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