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In this work we compute the solution of the linear PDE above with the use of the Feynman
- Kac lemma (Øksendal 2003). The Feynman- Kac lemma provides a connection between
stochastic differential equations and PDEs and therefore its use is twofold. On one side it
can be used to find probabilistic solutions of PDEs based on forward sampling of diffusions
while on the other side it can be used find solution of SDEs based on deterministic methods
that numerically solve PDEs. The solution of the PDE above can be found by evaluating
the expectation:
'(x, ti) =
!
e−
! tN
ti
1
" q(x)dt'(xtN )
#
" i
(4.15)
on sample paths " i = (xi, ..., xtN ) generated with the forward sampling of the diffusion
equation dx = f (x, t)dt + B(x, t)Ldw. Under the use of the Feynman Kac lemma the
stochastic optimal control problem has been transformed into an approximation problem
of a path integral. With a view towards a discrete time approximation, which will be
needed for numerical implementations, the solution (4.15) can be formulated as:
'(x, ti) = lim
dt&0
"
p (" i|xi) exp
0
2−
1
%
8
:&(x(tN)) +
N>−1
j=i
q(x, tj)dt
;
=
3
5d" i (4.16)
where " i = (xti , ....., xtN ) is a sample path (or trajectory piece) starting at state xti and
the term p (" i|xi) is the probability of sample path " i conditioned on the start state xti .
Since equation (4.16) provides the exponential cost to go 'ti in state xti , the integration
above is taken with respect to sample paths " i =
A
xti , xti+1, ....., xtN
B
. The differential
term d" i is defined as d" i = (dxti , ....., dxtN ). After the exponentiated value function
124
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 138 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | In this work we compute the solution of the linear PDE above with the use of the Feynman - Kac lemma (Øksendal 2003). The Feynman- Kac lemma provides a connection between stochastic differential equations and PDEs and therefore its use is twofold. On one side it can be used to find probabilistic solutions of PDEs based on forward sampling of diffusions while on the other side it can be used find solution of SDEs based on deterministic methods that numerically solve PDEs. The solution of the PDE above can be found by evaluating the expectation: '(x, ti) = ! e− ! tN ti 1 " q(x)dt'(xtN ) # " i (4.15) on sample paths " i = (xi, ..., xtN ) generated with the forward sampling of the diffusion equation dx = f (x, t)dt + B(x, t)Ldw. Under the use of the Feynman Kac lemma the stochastic optimal control problem has been transformed into an approximation problem of a path integral. With a view towards a discrete time approximation, which will be needed for numerical implementations, the solution (4.15) can be formulated as: '(x, ti) = lim dt&0 " p (" i|xi) exp 0 2− 1 % 8 :&(x(tN)) + N>−1 j=i q(x, tj)dt ; = 3 5d" i (4.16) where " i = (xti , ....., xtN ) is a sample path (or trajectory piece) starting at state xti and the term p (" i|xi) is the probability of sample path " i conditioned on the start state xti . Since equation (4.16) provides the exponential cost to go 'ti in state xti , the integration above is taken with respect to sample paths " i = A xti , xti+1, ....., xtN B . The differential term d" i is defined as d" i = (dxti , ....., dxtN ). After the exponentiated value function 124 |
