Transcript UdeM-McGill Machine Learning Seminar

Predictive State Representation
Masoumeh Izadi
School of Computer Science
McGill University
UdeM-McGill Machine Learning Seminar
Outline
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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
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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
q = a1o1...akok
Prediction for a test given a history:
p(q|h)=P(o1...ok |h, a1...ak)
Sufficient statistics: predictions for a set of
core tests, Q
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 }:
S = set of states
A = set of actions
 = set of observations
T = transition probability distribution for each action
O= observation probability distribution for each action-observation
R = reward functions for each action
Sufficient statistics: belief state (probability
distribution over S)
Belief State
Posterior probability distribution over states
|S|=3
b
a
1
b’
o
1
1
0 b(s)  1 for all sS and sS b(s) = 1
b’(s’) = O(s’,a,o)T(s,a,s’) b(s)/Pr(o | a,b)
Construct PSR from POMDP
Outcome function u (t):
the predictions for test t from all POMDP states.
u ( )  en  (1,1,...1)T
u (aot)  (T O u (t ) )
a
a ,o
T T
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.
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
all possible tests
model.
t1 t2
tj
s1
s2
si
sn
u(tj)
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
0.3
O1, O2
O1, O4
0.2
O3
0.7
O4
O2
O3, O2
0.8
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
ti=a1o1…akok
p(ti) = prob(o1…ok|a1…ak)
t1
t2
p(t1) p(t2)
ti
p(ti)
Prediction of all possible future events can be
generated having any precise model of the
system.
System Dynamics Matrix
tj=a1o1…akok
hi=a’1o’1…a’no’n
h1 =ε
h2
t1 t2
tj
hi
P(tj|hi)
p(tj|hi) = prob ( on+1= o1,…, on+k= ok|a’1o’1…a’no’n , a1…ak)
Linear dimension of a dynamical system is determined
by the rank of the system dynamics matrix.
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
b1
b2
t1 t2
tj
bi
P(tj|bi)
Core Test Discovery
 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
Tests (T)
 Keep extending until the
rank doesn’t change
Zij= P(tj|hi)
Histories (H)
System Dynamics Matrix
All possible extension of tests and histories needs
processing a huge matrix in large domains.
h1 =ε
h2
t1 t2
tj
hi
P(tj|hi)
Core Test Discovery
t1 t2
 millions of samples
required for a few state
problem.
h1 One-step
h2
histories/
tests
Repeat one-step extensions to
Qi till the rank doesn’t change
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
P(Q|h)
P(Q|hao)
1
o
0 ≤ P(qi ) ≤ 1
for all i’s
|Q|=3
1
1
P(qi |hao) = p(Q|h)T maoqi /p(Q|h)Tmao
Forward Search
Compare alternative future experiences.
a1
o1
a2
o
o1
o
2
o1
o2
o1
2
o2
o1
o2
a1
Exponential Complexity
a2
DP for Finite-Horizon POMDPs
a1
o1
a3
o2
a2
o1
2
p1
a2
o1
o
a2
a3
o1
o2
a1
a1
a2
o2
a2
o2
a1
o1
a2
a3
o2
a1
p1
p2,
p3,
s1 ,
o
s2 ,
2
p3
a1
2
p2
a3
a3
a1
o1
o1
o1
o
o1
a3
a2
o2
a3
The value function for a set
of trees is always piecewise
linear and convex (PWLC)
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
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)
Theorem: value function for a finite horizon is still
piecewise-linear and convex.
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]
Summary
 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.
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(?)