VIEWS: 10 PAGES: 54 POSTED ON: 6/23/2011 Public Domain
Hierarchical Reinforcement Learning [A Survey and Comparison of HRL techniques] Mausam The Outline of the Talk MDPs and Bellman’s curse of dimensionality. RL: Simultaneous learning and planning. Explore avenues to speed up RL. Illustrate prominent HRL methods. Compare prominent HRL methods. Discuss future research. Summarise Decision Making Environment What action next? Percept Action Slide courtesy Dan Weld Personal Printerbot States (S) : {loc,has-robot-printout, user-loc,has-user- printout},map Actions (A) :{moven,moves,movee,movew, extend-arm,grab-page,release-pages} Reward (R) : if h-u-po +20 else -1 Goal (G) : All states with h-u-po true. Start state : A state with h-u-po false. Episodic Markov Decision Process Episodic MDP ´ hS, A, P, R, G, s0i MDP with S : Set of environment states. absorbing goals A : Set of available actions. P : Probability Transition model. P(s’|s,a)* R : Reward model. R(s)* G : Absorbing goal states. s0 : Start state. * Markovian : Discount factor**. assumption. ** bounds R for infinite horizon. Goal of an Episodic MDP Find a policy (S ! A), which: maximises expected discounted reward for a a fully observable* Episodic MDP. if agent is allowed to execute for an indefinite horizon. * Non-noisy complete information perceptors Solution of an Episodic MDP Define V*(s) : Optimal reward starting in state s. Value Iteration : Start with an estimate of V*(s) and successively re-estimate it to converge to a fixed point. Complexity of Value Iteration Each iteration – polynomial in |S| Number of iterations – polynomial in |S| Overall – polynomial in |S| Polynomial in |S| - |S| : exponential in number of * Bellman’s features in the domain*. curse of dimensionality The Outline of the Talk MDPs and Bellman’s curse of dimensionality. RL: Simultaneous learning and planning. Explore avenues to speed up RL. Illustrate prominent HRL methods. Compare prominent HRL methods. Discuss future research. Summarise Learning Environment •Gain knowledge •Gain understanding •Gain skills •Modification of behavioural tendency Data Decision Making while Learning* Environment •Gain knowledge •Gain understanding •Gain skills •Modification of behavioural tendency What action Percepts next? Datum Action * Known as Reinforcement Learning Reinforcement Learning Unknown P and reward R. Learning Component : Estimate the P and R values via data observed from the environment. Planning Component : Decide which actions to take that will maximise reward. Exploration vs. Exploitation GLIE (Greedy in Limit with Infinite Exploration) Learning Model-based learning Learn the model, and do planning Requires less data, more computation Model-free learning Plan without learning an explicit model Requires a lot of data, less computation Q-Learning Instead of learning, P and R, learn Q* directly. Q*(s,a) : Optimal reward starting in s, if the first action is a, and after that the optimal policy is followed. Q* directly defines the optimal policy: Optimal policy is the action with maximum Q* value. Q-Learning Given an experience tuple hs,a,s’,ri Under suitable assumptions, and GLIE New Old estimate exploration Q-Learning estimate of Q value of Q value converges to optimal. Semi-MDP: When actions take time. The Semi-MDP equation: Semi-MDP Q-Learning equation: where experience tuple is hs,a,s’,r,Ni r = accumulated discounted reward while action a was executing. Printerbot Paul G. Allen Center has 85000 sq ft space Each floor ~ 85000/7 ~ 12000 sq ft Discretise location on a floor: 12000 parts. State Space (without map) : 2*2*12000*12000 --- very large!!!!! How do humans do the decision making? The Outline of the Talk MDPs and Bellman’s curse of dimensionality. RL: Simultaneous learning and planning. Explore avenues to speedup RL. Illustrate prominent HRL methods. Compare prominent HRL methods. Discuss future research. Summarise 1. The Mathematical Perspective A Structure Paradigm S : Relational MDP A : Concurrent MDP P : Dynamic Bayes Nets R : Continuous-state MDP G : Conjunction of state variables V : Algebraic Decision Diagrams : Decision List (RMDP) 2. Modular Decision Making 2. Modular Decision Making •Go out of room •Walk in hallway •Go in the room 2. Modular Decision Making Humans plan modularly at different granularities of understanding. Going out of one room is similar to going out of another room. Navigation steps do not depend on whether we have the print out or not. 3. Background Knowledge Classical Planners using additional control knowledge can scale up to larger problems. (E.g. : HTN planning, TLPlan) What forms of control knowledge can we provide to our Printerbot? First pick printouts, then deliver them. Navigation – consider rooms, hallway, separately, etc. A mechanism that exploits all three avenues : Hierarchies 1. Way to add a special (hierarchical) structure on different parameters of an MDP. 2. Draws from the intuition and reasoning in human decision making. 3. Way to provide additional control knowledge to the system. The Outline of the Talk MDPs and Bellman’s curse of dimensionality. RL: Simultaneous learning and planning. Explore avenues to speedup RL. Illustrate prominent HRL methods. Compare prominent HRL methods. Discuss future research. Summarise Hierarchy Hierarchy of : Behaviour, Skill, Module, SubTask, Macro-action, etc. picking the pages collision avoidance fetch pages phase walk in hallway HRL ´ RL with temporally extended actions Hierarchical Algos ´ Gating Mechanism Hierarchical Learning •Learning the gating function •Learning the individual behaviours •Learning both * g is a gate bi is a behaviour *Can be a multi- level hierarchy. Option : Movee until end of hallway Start : Any state in the hallway. Execute : policy as shown. Terminate : when s is end of hallway. Options [Sutton, Precup, Singh’99] An option is a well defined behaviour. o = h Io, o, o i Io : Set of states (IoµS) in which o can be initiated. o(s) : Policy (S!A*) when o is executing. o(s) : Probability that o terminates in s. *Can be a policy over lower level options. Learning An option is temporally extended action with well defined policy. Set of options (O) replaces the set of actions (A) Learning occurs outside options. Learning over options ´ Semi MDP Q- Learning. Machine: Movee + Collision Avoidance : End of hallway Call M1 Movee Choose Obstacle End of hallway Call M2 Return M1 Movew Moves Moves Return M2 Movew Moven Moven Return Hierarchies of Abstract Machines [Parr, Russell’97] A machine is a partial policy represented by a Finite State Automaton. Node : Execute a ground action. Call a machine as a subroutine. Choose the next node. Return to the calling machine. Hierarchies of Abstract Machines A machine is a partial policy represented by a Finite State Automaton. Node : Execute a ground action. Call a machine as subroutine. Choose the next node. Return to the calling machine. Learning Learning occurs within machines, as machines are only partially defined. Flatten all machines out and consider states [s,m] where s is a world state, and m, a machine node ´ MDP reduce(SoM) : Consider only states where machine node is a choice node ´ Semi-MDP. Learning ¼ Semi-MDP Q-Learning Task Hierarchy: MAXQ Decomposition [Dietterich’00] Root Children of a task are unordered Fetch Deliver Take Navigate(loc) Give Extend-arm Grab Release Extend-arm MovesMovewMovee Moven MAXQ Decomposition Augment the state s by adding the subtask i : [s,i]. Define C([s,i],j) as the reward received in i after j finishes. Q([s,Fetch],Navigate(prr)) = V([s,Navigate(prr)])+C([s,Fetch],Navigate(prr))* Reward V in Express receivedterms of Reward received C while navigating after navigation *Observe the Learn C, instead of learning Q context-free nature of Q-value The Outline of the Talk MDPs and Bellman’s curse of dimensionality. RL: Simultaneous learning and planning. Explore avenues to speedup RL. Illustrate prominent HRL methods. Compare prominent HRL methods. Discuss future research. Summarise 1. State Abstraction Abstract state : A state having fewer state variables; different world states maps to the same abstract state. If we can reduce some state variables, then we can reduce on the learning time considerably! We may use different abstract states for different macro-actions. State Abstraction in MAXQ Relevance : Only some variables are relevant for the task. Fetch : user-loc irrelevant Navigate(printer-room) : h-r-po,h-u-po,user-loc Fewer params for V of lower levels. Funnelling : Subtask maps many states to smaller set of states. Fetch : All states map to h-r-po=true, loc=pr.room. Fewer params for C of higher levels. State Abstraction in Options, HAM Options : Learning required only in states that are terminal states for some option. HAM : Original work has no abstraction. Extension: Three-way value decomposition*: Q([s,m],n) = V([s,n]) + C([s,m],n) + Cex([s,m]) Similar abstractions are employed. *[Andre,Russell’02] 2. Optimality Hierarchical Optimality vs. Recursive Optimality Optimality Options : Hierarchical Use (A [ O) : Global** Interrupt options HAM : Hierarchical* MAXQ : Recursive* Interrupt subtasks Use Pseudo-rewards * Can define eqns for both Iterate! optimalities **Adv. of using macro-actions maybe lost. 3. Language Expressiveness Option Can only input a complete policy HAM Can input a complete policy. Can input a task hierarchy. Can represent “amount of effort”. Later extended to partial programs. MAXQ Cannot input a policy (full/partial) 4. Knowledge Requirements Options Requires complete specification of policy. One could learn option policies – given subtasks. HAM Medium requirements MAXQ Minimal requirements 5. Models advanced Options : Concurrency HAM : Richer representation, Concurrency MAXQ : Continuous time, state, actions; Multi-agents, Average-reward. In general, more researchers have followed MAXQ Less input knowledge Value decomposition 6. Structure Paradigm S : Options, MAXQ A : All P : None R : MAXQ G : All V : MAXQ : All The Outline of the Talk MDPs and Bellman’s curse of dimensionality. RL: Simultaneous learning and planning. Explore avenues to speedup RL. Illustrate prominent HRL methods. Compare prominent HRL methods. Discuss future research. Summarise Directions for Future Research Bidirectional State Abstractions Hierarchies over other RL research Model based methods Function Approximators Probabilistic Planning Hierarchical P and Hierarchical R Imitation Learning Directions for Future Research Theory Bounds (goodness of hierarchy) Non-asymptotic analysis Automated Discovery Discovery of Hierarchies Discovery of State Abstraction Apply… Applications Toy Robot Flight Simulator AGV Scheduling Keepaway soccer P2 P1 D2 D1 Parts Ware- house Images courtesy Assemblies various sources D3 D4 P3 P4 Thinking Big… "... consider maze domains. Reinforcement learning researchers, including this author, have spent countless years of research solving a solved problem! Navigating in grid worlds, even with stochastic dynamics, has been far from rocket science since the advent of search techniques such as A*.” -- David Andre Use planners, theorem provers, etc. as components in big hierarchical solver. The Outline of the Talk MDPs and Bellman’s curse of dimensionality. RL: Simultaneous learning and planning. Explore avenues to speedup RL. Illustrate prominent HRL methods. Compare prominent HRL methods. Discuss future research. Summarise How to choose appropriate hierarchy Look at available domain knowledge If some behaviours are completely specified – options If some behaviours are partially specified – HAM If less domain knowledge available – MAXQ We can use all three to specify different behaviours in tandem. Main ideas in HRL community Hierarchies speedup learning Value function decomposition State Abstractions Greedy non-hierarchical execution Context-free learning and pseudo-rewards Policy improvement by re-estimation and re-learning.