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case. Recently (Todorov 2008) , the generalized duality was exploited when the backward
Zakai equation in nonlinear smoothing is considered. Essentially the backward Zakai
equation can be turned into a deterministic PDE and then a direct mapping between
the two PDEs can be made in the same way how it is made between the backward and
forward Riccati equations in linear control and filtering problems.
3.10 Conclusions
In this chapter we investigated the connection between SDEs, linear PDEs and Path
Integrals. My goal was to give an introduction to these mathematical concepts and their
connections by keeping a balance between a pedagogical and intuitive presentation and
a presentation that is characterized by rigor and mathematical precision.
In the next chapter, the path integral formalism is applied to stochastic optimal
control and reinforcement learning and the generalized path integral control is derived.
More precisely, the backward Chapman Kolmogorov is formulated and the Feynman-Kac
lemma is applied. Finally the path integral control is derived. Extensions of path integral
control to iterative and risk sensitive control are presented.
3.11 Appendix
We assume the stochastic differential equation dx = f (x, t)dt + B(x, t)dw . If the
drift f (x, t) and diffusion term B(x, t) satisfy the condition:||f (y, t)||2 + ||B(y, t)||2 <
K
$
1+max||y(s)||2
%
then
!
max0<s<t ||xs||2m
#
% C
$
1+
!
||xo||2m
#%
eCt, ,0 % t % T.
116
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 130 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text |
case. Recently (Todorov 2008) , the generalized duality was exploited when the backward
Zakai equation in nonlinear smoothing is considered. Essentially the backward Zakai
equation can be turned into a deterministic PDE and then a direct mapping between
the two PDEs can be made in the same way how it is made between the backward and
forward Riccati equations in linear control and filtering problems.
3.10 Conclusions
In this chapter we investigated the connection between SDEs, linear PDEs and Path
Integrals. My goal was to give an introduction to these mathematical concepts and their
connections by keeping a balance between a pedagogical and intuitive presentation and
a presentation that is characterized by rigor and mathematical precision.
In the next chapter, the path integral formalism is applied to stochastic optimal
control and reinforcement learning and the generalized path integral control is derived.
More precisely, the backward Chapman Kolmogorov is formulated and the Feynman-Kac
lemma is applied. Finally the path integral control is derived. Extensions of path integral
control to iterative and risk sensitive control are presented.
3.11 Appendix
We assume the stochastic differential equation dx = f (x, t)dt + B(x, t)dw . If the
drift f (x, t) and diffusion term B(x, t) satisfy the condition:||f (y, t)||2 + ||B(y, t)||2 <
K
$
1+max||y(s)||2
%
then
!
max0 |
