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Thus, we would update ) based on two terms. The first term is the average of *)ti ,
which is reasonable as it reflects the knowledge we gained from the exploration noise.
However, there would be a second update term due to the average over projected mean
parameters ) from every time step – it should be noted that Mti is a projection matrix
onto the range space of gti under the metric R−1, such that a multiplication with Mti
can only shrink the norm of ). From the viewpoint of having optimal parameters for
every time step, this update component is reasonable as it trivially eliminates the part
of the parameter vector that lies in the null space of gti and which contributes to the
command cost of a trajectory in a useless way. From the view point of a parameter vector
that is constant and time independent and that is updated iteratively, this second update
is undesirable, as the multiplication of the parameter vector ) with Mti in (6.19) and
the averaging operation over the time horizon reduces the L2 norm of the parameters at
every iteration, potentially in an uncontrolled way1. What we rather want is to achieve
convergence when the average of *)ti becomes zero, and we do not want to continue
updating due to the second term.
The problem is avoided by eliminating the projection matrix in the second term of
averaging, such that it become:
)(new) =
1
N
N>−1
i=0
*)ti +
1
N
N>−1
i=0
) =
1
N
N>−1
i=0
*)ti + )
The meaning of this reduced update is simply that we keep a component in ) that is
irrelevant and contributes to our trajectory cost in a useless way. However, this irrelevant
1To be precise, ! would be projected and continue shrinking until it lies in the intersection of all null
spaces of the gti basis function – this null space can easily be of measure zero.
189
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 203 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | Thus, we would update ) based on two terms. The first term is the average of *)ti , which is reasonable as it reflects the knowledge we gained from the exploration noise. However, there would be a second update term due to the average over projected mean parameters ) from every time step – it should be noted that Mti is a projection matrix onto the range space of gti under the metric R−1, such that a multiplication with Mti can only shrink the norm of ). From the viewpoint of having optimal parameters for every time step, this update component is reasonable as it trivially eliminates the part of the parameter vector that lies in the null space of gti and which contributes to the command cost of a trajectory in a useless way. From the view point of a parameter vector that is constant and time independent and that is updated iteratively, this second update is undesirable, as the multiplication of the parameter vector ) with Mti in (6.19) and the averaging operation over the time horizon reduces the L2 norm of the parameters at every iteration, potentially in an uncontrolled way1. What we rather want is to achieve convergence when the average of *)ti becomes zero, and we do not want to continue updating due to the second term. The problem is avoided by eliminating the projection matrix in the second term of averaging, such that it become: )(new) = 1 N N>−1 i=0 *)ti + 1 N N>−1 i=0 ) = 1 N N>−1 i=0 *)ti + ) The meaning of this reduced update is simply that we keep a component in ) that is irrelevant and contributes to our trajectory cost in a useless way. However, this irrelevant 1To be precise, ! would be projected and continue shrinking until it lies in the intersection of all null spaces of the gti basis function – this null space can easily be of measure zero. 189 |
