • Parameters: number of trials n and horizon/height h • -GreedyRollout: use -GreedyBandit • Parameters: number of trials n, and horizon/height h( often is a good choice)

… … … … … … … … … UniformRollout • Each action is tried roughly the same number of times (approximately times) s ak a1 a2 … SimQ(s,ai,π,h) trajectories Each simulates taking action ai then following π for h-1 steps. q11 q12 … q1w q21 q22 … q2w qk1 qk2 … qkw Samples of SimQ(s,ai,π,h)

… … … … … • Allocates a non-uniform number of trials across actions (focuses on more promising actions) s ak a1 a2 • For we might expect it to be better than UniformRollout for a same value of n. … q11 q12 … q1u q21 q22 … q2v qk1

… … … … … … … … … … … … … … … … Executing Rollout in Real World a2 ak Real worldstate/action sequence ak ak a1 a1 a2 a2 s … … run policy rollout run policy rollout … … Simulated experience

… … … … … … … … … Policy Rollout: # of Simulator Calls s ak a1 a2 … SimQ(s,ai,π,h) trajectories Each simulates taking action ai then following π for h-1 steps. • Total of n SimQ callseach using h calls to simulator • Total of hncalls to the simulator

Multi-Stage Rollout • A single call to Rollout[π,h,w](s) yields one iteration of policy improvement starting at policy π • We can use more computation time to yield multiple iterations of policy improvement via nesting calls to Rollout • Rollout[Rollout[π,h,w],h,w](s) returns the action for state s resulting from two iterations of policy improvement • Can nest this arbitrarily • Gives a way to use more time in order to improve performance

Practical Issues • There are three ways to speedup either rollout procedure • Use a faster policy • Decrease the number of trajectories n • Decrease the horizon h • Decreasing Trajectories: • If n is small compared to # of actions k, then performance could be poor since actions don’t get tried very often • One way to get away with a smaller n is to use an action filter • Action Filter: a function f(s) that returns a subset of the actions in state s that rollout should consider • You can use your domain knowledge to filter out obviously bad actions • Rollout decides among the rest

Practical Issues • There are three ways to speedup either rollout procedure • Use a faster policy • Decrease the number of trajectories n • Decrease the horizon h • Decrease Horizon h: • If h is too small compared to the “real horizon” of the problem, then the Q-estimates may not be accurate • One way to get away with a smaller h is to use a value estimation heuristic • Heuristic function: a heuristic function v(s) returns an estimate of the value of state s • SimQ is adjusted to run policy for h steps ending in state s’ and returning the sum of rewards adding in estimate v(s’)

… … … … … … … … … Multi-Stage Rollout s Each step requires nhsimulator callsfor Rollout policy ak a1 a2 … Trajectories of SimQ(s,ai,Rollout[π,h,w],h) • Two stage: compute rollout policy of “rollout policy of π” • Requires (nh)2calls to the simulator for 2 stages • In general exponential in the number of stages

Example: Rollout for Solitaire[Yan et al. NIPS’04] • Multiple levels of rollout can payoff but is expensive

… … … … … … … … … Rollout in 2-Player Games • SimQ simply uses the base policy to select moves for both players until the horizon • Is biased toward playing well against • Is this ok? s ak a1 a2 … p1 p2 q11 q12 … q1w q21 q22 … q2w qk1 qk2 … qkw

Another Useful Technique: Policy Switching • Suppose you have a set of base policies {π1, π2,…, πM} • Also suppose that the best policy to use can depend on the specific state of the system and we don’t know how to select. • Policy switching is a simple way to select which policy to use at a given step via a simulator

Another Useful Technique: Policy Switching s πM π 1 π 2 … Sim(s,π1,h) Sim(s,π2,h) Sim(s,πM,h) • The stochastic function Sim(s,π,h) simply samples the h-horizon value of π starting in state s • Implement by simply simulating π starting in s for h steps and returning discounted total reward • Use Bandit algorithm to select best policy and then select action chosen by that policy

… … … … … … … … … PolicySwitching PolicySwitch[{π1, π2,…, πM},h,n](s) • Define bandit with M arms giving rewards Sim(s,πi,h) • Let i* be index of the arm/policy selected by your favorite bandit algorithm using n trials • Return action πi*(s) s πM π 1 π 2 … Sim(s,πi,h) trajectories Each simulates following πifor h steps. v11 v12 … v1w v21 v22 … v2w vM1 vM2 … vMw Discounted cumulative rewards

… … … … … … … … … … … … … … … … Executing Policy Switching in Real World 𝜋2(s) 𝜋k(s’) Real worldstate/action sequence 1 𝜋1 𝜋k 𝜋k 𝜋2 𝜋2 s … … run policy rollout run policy rollout … … Simulated experience

Policy Switching: Quality • Let denote the ideal switching policy • Always pick the best policy indexat any state • The value of the switching policy is at least as good as the best single policy in the set • It will often perform better than any single policy in set. • For non-ideal case, were bandit algorithm only picks approximately the best arm we can add an error term to the bound. Theorem: For any state s, .

Policy Switching in 2-Player Games Now we give two policies sets one for each player: Max Player Policies (us) : Min Player Policies (them) : } These policy sets will often be the same, when players have the same actions sets. The encode our knowledge of what the possible effective strategies might be in the game

Minimax Policy Switching Build GameMatrix Current State s Game Simulator Each entry gives estimated value (for max player)of playing a policy pair against one another Each value estimated by averaging across w simulated games.

MinimaxSwitching Build GameMatrix Current State s Game Simulator MaxiMin Policy Select action MaxiMin Policy :

Monte-Carlo Planning: Policy Improvement