1. Apprenticeship learning for robotic control Pieter Abbeel Stanford University Joint work with Andrew Y. Ng, Adam Coates, Morgan Quigley.

2. This talk Dynamics Model P sa Reward Function R Reinforcement Learning Control policy  Recurring theme: Apprenticeship learning.

3. Motivation In practice reward functions are hard to specify, and people tend to tweak them a lot. Motivating example: helicopter tasks, e.g. flip. Another motivating example: Highway driving.

4. 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. –[Most closely related work: Ratliff et al. 2005, 2006.]

5. Algorithm For t = 1,2,… Inverse RL step: Estimate expert’s reward function R(s)= w T  (s) such that under R(s) the expert performs better than all previously found policies {  i }. RL step: Compute optimal policy  t for the estimated reward w. [Abbeel & Ng, 2004]

6. Maximize , w:||w|| 2 ≤ 1  s.t. U w (  E )  U w (  i ) +  i=1,…,t-1  = margin of expert’s performance over the performance of previously found policies. U w (  ) = E [  t=1 R(s t )|  ] = E [  t=1 w T  (s t )|  ] = w T E [  t=1  (s t )|  ] = w T  (  )  (  ) = E [  t=1  (s t )|  ] are the “feature expectations” Algorithm: IRL step T T T T

7. 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 |U w* (  E ) - U w* (  )| = |w* T  (  E ) - w* T  (  )|  ||w*|| 2 ||  (  E ) -  (  )|| 2  .

8. 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 kT 2 /  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., U w* (  )  U w* (  E ) - .

9. Case study: Highway driving Input: Driving demonstration Output: Learned behavior The only input to the learning algorithm was the driving demonstration (left panel). No reward function was provided.

10. 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. Driving demonstration Driving demonstration Learned behavior Learned behavior

11. 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. The algorithm exploits reward “simplicity” (vs. policy “simplicity” in previous approaches). Inverse reinforcement learning summary

12. The dynamics model Dynamics Model P sa Reward Function R Reinforcement Learning Control policy 

13. Collecting data to learn the dynamics model

14. Learning the dynamics model P sa from data Dynamics Model P sa Reward Function R Reinforcement Learning Control policy  For example, in discrete-state problems, estimate P sa (s’) to be the fraction of times you transitioned to state s’ after taking action a in state s. Challenge: Collecting enough data to guarantee that you can model the entire flight envelop. Estimate P sa from data

15. Collecting data to learn dynamical model State-of-the-art: E 3 algorithm (Kearns and Singh, 2002) Have good model of dynamics? YES “Exploit” NO “Explore”

16. Aggressive exploration (Manual flight) Aggressively exploring the edges of the flight envelope isn’t always a good idea.

17. Learning the dynamics (a 1, s 1, a 2, s 2, a 3, s 3, ….) Expert human pilot flight Learn P sa Dynamics Model P sa Reward Function R Reinforcement Learning Control policy  (a 1, s 1, a 2, s 2, a 3, s 3, ….) Autonomous flight Learn P sa

18. Apprenticeship learning of model Theorem. Suppose that we obtain m = O(poly(S, A, T, 1/  )) examples from a human expert demonstrating the task. Then after a polynomial number k of iterations of testing/re-learning, with high probability, we will obtain a policy  whose performance is comparable to the expert’s: U(  )  U(  E ) -  Thus, so long as a demonstration is available, it isn’t necessary to explicitly explore. In practice, k=1 or 2 is almost always enough. [Abbeel & Ng, 2005]

19. Proof idea From initial pilot demonstrations, our model/simulator P sa will be accurate for the part of the flight envelop (s,a) visited by the pilot. Our model/simulator will correctly predict the helicopter’s behavior under the pilot’s policy  E. Consequently, there is at least one policy (namely  E ) that looks like it’s able to fly the helicopter in our simulation. Thus, each time we solve the MDP using the current simulator P sa, we will find a policy that successfully flies the helicopter according to P sa. If, on the actual helicopter, this policy fails to fly the helicopter---despite the model P sa predicting that it should---then it must be visiting parts of the flight envelop that the model is failing to accurately model. Hence, this gives useful training data to model new parts of the flight envelop.

20. Configurations flown (exploitation only)

21. Tail-in funnel

22. Nose-in funnel

23. In-place rolls

24. In place flips

25. Acknowledgements Andrew Ng, Adam Coates, Morgan Quigley

26. Thank You!