Pieter Abbeel and Andrew Y. Ng Apprenticeship Learning via Inverse Reinforcement Learning Pieter...
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Transcript of Pieter Abbeel and Andrew Y. Ng Apprenticeship Learning via Inverse Reinforcement Learning Pieter...
Pieter Abbeel and Andrew Y. Ng
Apprenticeship Learning via Inverse Reinforcement Learning
Pieter Abbeel
Stanford University
[Joint work with Andrew Ng.]
Pieter Abbeel and Andrew Y. Ng
Overview
• Reinforcement Learning (RL)
• Motivation for Apprenticeship Learning
• Proposed algorithm
• Theoretical results
• Experimental results
• Conclusion
Pieter Abbeel and Andrew Y. Ng
Example of Reinforcement Learning Problem
Highway driving.
Pieter Abbeel and Andrew Y. Ng
RL formalism
• Assume that at each time step, our system is in some state st.
• Upon taking an action at, our system randomly transitions to some new state st+1.
• We are also given a reward function R.
• The goal: Pick actions over time so as to maximize the expected sum of rewards E[R(s0) + R(s1) + … + R(sT)].
Systemdynamics
s0
s1
Systemdynamics
…System
dynamicssT-1
sT
s2
R(s0) R(s2) R(sT-1)R(s1) R(sT)+ ++…++
Pieter Abbeel and Andrew Y. Ng
RL formalism
• Markov Decision Process (S,A,P,s0,R)
• W.l.o.g. we assume
• Policy
• Utility of a policy for reward R=wT
Pieter Abbeel and Andrew Y. Ng
Motivation for Apprenticeship Learning
Reinforcement learning (RL) gives powerful tools for solving MDPs. It can be difficult to specify the reward function. Example: Highway driving.
Pieter Abbeel and Andrew Y. Ng
Apprenticeship Learning
• Learning from observing an expert.
• Previous work:
– Learn to predict expert’s actions as a function of states.
– Usually lacks strong performance guarantees.
– (E.g.,. Pomerleau, 1989; Sammut et al., 1992; Kuniyoshi et al., 1994; Demiris & Hayes, 1994; Amit & Mataric, 2002; Atkeson & Schaal, 1997; …)
• Our approach:
– Based on inverse reinforcement learning (Ng & Russell, 2000).
– Returns policy with performance as good as the expert as measured according to the expert’s unknown reward function.
Pieter Abbeel and Andrew Y. Ng
Algorithm
For i = 1,2,…
Inverse RL step:
Estimate expert’s reward function R(s)= wT(s) such that under R(s) the expert performs better than all previously found policies {j}.
RL step:
Compute optimal policy i for
the estimated reward w.
Pieter Abbeel and Andrew Y. Ng
Algorithm: Inverse RL step
Pieter Abbeel and Andrew Y. Ng
Algorithm: Inverse RL step
Quadratric programming problem. (same as for SVM)
Pieter Abbeel and Andrew Y. Ng
Algorithm
1
(0)
w(1)
w(2)(1)
(2)
2
w(3)
(E)
Pieter Abbeel and Andrew Y. Ng
Feature Expectation Closeness and Performance
If we can find a policy such that
||(E) - ()||2 ,
then for any underlying reward R*(s) =w*T(s),
we have that
|Uw*(E) - Uw*()| = |w*T (E) - w*T ()|
||w*||2 ||(E) - ()||2
.
Pieter Abbeel and Andrew Y. Ng
Theoretical Results: Convergence
Theorem. Let an MDP (without reward function), a k-dimensional feature vector and the expert’s feature expectations (E) be given. Then after at most
k T2/2
iterations, the algorithm outputs a policy that performs nearly as well as the expert, as evaluated on the unknown reward function R*(s)=w*T(s), i.e.,
Uw*() Uw*(E) - .
Pieter Abbeel and Andrew Y. Ng
Theoretical Results: Sampling
In practice, we have to use sampling to estimate the feature expectations of the expert. We still have -optimal performance with high probability if the number of observed samples is at least
O(poly(k,1/)).
Note: the bound has no dependence on the “complexity” of the policy.
Pieter Abbeel and Andrew Y. Ng
Gridworld Experiments
Reward function is piecewise constant over small regions.Features for IRL are these small regions.
128x128 grid, small regions of size 16x16.
Pieter Abbeel and Andrew Y. Ng
Gridworld Experiments
Pieter Abbeel and Andrew Y. Ng
Gridworld Experiments
Pieter Abbeel and Andrew Y. Ng
Gridworld Experiments
Pieter Abbeel and Andrew Y. Ng
Gridworld Experiments
Pieter Abbeel and Andrew Y. Ng
Case study: Highway driving
The only input to the learning algorithm was the driving demonstration (left panel). No reward function was provided.
Input: Driving demonstration Output: Learned behavior
Pieter Abbeel and Andrew Y. Ng
More driving examples
In each video, the left sub-panel shows a demonstration of a different driving “style”, and the right sub-panel shows the behavior learned from watching the demonstration.
Pieter Abbeel and Andrew Y. Ng
Car driving results
CollisionLeft Shoulder
Left Lane
Middle Lane
Right Lane
Right Shoulder
(expert) 0 0 0.13 0.20 0.60 0.07
1 (learned) 0 0 0.09 0.23 0.60 0.08
w (learned) -0.08 -0.04 0.01 0.01 0.03 -0.01
(expert) 0.12 0 0.06 0.47 0.47 0
2 (learned) 0.13 0 0.10 0.32 0.58 0
w (learned) 0.23 -0.11 0.01 0.05 0.06 -0.01
(expert) 0 0 0 0.01 0.70 0.29
3 (learned) 0 0 0 0 0.74 0.26
w (learned) -0.11 -0.01 -0.06 -0.04 0.09 0.01
Pieter Abbeel and Andrew Y. Ng
Different Formulation
LP formulation for RL problem
max. s,a (s,a) R(s)
s.t.
s a (s,a) = s’,a P(s|s’,a) (s’,a)
QP formulation for Apprenticeship Learning
min. , i (E,i - i)2
s.t.
s a (s,a) = s’,a P(s|s’,a) (s’,a)
i i = s,a i(s) (s,a)
Pieter Abbeel and Andrew Y. Ng
Different Formulation (ctd.)
Our algorithm is equivalent to iteratively
linearizing QP at current point (Inverse RL step),
solve resulting LP (RL step).
Why not solving QP directly? Typically only possible for very small toy problems (curse of dimensionality). [Our algorithm makes use of existing RL solvers to deal with the curse of dimensionality.]
Pieter Abbeel and Andrew Y. Ng
Our algorithm returns a policy with performance as good as the expert as evaluated according to the expert’s unknown reward function.
Algorithm is guaranteed to converge in poly(k,1/) iterations.
Sample complexity poly(k,1/).
The algorithm exploits reward “simplicity” (vs. policy “simplicity” in previous approaches).
Conclusions
Pieter Abbeel and Andrew Y. Ng
Proof (sketch)
1(0)
w(1)
(1)
2
(1)
(E)
d0 d1
Pieter Abbeel and Andrew Y. Ng
Proof (sketch)
Pieter Abbeel and Andrew Y. Ng
More driving examples
In each video, the left sub-panel shows a demonstration of a different driving “style”, and the right sub-panel shows the behavior learned from watching the demonstration.
Pieter Abbeel and Andrew Y. Ng
Additional slides for poster
(slides to come are additional material, not included in the talk, in particular: projection (vs. QP) version of the Inverse RL step; another formulation of the apprenticeship learning problem, and its relation to our algorithm)
Pieter Abbeel and Andrew Y. Ng
Simplification of Inverse RL step: QP Euclidean projection
• In the Inverse RL step
– set (i-1) = orthogonal projection of E onto line through { (i-1),((i-1)) }
– set w(i) = E - (i-1)
• Note: the theoretical results on convergence and sample complexity hold unchanged for the simpler algorithm.
Pieter Abbeel and Andrew Y. Ng
Algorithm (projection version)
1
E
(0)
w(1)
(1)
2
Pieter Abbeel and Andrew Y. Ng
Algorithm (projection version)
1
E
(0)
w(1)
w(2)(1)
(2)
2
(1)
Pieter Abbeel and Andrew Y. Ng
Algorithm (projection version)
1
E
(0)
w(1)
w(2)(1)
(2)
2
w(3)
(1)
(2)
Pieter Abbeel and Andrew Y. Ng
Appendix: Different View
Bellman LP for solving MDPs
Min. V c’V s.t.
s,a V(s) R(s,a) + s’ P(s,a,s’)V(s’)
Dual LP
Max. s,a (s,a)R(s,a) s.t.
s c(s) - a (s,a) + s’,a P(s’,a,s) (s’,a) =0
Apprenticeship Learning as QP
Min. i (E,i - s,a (s,a)i(s))2 s.t.
s c(s) - a (s,a) + s’,a P(s’,a,s) (s’,a) =0
Pieter Abbeel and Andrew Y. Ng
Different View (ctd.)
Our algorithm is equivalent to iteratively
linearize QP at current point (Inverse RL step),
solve resulting LP (RL step).
Why not solving QP directly? Typically only possible for very small toy problems (curse of dimensionality). [Our algorithm makes use of existing RL solvers to deal with the curse of dimensionality.]
Pieter Abbeel and Andrew Y. Ng
Slides that are different for poster
(slides to come are slightly different for poster, but already “appeared” earlier)
Pieter Abbeel and Andrew Y. Ng
Algorithm (QP version)
1
(0)
w(1)
(1)
2
Uw() = wT()
(E)
Pieter Abbeel and Andrew Y. Ng
Algorithm (QP version)
1
(0)
w(1)
w(2)(1)
(2)
2
Uw() = wT()
(E)
Pieter Abbeel and Andrew Y. Ng
Algorithm (QP version)
1
(0)
w(1)
w(2)(1)
(2)
2
w(3)
Uw() = wT()
(E)
Pieter Abbeel and Andrew Y. Ng
Gridworld Experiments
Pieter Abbeel and Andrew Y. Ng
Case study: Highway driving
(Videos available.)
Input: Driving demonstration Output: Learned behavior
Pieter Abbeel and Andrew Y. Ng
More driving examples
(Videos available.)
Collision
Offroad Left
Left Lane
Middle
Lane Right
Lane Offroad
Right
1 Feature Distr. Expert 0 0 0.1325 0.2033 0.5983 0.0658
Feature Distr. Learned 5.00E-05 0.0004 0.0904 0.2286 0.604 0.0764
Weights Learned -0.0767 -0.0439 0.0077 0.0078 0.0318 -0.0035
2 Feature Distr. Expert 0.1167 0 0.0633 0.4667 0.47 0
Feature Distr. Learned 0.1332 0 0.1045 0.3196 0.5759 0
Weights Learned 0.234 -0.1098 0.0092 0.0487 0.0576 -0.0056
3 Feature Distr. Expert 0 0 0 0.0033 0.7058 0.2908
Feature Distr. Learned 0 0 0 0 0.7447 0.2554
Weights Learned -0.1056 -0.0051 -0.0573 -0.0386 0.0929 0.0081
4 Feature Distr. Expert 0.06 0 0 0.0033 0.2908 0.7058
Feature Distr. Learned 0.0569 0 0 0 0.2666 0.7334
Weights Learned 0.1079 -0.0001 -0.0487 -0.0666 0.059 0.0564
5 Feature Distr. Expert 0.06 0 0 1 0 0
Feature Distr. Learned 0.0542 0 0 1 0 0
Weights Learned 0.0094 -0.0108 -0.2765 0.8126 -0.51 -0.0153
Car driving results (more detail)
Pieter Abbeel and Andrew Y. Ng
Proof (sketch)
Pieter Abbeel and Andrew Y. Ng
Apprenticeship Learning via Inverse Reinforcement Learning
Pieter Abbeel and Andrew Y. Ng
Stanford University