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Reinforcement Learning Policy Op5miza5on Pieter Abbeel UC Berkeley EECS Policy Op>miza>on n Consider control policy parameterized by parameter vector H X max E[ R ( s t ) | ] t =0 n O<en stochas>c policy class

  1. Reinforcement Learning – Policy Op5miza5on Pieter Abbeel UC Berkeley EECS

  2. Policy Op>miza>on n Consider control policy parameterized by parameter vector θ H X max E[ R ( s t ) | π θ ] θ t =0 n O<en stochas>c policy class (smooths out the problem): n probability of taking ac>on u in state s π θ ( u | s )

  3. Learning to Trot/Run A<er learning Before learning (hand-tuned) [Policy search was done through trials on the actual robot.] Kohl and Stone, ICRA2004

  4. Learning to Trot/Run 12 parameters define the Aibo’s gait: n n The front locus (3 parameters: height, x-pos., y-pos.) n The rear locus (3 parameters) n Locus length n Locus skew mul>plier in the x-y plane (for turning) n The height of the front of the body n The height of the rear of the body n The >me each foot takes to move through its locus n The frac>on of >me each foot spends on the ground Kohl and Stone, ICRA2004

  5. [Policy search was done in simulation] [Ng + al, ISER 2004]

  6. Learning to Hover

  7. Ball-In-A-Cup [Kober and Peters, 2009]

  8. Learning to Walk in 20 Minutes [Tedrake, Zhang, Seung 2005]

  9. Learning to Walk in 20 Minutes passive hip joint [1DOF] Arms: coupled to opposite leg to reduce yaw moment freely swinging load [1DOF] 44 cm 2 x 2 (roll, pitch) posi>on controlled servo motors [4 DOF] 9DOFs: * 6 internal DOFs * 3 DOFs for the robot’s orienta>on (always assumed in contact with ground at a single point, absolute (x,y) ignored) Natural gait down 0.03 radians ramp: 0.8Hz, 6.5cm steps [Tedrake, Zhang, Seung 2005]

  10. Learning to Walk in 20 Minutes [Tedrake, Zhang, Seung 2005]

  11. Gradient-Free Methods H X max U ( θ ) = max E[ R ( s t ) | π θ ] θ θ t =0 n Cross-Entropy Method (CEM) n Covariance Matrix Adapta>on (CMA) n Dynamics model: stochas>c: OK; unknown: OK n Policy class: stochas>c: OK n Downside: gradient-free methods slower than gradient-based methods à in prac>ce OK if low-dimensional θ and willing to do do many runs

  12. Gradient-Based Policy Op>miza>on H X max U ( θ ) = max E[ R ( s t ) | π θ ] θ θ t =0 f f s0 s1 s2 π θ π θ π θ u0 u1 u2 R R R r0 r1 r2

  13. Overview of Methods / Seings Dynamics Policy D+K D+U S+K+R S+K S+U D S+R S + + + + PD + + + + + + + LR D: determinis>c; S: stochas>c; K: known; U: unknown; R: reparameterizable; PD: path deriva>ve (=perturba>on analysis) LR: likelihood ra>o (=score func>on)

  14. Ques>ons n When more than one is applicable, which one is best? n When dynamics is only available as black-box, but deriva>ves aren’t available – finite differences based deriva>ves? n Vs. directly finite differences / gradient-free on the policy n Note: finite difference tricky (imprac>cal?) when can’t control random seed n What if model is unknown, but es>mate available

  15. Gradient Computa>on – Unknown Model – Finite Differences

  16. Noise Can Dominate

  17. Finite Differences and Noise n Solu>on 1: Average over many samples n Solu>on 2: Fix the randomness (if possible) n Intui>on by example: wind influence on a helicopter is stochas>c, but if we assume the same wind paqern across trials, this will make the different choices of θ more readily comparable n General instan>a>on: Fix the random seed; and the result is determinis>c system n Ng & Jordan, 2000 provide theore>cal analysis of gains from fixing randomness

  18. Path Deriva>ve for Dynamics: D+K; Policy: D Reminder of op>miza>on objec>ve: n H X max U ( θ ) = max E[ R ( s t ) | π θ ] θ θ t =0 Can compute gradient es>mate along current roll-out: n H ∂ U ∂ R ∂ s ( s t ) ∂ s t X = ∂θ i ∂θ i t =0 ∂ s t = ∂ f ∂ s ( s t − 1 , u t − 1 ) ∂ s t − 1 + ∂ f ∂ s ( s t − 1 , u t − 1 ) ∂ u t − 1 ∂θ i ∂θ i ∂θ i ∂ u t = ∂π θ ( s t , θ ) + ∂π θ ∂ s ( s t , θ ) ∂ s t ∂θ i ∂θ i ∂θ i

  19. Path Deriva>ve for Dynamics: S+K+R; Policy: S+R Reminder of op>miza>on objec>ve: n H X max U ( θ ) = max E[ R ( s t ) | π θ ] θ θ t =0 (draw reparameterized graph on board) n + average over mul>ple samples n

  20. Overview of Methods / Seings Dynamics Policy D+K D+U S+K+R S+K S+U D S+R S + + + + PD + + + + + + + LR D: determinis>c; S: stochas>c; K: known; U: unknown; R: reparameterizable; PD: path deriva>ve (=perturba>on analysis) LR: likelihood ra>o (=score func>on)

  21. Gradient Computa>on – Unknown Model – Likelihood Ra>o

  22. Likelihood Ra>o Gradient [Note: Can also be derived/generalized through an importance sampling deriva>on – Tang and Abbeel, 2011]

  23. Importance Sampling n On board..

  24. Likelihood Ra>o Gradient Es>mate

  25. Likelihood Ra>o Gradient Es>mate

  26. Likelihood Ra>o Gradient Es>mate n As formulated thus far: unbiased but very noisy n Fixes that lead to real-world prac>cality n Baseline n Temporal structure n Also: KL-divergence trust region / natural gradient (= general trick, equally applicable to perturba>on analysis and finite differences)

  27. Likelihood Ra>o with Baseline n Gradient es>mate with baseline: m g = 1 X r θ log P ( τ ( i ) ; θ )( R ( τ ( i ) ) � b ) ˆ m i =1 n Crudely, increasing log-likelihood of paths with higher than baseline reward and decreasing log-likelihood of paths with lower than baseline reward " m # 1 n S>ll unbiased? Yes! X r θ log P ( τ ( i ) ; θ ) b E = 0 m i =1

  28. Likelihood Ra>o and Temporal Structure n Current es>mate: m g = 1 X r θ log P ( τ ( i ) ; θ )( R ( τ ( i ) ) � b ) ˆ m i =1 H − 1 ! H − 1 ! m = 1 r θ log π θ ( u ( i ) t | s ( i ) R ( s ( i ) t , u ( i ) X X X t ) t ) � b m i =1 t =0 t =0 n Future ac>ons do not depend on past rewards, hence can lower variance by instead using: H − 1 H − 1 m ! 1 r θ log π θ ( u ( i ) t | s ( i ) R ( s ( i ) k , u ( i ) X X X t ) k ) � b m i =1 t =0 k = t

  29. Step-sizing and Trust Regions n Naïve step-sizing: Line search n Step-sizing necessary as gradient is only first-order approxima>on n Line search in the direc>on of gradient n Simple, but expensive (evalua>ons along the line) n Naïve: ignores where the first-order approxima>on is good/poor

  30. Step-sizing and Trust Regions n Advanced step-sizing: Trust regions n First-order approxima>on from gradient is a good approxima>on within “trust region” à Solve for best point within trust region: g > δθ max ˆ δθ s . t . KL ( P ( τ ; θ ) || P ( τ ; θ + δθ )) ≤ ε

  31. KL Trust Region (a.k.a. natural gradient) n Solve for best point within trust region: g > δθ max ˆ δθ s . t . KL ( P ( τ ; θ ) || P ( τ ; θ + δθ )) ≤ ε n KL can be approximated efficiently with 2 nd order expansion: G: Fisher Informa>on Matrix

  32. Experiments in Locomo>on [Schulman, Levine, Abbeel, 2014]

  33. Actor-Cri>c Variant n Current es>mate: H − 1 H − 1 m ! 1 r θ log π θ ( u ( i ) t | s ( i ) R ( s ( i ) k , u ( i ) X X X t ) k ) � b m i =1 t =0 k = t Sample based estimate of Q ( s ( i ) k , u ( i ) k ) n Actor-cri>c algorithms in parallel run an es>mator for the Q- func>on, and subs>tute in the es>mated Q value

  34. Learning Locomo>on [Schulman, Moritz, Levine, Jordan, Abbeel, 2015]

  35. In Contrast: Darpa Robo>cs Challenge

  36. Thank you

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