Predictive State Representation
Masoumeh IzadiSchool of Computer Science
McGill University
UdeM-McGill Machine Learning Seminar
Outline
Predictive Representations PSR model specifications Learning PSR Using PSR in Control Problem Conclusion Future Directions
Motivation
In a dynamical system:
Knowing the exact state of the system is mostly an unrealistic assumption.
Real world tasks exhibit uncertainty POMDPs maintain belief b=(p(s0)….p(sn)) over
hidden variables si as the state. Beliefs are not verifiable! POMDPs are hard to learn and to solve.
Motivation
Potential alternatives: K-Markov Model not general!
Predictive Representations
Predictive Representations
State representation is in terms of experience. Status (state) is represented by predictions
made from it. Predictions represent cause and effect. Predictions are testable, maintainable, and
learnable.
No explicit notion of topological relationships.
Predictive State Representation
Test: a sequence of action-observation pairs
Prediction for a test given a history: Sufficient statistics: predictions for a set of
core tests, Q
q = a1o1...akok
p(q|h)=P(o1...ok |h, a1...ak)
Core Tests
A set of tests Q is a core tests set if its prediction forms a sufficient statistic for the dynamical system.
p(Q|h)=[p(q1|h) ...p(qn |h)]
For any test t: p(t|h)=f_t (p(Q|h))
Linear PSR Model
For any test q, there exists a projection vector mq s.t:
p(q|h) = p(Q|h)T mq
Given a new action-observation pair, ao, the prediction vector for each qi є Q is updated by:
P(qi |hao) = p(aoqi|h) / p(ao|h) = p(Q|h)T maoqi/ p(Q|h)Tmao
PSR Model Parameters
The set of core tests: Q={q1….qn}
Projection vectors for one step tests :mao (for all ao pairs )
Projection vectors for one step extension of core tests maoqi (for all ao pairs )
Linear PSR vs. POMDP
A linear PSR representation can be more compact than the POMDP representation.
A POMDP with n nominal states can represent a dynamical system of
dimensions ≤ n
POMDP Model
The model is an n-tuple { S, A, , T, O, R }:
Sufficient statistics: belief state (probability distribution over S)
S = set of states A = set of actions = set of observationsT = transition probability distribution for each actionO= observation probability distribution for each action-observationR = reward functions for each action
Belief State
Posterior probability distribution over states
b b’a
o1
|S|=3
1
1
b’(s’) = O(s’,a,o)T(s,a,s’) b(s)/Pr(o | a,b)
0 b(s) 1 for all sS and sS b(s) = 1
Construct PSR from POMDP
Outcome function u (t):the predictions for test t from all POMDP states.
Definition: A test t is said to be independent of a set of tests T if its outcome vector is linearly independent of the predictions for tests in T.
TToaa
Tn
tuOTaotu
eu
))(()(
)1,...1,1()(,
State Prediction Matrix
Rank of the matrix determines the size of Q.
Core tests corresponds to linearly independent columns.
Entries are computed using the POMDP model.
u(tj)
t1 t2 all possible teststj
s2
s1
si
sn
Linearly Independent States
Definition: A linearly dependent state of an MDP is a state for which any action transition function is a linear combination of the transition functions from other states.
Having the same dynamical structure is a special case of linear dependency.
Example
0.2
0.8
0.7
0.3O1, O2
O3, O2
O1, O4
O3
O4
O2
Linear PSR needs only two tests to represent the system
e.g.: ao1, ao4 can predict any other tests
State Space Compression
Theorem For any controlled dynamical system :
linearly dependent states in the underlying MDP
more compact PSR than the corresponding POMDP.Reverse direction is not always the case due to possible structure in the observations
Exploiting Structure
PSR exploits linear independence structure in the dynamics of a system.
PSR also exploits regularities in dynamics.
Lossless compression needs invariance of state representation in terms of values as well as dynamics.
Including reward as part of observation makes linear PSR similar to linear lossless compressions for POMDPs.
POMDP Example
States: 20 (directions , grid state)Actions: 3(turn left, turn right, move);Observations: 2 (wall, nothing);
Structure Captured by PSR
Alias states (by immediate observation)
Predictive classes (by PSR core tests)
Generalization
• Good generalization results when similar situations have similar representations.• A good generalization makes it possible to learn with small amount of experience.
• Predictive representation: generalizes the state space well. makes the problem simpler and yet precise. assists reinforcement learning algorithms. [Rafols et al 2005]
Learning the PSR Model
The set of core tests: Q={q1….q|Q|}
Projection vectors for one step tests :mao (for all ao pairs )
Projection vectors for one step extension of core tests maoqi (for all ao pairs )
System Dynamics Vector
Prediction of all possible future events can be generated having any precise model of the system.
t1 t2
p(t1) p(t2) p(ti)ti
ti=a1o1…akok
p(ti) = prob(o1…ok|a1…ak)
System Dynamics Matrix
Linear dimension of a dynamical system is determined by the rank of the system dynamics matrix.
P(tj|hi)
t1 t2 tjh1 =ε
hi
h2
tj=a1o1…akok
hi=a’1o’1…a’no’n
p(tj|hi) = prob ( on+1= o1,…, on+k= ok|a’1o’1…a’no’n , a1…ak)
POMDP in System Dynamics Matrix
Any model must be able to generate System Dynamic Matrix.Core beliefs B = {b1 b2 … qN} :
Span the reachable subspace of continuous belief space; Can be beneficial in POMDP solution methods [Izadi et al 2005]
Represent reduced state space dimensions in structured domains
P(tj|bi)
t1 t2 tjb1
bi
b2
Core Test Discovery
Zij= P(tj|hi)
Extend tests and histories one-step and estimate entries of Z (counting data samples).
Find the rank and keep the linearly independent tests and histories
Keep extending until the rank doesn’t change
Tests (T)
Histories (H)
System Dynamics Matrix
P(tj|hi)
t1 t2 tjh1 =ε
hi
h2
All possible extension of tests and histories needs processing a huge matrix in large domains.
Core Test Discovery
t1 t2 h1 h2
One-step histories/ tests
Repeat one-step extensions to Qi till the rank doesn’t change
millions of samples required for a few state problem.
PSR Learning
Structure Learning:
which tests to choose for Q from data
Parameter Learning:
how to tune m-vectors given the structure and experience data
Learning Parameters
PSR Gradient algorithm [Singh et al. 2003]
Principle-Component based algorithm for TPSR (uncontrolled system) [Rosencrantz et al. 20004]
Suffix-History Algorithm [ James et al.2004]
POMDP EM
Results on PSR Model Learning
Planning
States expressed in predictive form.
Planning and reasoning should be in terms of experience.
Rewards treated as part of observations.
Tests are of the form: t=a1(o1r1)….an(onrn).
General POMDP methods (e.g. dynamic programming) can be used.
Predictive Space
1
1
1|Q|=3
P(Q|h) P(Q|hao)
o
P(qi |hao) = p(Q|h)T maoqi /p(Q|h)Tmao
0 ≤ P(qi ) ≤ 1 for all i’s
Forward Search
a2
o1
o1 o2 a1 a2
o2o1
o1 o2 o1 o2
o2
a1
Exponential Complexity
Compare alternative future experiences.
DP for Finite-Horizon POMDPs
The value function for a set of trees is always piecewise linear and convex (PWLC)
p1
p2,
s1, s2,
a2
a3 a3
a3 a1 a2 a1
o1
o1 o2 o1 o2
o2
a1
a2 a3
a3 a2 a1 a1
o1
o1 o2 o1 o2
o2
a1
a1 a2
a2 a2 a3 a3
o1
o1 o2 o1 o2
o2
p1 p2
p3
p3,
Value Iteration in POMDPs
Value iteration: Initialize value function
V(b) = max_a Σ_s R(s,a) b(s) This produces 1 alpha-vector per action.
Compute the value function at the next iteration using Bellman’s equation:
V(b)= max_a [Σ_s R(s,a)b(s)+Σ_s’[T(s,a,s’)O(s’,a,z)α(s’)]]
DP for Finite-Horizon PSRs
Theorem: value function for a finite horizon is still piecewise-linear and convex.
There’s a scaler reward for each test. R(ht,a)= Σ_r prob (r |ht , a)
Value of a policy tree is a linear function of prediction vector.
Vp(p(Q|h)=PT(Q|h)( n_a + Σ_o Mao w)
Value Iteration in PSRs
Value iteration just as in POMDPs V(p(Q|h)) = max _α [Vα(p(Q|h))]
Represent any finite-horizon solution by a finite set of alpha-vectors (policy trees).
Results on PSR Control
James etal.2004
Results on PSR Control
• Current PSR planning algorithms are not advantageous to POMDP planning ([Izadi & Precup 2003], [James et al. 2004]).
• Planning Requires precise definition of predictive space.
• It is important to analyze the impact of PSR planning on structured domains.
Predictive Representations
Linear PSR EPSR action sequence +last observation
[Rudary and Singh 2004]
mPSR augmented with history [James et al 2005]
TD Networks temporal difference learning with network of interrelated predictions [Tanner
and Sutton 2004]
A good state representation should be: compact useful for planning efficiently learnable
Predictive state representation provide a lossless compression which reflects the underlying structure.
PSR generalizes the space and facilitate planning.
Summary
Limitations
Learning and Discovery in PSRs still lack efficient algorithms.
Current algorithms need way too data samples.
Experiments on many ideas can only be done on toy problems so far due to model learning limitation.
Future Work
Theory of PSR and possible extensions
Efficient algorithms for learning predictive models
More on combining temporal abstraction with PSR
More on planning algorithms for PSR and EPSR
Approximation methods are yet to be developed
PSR for continuous systems
Generalization across states in stochastic systems
Non linear PSRs and exponential compression(?)