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Table 6.1: Pseudocode of the PI2 algorithm for a 1D Parameterized Policy (Note that
the discrete time step dt was absobed as a constant multiplier in the cost terms).
• Given:
– An immediate cost function rt = qt + )Tt
R)t
– A terminal cost term &tN (cf. 4.25)
– A stochastic parameterized policy at = gT
t () + !t)
– The basis function gti from the system dynamics (cf. 4.2)
– The variance %! of the mean-zero noise !t
– The initial parameter vector )
• Repeat until convergence of the trajectory cost R:
– Create K roll-outs of the system from the same start state x0 using stochstic
parameters ) + !t at every time step
– For k = 1...K, compute:
& P (" i,k) = e−1"
S(" i,k)
"Kk
=1[e−1"
S(" i,k)]
& S(" i,k) = &tN,k+
CN−1
j=i qtj ,k+ 12
CN−1
j=i+1()+Mtj ,k!tj ,k)TR()+Mtj ,k!tj ,k)
& Mtj ,k =
R−1gtj ,k gT
tj ,k
gT
tj ,kR−1gtj ,k
– For i = 1...(N − 1), compute:
& *)ti =
CKk
=1 [P (" i,k)Mti,k !ti,k]
– Compute [*)]j =
"N−1
i=0 (N−i) wj,ti [$)ti ]j "N−1
i=0 wj,ti (N−i)
– Update ) 6 ) + *)
– Create one noiseless roll-out to check the trajectory cost R = &tN +
CN−1
i=0 rti .
In case the noise cannot be turned off, i.e., a stochastic system, multiple roll-outs
need be averaged.
193
Object Description
| Title | Iterative path integral stochastic optimal control: theory and applications to motor control |
| Author | Theodorou, Evangelos A. |
| Author email | etheodor@usc.edu; theo0027@umn.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Computer Science |
| School | Viterbi School of Engineering |
| Date defended/completed | 2011-01-11 |
| Date submitted | 2011 |
| Restricted until | Unrestricted |
| Date published | 2011-04-29 |
| Advisor (committee chair) | Schaal, Stefan |
| Advisor (committee member) |
Valero-Cuevas, Francisco Sukhatme, Gaurav S. Todorov, Emo Schweighofer, Nicolas |
| Abstract | Motivated by the limitations of current optimal control and reinforcement learning methods in terms of their efficiency and scalability, this thesis proposes an iterative stochastic optimal control approach based on the generalized path integral formalism. More precisely, we suggest the use of the framework of stochastic optimal control with path integrals to derive a novel approach to RL with parameterized policies. While solidly grounded in value function estimation and optimal control based on the stochastic Hamilton Jacobi Bellman (HJB) equation, policy improvements can be transformed into an approximation problem of a path integral which has no open algorithmic parameters other than the exploration noise. The resulting algorithm can be conceived of as model-based, semi-model-based, or even model free, depending on how the learning problem is structured. The new algorithm, Policy Improvement with Path Integrals (PI2), demonstrates interesting similarities with previous RL research in the framework of probability matching and provides intuition why the slightly heuristically motivated probability matching approach can actually perform well. Applications to high dimensional robotic systems are presented for a variety of tasks that require optimal planning and gain scheduling.; In addition to the work on generalized path integral stochastic optimal control, in this thesis we extend model based iterative optimal control algorithms to the stochastic setting. More precisely we derive the Differential Dynamic Programming algorithm for stochastic systems with state and control multiplicative noise. Finally, in the last part of this thesis, model based iterative optimal control methods are applied to bio-mechanical models of the index finger with the goal to find the underlying tendon forces applied for the movements of, tapping and flexing. |
| Keyword | stochastic optimal control; reinforcement learning,; robotics |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m3804 |
| Contributing entity | University of Southern California |
| Rights | Theodorou, Evangelos A. |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | cisadmin@lib.usc.edu |
| Filename | etd-Theodorou-4581 |
| Archival file | uscthesesreloadpub_Volume14/etd-Theodorou-4581.pdf |
Description
| Title | Page 207 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | Table 6.1: Pseudocode of the PI2 algorithm for a 1D Parameterized Policy (Note that the discrete time step dt was absobed as a constant multiplier in the cost terms). • Given: – An immediate cost function rt = qt + )Tt R)t – A terminal cost term &tN (cf. 4.25) – A stochastic parameterized policy at = gT t () + !t) – The basis function gti from the system dynamics (cf. 4.2) – The variance %! of the mean-zero noise !t – The initial parameter vector ) • Repeat until convergence of the trajectory cost R: – Create K roll-outs of the system from the same start state x0 using stochstic parameters ) + !t at every time step – For k = 1...K, compute: & P (" i,k) = e−1" S(" i,k) "Kk =1[e−1" S(" i,k)] & S(" i,k) = &tN,k+ CN−1 j=i qtj ,k+ 12 CN−1 j=i+1()+Mtj ,k!tj ,k)TR()+Mtj ,k!tj ,k) & Mtj ,k = R−1gtj ,k gT tj ,k gT tj ,kR−1gtj ,k – For i = 1...(N − 1), compute: & *)ti = CKk =1 [P (" i,k)Mti,k !ti,k] – Compute [*)]j = "N−1 i=0 (N−i) wj,ti [$)ti ]j "N−1 i=0 wj,ti (N−i) – Update ) 6 ) + *) – Create one noiseless roll-out to check the trajectory cost R = &tN + CN−1 i=0 rti . In case the noise cannot be turned off, i.e., a stochastic system, multiple roll-outs need be averaged. 193 |
