Factored Approches for MDP & RL (Some Slides taken from Alan Ferns course) Factored MDP/RL Representations Advantages States made of features Specification: is far easier Inference: Novel lifted versions of the Value and Policy iterations possible Boolean vs. Continuous Actions modify the
features (probabilistically) Representations include Probabilistic STRIPS, 2-Timeslice Dynamic Bayes Nets etc. Reward and Value functions Representations include ADDs, linear weighted sums of features etc. Bellman backup directly in terms of ADDs Policy gradient approach where you do direct search in the policy space Learning: Generalization possibilities Q-learning etc. will now directly update the factored representations
(e.g. weights of the features) Thus giving implicit generalization Approaches such as FF-HOP can recognize and reuse common substructure Problems with transition systems Transition systems are a great conceptual tool to understand the differences between the various planning problems However direct manipulation of transition systems tends to be too cumbersome The size of the explicit graph corresponding to a transition system is often very large The remedy is to provide compact representations for transition systems
Start by explicating the structure of the states e.g. states specified in terms of state variables Represent actions not as incidence matrices but rather functions specified directly in terms of the state variables An action will work in any state where some state variables have certain values. When it works, it will change the values of certain (other) state variables State Variable Models World is made up of states which are defined in terms of state variables Can be boolean (or multi-ary or continuous) States are complete assignments over state variables So, k boolean state variables can represent how many states? Actions change the values of the state variables
Applicability conditions of actions are also specified in terms of partial assignments over state variables Blocks world Init: Ontable(A),Ontable(B), Clear(A), Clear(B), hand-empty Goal: State variables: Ontable(x) On(x,y) Clear(x) hand-empty holding(x) ~clear(B), hand-empty STRIPS ASSUMPTION: If an action changes Goal state: a state variable,
A partial specification of the desired state variable/value combinations this must be --desired values can be both positive and negative explicitly mentioned in its kup(x) Putdown(x) effects ec: hand-empty,clear(x),ontable(x) Prec: holding(x) ff: holding(x),~ontable(x),~hand-empty,~Clear(x) eff: Ontable(x), hand-empty,clear(x),~holdin Initial state: Complete specification of T/F values to state variables --By convention, variables with F values are omitted Unstack(x,y) Stack(x,y) Prec: on(x,y),hand-empty,cl(x) Prec: holding(x), clear(y)
eff: holding(x),~clear(x),clear(y),~hand-emp eff: on(x,y), ~cl(y), ~holding(x), hand-empty All the actions here have only positive preconditions; but this is not necessary Why is STRIPS representation compact? (than explicit transition systems) In explicit transition systems actions are represented as state-to-state transitions where in each action will be represented by an incidence matrix of size |S|x|S| In state-variable model, actions are represented only in terms of state variables whose values they care about, and whose value they affect. Consider a state space of 1024 states. It can be represented by log21024=10 state variables. If an action needs variable v1 to be true and makes v7 to
be false, it can be represented by just 2 bits (instead of a 1024x1024 matrix) Of course, if the action has a complicated mapping from states to states, in the worst case the action rep will be just as large The assumption being made here is that the actions will have effects on a small number of state variables. First order Sit. Calc Rel/ Prop STRIPS rep Atomic
Transition rep Factored Representations fo MDPs: 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 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 Ideas for Efficient Algorithms.. Use search Useheuristic factored representations (and reachability Factored representations for Actions, Reward Functions, information) Values and Policies LAO*, RTDPmanipulating factored representations during Directly Bellman update Usethe
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. Probabilistic Planning --The competition (IPPC) --The Action language.. PPDDL was based on PSO A new standard RDDL is based on 2-TBN Not
ergodic Reducing Heuristic Computation Cost by exploiting factored representations The heuristics computed for a state might give us an idea about the heuristic value of other similar states Similarity is possible to determine in terms of the state structure Exploit overlapping structure of heuristics for different states E.g. SAG idea for McLUG E.g. Triangle tables idea for plans (c.f. Kolobov) A Plan is a Terrible Thing to Waste Suppose we have a plan s0a0s1a1s2a2s3ansG We realized that this tells us not just the estimated value of s0, but also of s1,s2sn
So we dont need to compute the heuristic for them again Is that all? If we have states and actions in factored representation, then we can explain exactly what aspects of si are relevant for the plans success. The explanation is a proof of correctness of the plan Can be based on regression (if the plan is a sequence) or causal proof (if the plan is a partially ordered one. The explanation will typically be just a subset of the literals making up the state That means actually, the plan suffix from si may actually be relevant in many more states that are consistent with that explanation Triangle Table Memoization Use triangle tables / memoization C B A
A B C If the above problem is solved, then we dont need to call FF again for the below: B A A B Explanation-based Generalization (of Successes and Failures) Suppose we have a plan P that solves a problem [S, G].
We can first find out what aspects of S does this plan actually depend on Explain (prove) the correctness of the plan, and see which parts of S actually contribute to this proof Now you can memoize this plan for just that subset of S Relaxations for Stochastic Planning Determinizations can also be used as a basis for heuristics to initialize the V for value iteration [mGPT; GOTH etc] Heuristics come from relaxation We can relax along two separate dimensions: Relax ve interactions Consider +ve interactions alone using relaxed planning graphs Relax uncertainty Consider determinizations
Or a combination of both! --Factored TD and Q-learning --Policy search (has to be factored..) Large State Spaces When a problem has a large state space we can not longer represent the V or Q functions as explicit tables Even if we had enough memory Never enough training data! Learning takes too long What to do?? 32 [Slides from Alan Fern] Function Approximation
Never enough training data! Must generalize what is learned from one situation to other similar new situations Idea: Instead of using large table to represent V or Q, use a parameterized function The number of parameters should be small compared to number of states (generally exponentially fewer parameters) Learn parameters from experience When we update the parameters based on observations in one state, then our V or Q estimate will also change for other similar states I.e. the parameterization facilitates generalization of experience 33 Linear Function Approximation Define a set of state features f1(s), , fn(s)
The features are used as our representation of states States with similar feature values will be considered to be similar A common approximation is to represent V(s) as a weighted sum of the features (i.e. a linear approximation) V ( s ) 0 1 f1 ( s ) 2 f 2 ( s ) ... n f n ( s ) The approximation accuracy is fundamentally limited by the information provided by the features Can we always define features that allow for a perfect linear approximation? Yes. Assign each state an indicator feature. (I.e. ith feature is 1 iff ith state is present and i represents value of ith state) Of course this requires far to many features and gives no generalization. 34 Example Consider grid problem with no obstacles, deterministic actions
U/D/L/R (49 states) Features for state s=(x,y): f1(s)=x, f2(s)=y (just 2 features) V(s) = 0 + 1 x + 2 y 6 Is there a good linear approximation? Yes. 0 =10, 1 = -1, 2 = -1 (note upper right is origin) 0 10 0 V(s) = 10 - x - y subtracts Manhattan dist. from goal reward 6 35
But What If We Change Reward V(s) = 0 + 1 x + 2 y Is there a good linear approximation? No. 0 0 10 36 But What If We Change Reward V(s) = 0 + 1 x + 2 y Is there a good linear approximation? No. 0 0
10 37 But What If V(s) = 0 + 1 x + 2 y + 3 z h Include new feature z 3 0 0 5 z= |3-x| + |3-y| 5 z is dist. to goal location h Does this allow a good linear approx?
10 3 5 0 =10, 1 = 2 = 0, 0 = -1 Feature Engineering. 38 Linear Function Approximation Define a set of features f1(s), , fn(s) The features are used as our representation of states States with similar feature values will be treated similarly More complex functions require more complex features V ( s ) 0 1 f1 ( s) 2 f 2 ( s) ... n f n ( s )
Our goal is to learn good parameter values (i.e. feature weights) that approximate the value function well How can we do this? Use TD-based RL and somehow update parameters based on each experience. 41 TD-based RL for Linear Approximators 1. Start with initial parameter values 2. Take action according to an explore/exploit policy (should converge to greedy policy, i.e. GLIE) 3. Update estimated model 4. Perform TD update for each parameter i ? 5. Goto 2 What is a TD update for a parameter? 42
Aside: Gradient Descent Given a function f(1,, n) of n real values =(1,, n) suppose we want to minimize f with respect to A common approach to doing this is gradient descent The gradient of f at point , denoted by f(), is an n-dimensional vector that points in the direction where f increases most steeply at point Vector calculus tells us that f() is just a vector of partial derivatives f ( ) f ( ) f ( ) , , 1
n where f ( ) i lim 0 This will be used Again with Graphical Model Learning f (1 , i 1 , i , i 1 , , n ) f ( ) can decrease f by moving in negative gradient direction 43
Aside: Gradient Descent for Squared Error Suppose that we have a sequence of states and target values for s1 , v( s1 ) , s2 , v( s2 ) , each state E.g. produced by the TD-based RL loop Our goal is to minimize the sum of squared errors between our estimated function and each target value: 1 E j V ( s j ) v ( s j ) 2 squared error of example j our estimated value for jth state
2 target value for jth state After seeing jth state the gradient descent rule tells us that we can decrease error by updating parameters by: i i learning rate E j i , E j V ( s j ) i V ( s j ) i
E j 44 Aside: continued i i E j i i V ( s j ) v( s j ) V ( s j ) E j
V ( s j ) i depends on form of approximator For a linear approximation function: V ( s) 1 1 f1 ( s) 2 f 2 ( s) ... n f n ( s) Thus the update becomes: V ( s j ) i fi (s j )
i i v( s j ) V ( s j ) f i ( s j ) For linear functions this update is guaranteed to converge to best approximation for suitable learning rate schedule 45 TD-based RL for Linear Approximators 1. 2. 3. 4. Start with initial parameter values Take action according to an explore/exploit policy (should converge to greedy policy, i.e. GLIE) Transition from s to s Update estimated model Perform TD update for each parameter
i i v( s ) V ( s ) f i ( s ) 5. Goto 2 What should we use for target value v(s)? Use the TD prediction based on the next state s Note that we are generalizing w.r.t. possibly faulty data.. (the neighbors value may not be correct yet..) this is the same as previous TD method only with approximation
v ( s ) R ( s ) V ( s ' )
46 TD-based RL for Linear Approximators 1. 2. 3. 4. Start with initial parameter values Take action according to an explore/exploit policy (should converge to greedy policy, i.e. GLIE) Update estimated model Perform TD update for each parameter i i R( s) V ( s' ) V ( s) f i ( s) 5.
Goto 2 Step 2 requires a model to select greedy action For applications such as Backgammon it is easy to get a simulation-based model For others it is difficult to get a good model But we can do the same thing for model-free Q-learning 47 Q-learning with Linear Approximators Q ( s, a ) 0 1 f1 ( s, a ) 2 f 2 ( s, a ) ... n f n ( s, a ) Features are a function of states and actions. 1. 2. 3. 4. Start with initial parameter values
Take action a according to an explore/exploit policy (should converge to greedy policy, i.e. GLIE) transitioning from s to s Perform TD update for each parameter i i R( s ) max Q ( s ' , a' ) Q ( s, a) f i ( s, a) Goto 2 a' For both Q and V, these algorithms converge to the closest linear approximation to optimal Q or V. 48 Policy Gradient Ascent Let () be the expected value of policy .
() is just the expected discounted total reward for a trajectory of . For simplicity assume each trajectory starts at a single initial state. Our objective is to find a that maximizes () Policy gradient ascent tells us to iteratively update parameters via: ( ) Problem: () is generally very complex and it is rare that we can compute a closed form for the gradient of (). We will instead estimate the gradient based on experience 56 Gradient Estimation Concern: Computing or estimating the gradient of discontinuous functions can be problematic. For our example parametric policy ( s ) arg max Q ( s, a) a
is () continuous? No. There are values of where arbitrarily small changes, cause the policy to change. Since different policies can have different values this means that changing can cause discontinuous jump of (). 57 Example: Discontinous () ( s ) arg max Q ( s, a) 1 f1 ( s, a) a Consider a problem with initial state s and two actions a1 and a2 a1 leads to a very large terminal reward R1 a2 leads to a very small terminal reward R2 Fixing 2 to a constant we can plot the ranking assigned to each action by Q and the corresponding value ()
Discontinuity in () when ordering of a1 and a2 change Q ( s, a1) R1 () Q ( s, a 2) 1 R2 1 58 Probabilistic Policies We would like to avoid policies that drastically change with small parameter changes, leading to discontinuities A probabilistic policy takes a state as input and returns a
Aka Mixed Policy distribution over actions (not needed for Given a state s (s,a) returns the probability that selects action a in s Optimality) Note that () is still well defined for probabilistic policies Now uncertainty of trajectories comes from environment and policy Importantly if (s,a) is continuous relative to changing then () is also continuous relative to changing A common form for probabilistic policies is the softmax function or Boltzmann exploration function exp Q ( s , a )
( s , a ) Pr( a | s ) (s, a ' ) exp Q a ' A 59 Empirical Gradient Estimation Our first approach to estimating () is to simply compute empirical gradient estimates ( ) ( ) Recall that = (1,, n) and ( )
, , 1 n so we can compute the gradient by empirically estimating each partial derivative (1 , i 1 , i , i 1 , , n ) ( ) ( ) lim 0 i So for small we can estimate the partial derivatives by (1 , i 1 , i , i 1 , , n ) ( )
This requires estimating n+1 values: ( ), (1 , i 1 , i , i 1 , , n ) | i 1,..., n 60 Empirical Gradient Estimation How do we estimate the quantities ( ), (1 , i 1 , i , i 1 , , n ) | i 1,..., n For each set of parameters, simply execute the policy for N trials/episodes and average the values achieved across the Doable without permanent damage if there is a simulator trials
This requires a total of N(n+1) episodes to get gradient estimate For stochastic environments and policies the value of N must be relatively large to get good estimates of the true value Often we want to use a relatively large number of parameters Often it is expensive to run episodes of the policy So while this can work well in many situations, it is often not a practical approach computationally Better approaches try to use the fact that the stochastic policy is differentiable. Can get the gradient by just running the current policy multiple times 61 Applications of Policy Gradient Search Policy gradient techniques have been used to create controllers for difficult helicopter maneuvers For example, inverted helicopter flight.
A planner called FPG also won the 2006 International Planning Competition If you dont count FF-Replan 62 Policy Gradient Recap When policies have much simpler representations than the corresponding value functions, direct search in policy space can be a good idea Allows us to design complex parametric controllers and optimize details of parameter settings For baseline algorithm the gradient estimates are unbiased (i.e. they will converge to the right value) but have high variance Can require a large N to get reliable estimates h OLPOMDP offers can trade-off bias and variance via the discount parameter [Baxter & Bartlett, 2000]
Can be prone to finding local maxima Many ways of dealing with this, e.g. random restarts. 64 Gradient Estimation: Single Step Problems For stochastic policies it is possible to estimate () directly from trajectories of just the current policy Idea: take advantage of the fact that we know the functional form of the policy First consider the simplified case where all trials have length 1 For simplicity assume each trajectory starts at a single initial state and reward only depends on action choice () is just the expected reward of action selected by . ( ) ( so , a) R(a ) a
where s0 is the initial state and R(a) is reward of action a The gradient of this becomes ( ) ( so , a) R(a) ( so , a ) R(a) a a How can we estimate this by just observing the execution of ? 65 Gradient Estimation: Single Step Problems Rewriting ( ) ( so , a ) R (a ) a ( so
( so , a ) , a) R(a) ( so , a ) a ( so , a ) log ( so , a ) R (a ) a can get closed form g(s0,a) The gradient is just the expected value of g(s0,a)R(a) over execution trials of Can estimate by executing for N trials and averaging samples 1 ( ) N
N g (s , a )R(a ) o j j j 1 aj is action selected by policy on jth episode Only requires executing for a number of trials that need not depend on the number of parameters 66 Gradient Estimation: General Case So for the case of a length 1 trajectories we got:
1 ( ) N N g ( s , a )R(a) o j j 1 For the general case where trajectories have length greater than one and reward depends on state we can do some work and get: length of trajectory j 1
( ) N # of trajectories of current policy N Tj g (s j ,t , a j , t ) R j ( s j ,t ) j 1 t 1 Observed total reward in trajectory j from step t to end
sjt is tth state of jth episode, ajt is tth action of epidode j The derivation of this is straightforward but messy. 67 How to interpret gradient expression? Direction to move parameters in order to increase the probability that policy selects ajt in state sjt g ( s, a) log ( s, a) 1 ( ) N N Tj g (s
j ,t , a j , t ) R j ( s j ,t ) j 1 t 1 Total reward observed after taking ajt in state sjt So the overall gradient is a reward weighted combination of individual gradient directions For large Rj(sj,t) will increase probability of aj,t in sj,t For negative Rj(sj,t) will decrease probability of aj,t in sj,t Intuitively this increases probability of taking actions that typically are followed by good reward sequences 68 Basic Policy Gradient Algorithm Repeat until stopping condition
1. Execute for N trajectories while storing the state, action, reward sequences 1 N N Tj g (s j ,t , a j , t ) R j ( s j ,t ) j 1 t 1
One disadvantage of this approach is the small number of updates per amount of experience Also requires a notion of trajectory rather than an infinite sequence of experience Online policy gradient algorithms perform updates after each step in environment (often learn faster) 69 Computing the Gradient of Policy Both algorithms require computation of g ( s, a) log ( s, a) For the Boltzmann distribution with linear approximation we have:
where exp Q ( s , a ) (s, a ) exp Q ( s , a ' ) a ' A Q ( s, a ) 0 1 f1 ( s, a ) 2 f 2 ( s, a ) ... n f n ( s, a ) Here the partial derivatives needed for g(s,a) are: log ( s, a)
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