4/3

Outline… Talk about SSSP problem Talk about DP vs. A* Talk about heuristic—and how the “deterministic plan” can be an admissible heuristic –What happens if the determinization is unlikely? (won’t the cost be high?) –Need all outcome determinization? Talk about PPDDL Sampling approaches.. Q-values.. (value of state-action combinations) –Value of a state is max of its Q values (w.r.t all possible actions) Add stuff on (1) getting the admissible heuristic (2) the stuff on simulation vs. execution.. (sort of montecarlo vs. actual evaluation)

(FO)MDPs: The plan General model has no initial state; complex cost and reward functions, and finite/infinite/indefinite horizons Standard algorithms are Value and Policy iteration –Have to look at the entire state space Can be made even more general with –Partial observability (POMDPs) –Continuous state spaces –Multiple agents (DECPOMDPS/MDPS) –Durative actions Conurrent MDPs Semi-MDPs Directions –Efficient algorithms for special cases TODAY & 4/10 –Combining “Learning” of the model and “planning” with the model Reinforcement Learning—4/8

Markov Decision Process (MDP)  S : A set of states  A : A set of actions  P r(s’|s,a): transition model (aka M a s,s’ )  C (s,a,s’): cost model  G : set of goals  s 0 : start state   : discount factor  R ( s,a,s’): reward model Value function: expected long term reward from the state Q values: Expected long term reward of doing a in s V(s) = max Q(s,a) Greedy Policy w.r.t. a value function Value of a policy Optimal value function

Examples of MDPs  Goal-directed, Indefinite Horizon, Cost Minimization MDP Most often studied in planning community  Infinite Horizon, Discounted Reward Maximization MDP Most often studied in reinforcement learning  Goal-directed, Finite Horizon, Prob. Maximization MDP Also studied in planning community  Oversubscription Planning: Non absorbing goals, Reward Max. MDP Relatively recent model

SSPP—Stochastic Shortest Path Problem An MDP with Init and Goal states MDPs don’t have a notion of an “initial” and “goal” state. (Process orientation instead of “task” orientation) –Goals are sort of modeled by reward functions Allows pretty expressive goals (in theory) –Normal MDP algorithms don’t use initial state information (since policy is supposed to cover the entire search space anyway). Could consider “envelope extension” methods –Compute a “deterministic” plan (which gives the policy for some of the states; Extend the policy to other states that are likely to happen during execution –RTDP methods SSSP are a special case of MDPs where –(a) initial state is given –(b) there are absorbing goal states –(c) Actions have costs. All states have zero rewards A proper policy for SSSP is a policy which is guaranteed to ultimately put the agent in one of the absorbing states For SSSP, it would be worth finding a partial policy that only covers the “relevant” states (states that are reachable from init and goal states on any optimal policy) –Value/Policy Iteration don’t consider the notion of relevance –Consider “heuristic state search” algorithms Heuristic can be seen as the “estimate” of the value of a state.

  Define J*(s) {optimal cost} as the minimum expected cost to reach a goal from this state.  J* should satisfy the following equation: Bellman Equations for Cost Minimization MDP (absorbing goals)[also called Stochastic Shortest Path] Q*(s,a)

  Define V*(s) {optimal value} as the maximum expected discounted reward from this state.  V* should satisfy the following equation: Bellman Equations for infinite horizon discounted reward maximization MDP

  Define P*(s,t) {optimal prob.} as the maximum probability of reaching a goal from this state at t th timestep.  P* should satisfy the following equation: Bellman Equations for probability maximization MDP

Modeling Softgoal problems as deterministic MDPs Consider the net-benefit problem, where actions have costs, and goals have utilities, and we want a plan with the highest net benefit How do we model this as MDP? –(wrong idea): Make every state in which any subset of goals hold into a sink state with reward equal to the cumulative sum of utilities of the goals. Problem—what if achieving g1 g2 will necessarily lead you through a state where g1 is already true? –(correct version): Make a new fluent called “done” dummy action called Done-Deal. It is applicable in any state and asserts the fluent “done”. All “done” states are sink states. Their reward is equal to sum of rewards of the individual states.

Ideas for Efficient Algorithms.. Use heuristic search (and reachability information) –LAO*, RTDP Use execution and/or Simulation –“Actual Execution” Reinforcement learning (Main motivation for RL is to “learn” the model) –“Simulation” –simulate the given model to sample possible futures Policy rollout, hindsight optimization etc. Use “factored” representations –Factored representations for Actions, Reward Functions, Values and Policies –Directly manipulating factored representations during the Bellman update

Heuristic Search vs. Dynamic Programming (Value/Policy Iteration) VI and PI approaches use Dynamic Programming Update Set the value of a state in terms of the maximum expected value achievable by doing actions from that state. They do the update for every state in the state space –Wasteful if we know the initial state(s) that the agent is starting from Heuristic search (e.g. A*/AO*) explores only the part of the state space that is actually reachable from the initial state Even within the reachable space, heuristic search can avoid visiting many of the states. –Depending on the quality of the heuristic used.. But what is the heuristic? –An admissible heuristic is a lowerbound on the cost to reach goal from any given state –It is a lowerbound on V*!

Connection with Heuristic Search s0s0 G s0s0 G ?? s0s0 G ?? regular graph acyclic AND/OR graph cyclic AND/OR graph

Connection with Heuristic Search s0s0 G s0s0 G ?? s0s0 G ?? regular graph soln:(shortest) path A* acyclic AND/OR graph soln:(expected shortest) acyclic graph AO* [Nilsson’71] cyclic AND/OR graph soln:(expected shortest) cyclic graph LAO* [Hansen&Zil.’98] All algorithms able to make effective use of reachability information!

LAO* [Hansen&Zilberstein’98] 1.add s 0 in the fringe and in greedy graph 2.repeat  expand a state on the fringe (in greedy graph)  initialize all new states by their heuristic value  perform value iteration for all expanded states  recompute the greedy graph 3.until greedy graph is free of fringe states 4.output the greedy graph as the final policy

LAO* [Iteration 1] s0s0 G ?? s0s0 add s 0 in the fringe and in greedy graph

LAO* [Iteration 1] s0s0 G ?? s0s0 expand a state on fringe in greedy graph ??

LAO* [Iteration 1] s0s0 G ?? s0s0  initialise all new states by their heuristic values  perform VI on expanded states ?? h hhh J1J1

LAO* [Iteration 1] s0s0 G ?? s0s0 recompute the greedy graph ?? h hhh J1J1

LAO* [Iteration 2] s0s0 G ?? s0s0 expand a state on the fringe initialise new states ?? h hhh J1J1 h h

LAO* [Iteration 2] s0s0 G ?? s0s0 perform VI compute greedy policy ?? h hh J2J2 h h J2J2

LAO* [Iteration 3] s0s0 G ?? s0s0 expand fringe state ?? h h J2J2 h h J2J2 G

LAO* [Iteration 3] s0s0 G ?? s0s0 perform VI recompute greedy graph ?? h h J3J3 h h J3J3 G J3J3

LAO* [Iteration 4] s0s0 G ?? s0s0 ?? h J4J4 h h J4J4 G J4J4 h J4J4

s0s0 G ?? s0s0 ?? h J4J4 h h J4J4 G J4J4 h J4J4 Stops when all nodes in greedy graph have been expanded

Comments  Dynamic Programming + Heuristic Search  admissible heuristic ⇒ optimal policy  expands only part of the reachable state space  outputs a partial policy one that is closed w.r.t. to P r and s 0 Speedups expand all states in fringe at once perform policy iteration instead of value iteration perform partial value/policy iteration weighted heuristic: f = (1-w).g + w.h ADD based symbolic techniques (symbolic LAO*)

Real Time Dynamic Programming [Barto, Bradtke, Singh’95]  Trial: simulate greedy policy starting from start state; perform Bellman backup on visited states  RTDP: repeat Trials until cost function converges

AO* search for solving SSP problems Main issues: -- Cost of a node is expected cost of its children -- The And tree can have LOOPS  Cost backup is complicated Intermediate nodes given admissible heuristic estimates --can be just the shortest paths (or their estimates)

LAO*--turning bottom-up labeling into a full DP

RTDP Approach: Interleave Planning & Execution (Simulation) Start from the current state S. Expand the tree (either uniformly to k-levels, or non-uniformly—going deeper in some branches) Evaluate the leaf nodes; back-up the values to S. Update the stored value of S. Pick the action that leads to best value Do it {or simulate it}. Loop back. Leaf nodes evaluated by Using their “cached” values  If this node has been evaluated using RTDP analysis in the past, you use its remembered value else use the heuristic value  If not use heuristics to estimate a. Immediate reward values b. Reachability heuristics Sort of like depth-limited game-playing (expectimax) --Who is the game against? Can also do “reinforcement learning” this way  The M ij are not known correctly in RL

Greedy “On-Policy” RTDP without execution  Using the current utility values, select the action with the highest expected utility (greedy action) at each state, until you reach a terminating state. Update the values along this path. Loop back—until the values stabilize

Min ? ? s0s0 JnJn JnJn JnJn JnJn JnJn JnJn JnJn Q n+1 (s 0,a) J n+1 (s 0 ) a greedy = a 2 Goal a1a1 a2a2 a3a3 RTDP Trial ?

Comments  Properties if all states are visited infinitely often then J n → J*  Advantages Anytime: more probable states explored quickly  Disadvantages complete convergence is slow! no termination condition

Labeled RTDP [Bonet&Geffner’03]  Initialise J 0 with an admissible heuristic ⇒ J n monotonically increases  Label a state as solved if the J n for that state has converged  Backpropagate ‘solved’ labeling  Stop trials when they reach any solved state  Terminate when s 0 is solved sG high Q costs best action ) J(s) won’t change! sG ? t both s and t get solved together high Q costs

Properties  admissible J 0 ⇒ optimal J*  heuristic-guided explores a subset of reachable state space  anytime focusses attention on more probable states  fast convergence focusses attention on unconverged states  terminates in finite time

Recent Advances: Bounded RTDP [McMahan, Likhachev & Gordon’05]  Associate with each state Lower bound (lb): for simulation Upper bound (ub): for policy computation gap(s) = ub(s) – lb(s)  Terminate trial when gap(s) <   Bias sampling towards unconverged states proportional to P r(s’|s,a).gap(s’)  Perform backups in reverse order for current trajectory.

Recent Advances: Focused RTDP [Smith&Simmons’06] Similar to Bounded RTDP except a more sophisticated definition of priority that combines gap and prob. of reaching the state adaptively increasing the max-trial length Recent Advances: Learning DFS [Bonet&Geffner’06]  Iterative Deepening A* equivalent for MDPs  Find strongly connected components to check for a state being solved.

Other Advances  Ordering the Bellman backups to maximise information flow. [Wingate & Seppi’05] [Dai & Hansen’07]  Partition the state space and combine value iterations from different partitions. [Wingate & Seppi’05] [Dai & Goldsmith’07]  External memory version of value iteration [Edelkamp, Jabbar & Bonet’07]  …

Policy Gradient Approaches [Williams’92]  direct policy search parameterised policy Pr(a|s,w) no value function flexible memory requirements  policy gradient J(w)=E w [  t=0.. 1  t c t ] gradient descent (wrt w) reaches a local optimum continuous/discrete spaces parameterised policy Pr(a|s.w) …… parameters w state s action a Pr(a=a 1 |s,w) Pr(a=a 2 |s,w) Pr(a=a k |s,w) …. non-stationary

Policy Gradient Algorithm  J(w)=E w [  t=0.. 1  t c t ] (failure prob.,makespan, …)  minimise J by computing gradient stepping the parameters away w t+1 = w t −  r J(w)  until convergence  Gradient Estimate [Sutton et.al.’99, Baxter & Bartlett’01]  Monte Carlo estimate from trace s 1, a 1, c 1, …, s T, a T, c T e t+1 = e t + r w log Pr(a t+1 |s t,w t ) w t+1 = w t -  t c t e t+1

Policy Gradient Approaches  often used in reinforcement learning partial observability model free ( P r(s’|s,a), P r(o|s) are unknown) to learn a policy from observations and costs Reinforcement Learner …… Pr(a|o,w) Pr(a=a 1 |o,w) Pr(a=a 2 |o,w) Pr(a=a k |o,w) …. world/simulator P r(s’|s,a) P r(o|s) observation o cost c action a

Modeling Complex Problems  Modeling time continuous variable in the state space discretisation issues large state space  Modeling concurrency many actions may execute at once large action space  Modeling time and concurrency large state and action space!! J(s) t t

Ideas for Efficient Algorithms.. Use heuristic search (and reachability information) –LAO*, RTDP Use execution and/or Simulation –“Actual Execution” Reinforcement learning (Main motivation for RL is to “learn” the model) –“Simulation” –simulate the given model to sample possible futures Policy rollout, hindsight optimization etc. Use “factored” representations –Factored representations for Actions, Reward Functions, Values and Policies –Directly manipulating factored representations during the Bellman update

Factored Representations: Actions Actions can be represented directly in terms of their effects on the individual state variables (fluents). The CPTs of the BNs can be represented compactly too! –Write a Bayes Network relating the value of fluents at the state before and after the action Bayes networks representing fluents at different time points are called “Dynamic Bayes Networks” We look at 2TBN (2-time-slice dynamic bayes nets) Go further by using STRIPS assumption –Fluents not affected by the action are not represented explicitly in the model –Called Probabilistic STRIPS Operator (PSO) model

Action CLK

Envelope Extension Methods For each action, take the most likely outcome and discard the rest. Find a plan (deterministic path) from Init to Goal state. This is a (very partial) policy for just the states that fall on the maximum probability state sequence. Consider states that are most likely to be encountered while traveling this path. Find policy for those states too. Tricky part is to show that we can converge to the optimal policy

Factored Representations: Reward, Value and Policy Functions Reward functions can be represented in factored form too. Possible representations include –Decision trees (made up of fluents) –ADDs (Algebraic decision diagrams) Value functions are like reward functions (so they too can be represented similarly) Bellman update can then be done directly using factored representations..

SPUDDs use of ADDs

Direct manipulation of ADDs in SPUDD

Policy Rollout: Time Complexity …… Following PR[π,h,w] s s … … … … … Trajectories under  a1 an To compute PR[π,h,w](s) for each action we need to compute w trajectories of length h Total of |A|hw calls to the simulator

Policy Rollout  Often π’ is significantly better than π. I.e. one step of policy iteration can provide substantial improvement  Using simulation to approximate π’ is known as policy rollout PolicyRollout(s, π, h, w)  FOR each action a, Q’(a) = EstimateQ(s, a, π, h, w)  RETURN arg max a Q’(a)  We will denote the rollout policy by PR[π,h,w]  Note that PR[π,h,w] is stochastic  Questions:  What is the complexity of computing PR[π,h,w](s)?  Can we approximate k iterations of policy iteration using sampling?  How good is the rollout policy compared to the policy iteration improvement?

Multi-Stage Policy Rollout …… Following PR[PR[π,h,w],h,w] s s … … … … … Trajectories under PR[ π,h,w] a1 an Approximates policy resulting from two steps of PI. Requires (|A|hw) 2 calls to the simulator In general exponential in the number of PI iterations Each step requires |A|hw simulator calls

Policy Rollout: Quality  How good is PR[π,h,w](s) compared to π’?  In general for a fixed h and w there is always an MDP such that the quality of the rollout policy is arbitrarily worse than π’.  If we make an assumption about the MDP, then it is possible to select h and w so that the rollout quality is close to π’.  This is a bit involved.  In your homework you will solve a related problem.  Choose h and w so that with high probability PR[π,h,w](s) selects an action that maximizes Q π (s,a)

Rollout Summary  We often are able to write simple, mediocre policies  Network routing policy  Policy for card game of Hearts  Policy for game of Backgammon  Solitaire playing policy  Policy rollout is a general and easy way to improve upon such policies  Often observe substantial improvement, e.g.  Compiler instruction scheduling  Backgammon  Network routing  Combinatorial optimization  Game of GO  Solitaire

Example: Rollout for Go  Go is a popular board game for which there are still no highly-ranked computer programs  High branching factor  Difficult to design heuristics  Unlike Chess where the best computers compete with the best humans  What if we play Go using level-1 rollouts of the random policy?  With a few additional tweaks you get the best go program to date! (CrazyStone, winner of 11 th Computer Go Olympiad)