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Essentially, (6.22) computes a discrete probability at time ti of each trajectory roll-out
with the help of the cost (6.23). For every time step of the trajectory, a parameter
update is computed in (6.24) based on a probability weighted average over trajectories.
The parameter updates at every time step are finally averaged in (6.25). Note that
we chose a weighted average by giving every parameter update a weight2 according to
the time steps left in the trajectory and the activation of the kernel in (6.3). This
average can be interpreted as using a function approximator with only a constant (offset)
parameter vector to approximate the time dependent parameters. Giving early points in
the trajectory a higher weight is useful since their parameters affect a large time horizon
and thus higher trajectory costs. Other function approximation (or averaging) schemes
could be used to arrive at a final parameter update – we preferred this simple approach
as it gave very good learning results. The final parameter update is )(new) = )(old) + *).
The parameter % regulates the sensitivity of the exponentiated cost and can auto-matically
be optimized for every time step i to maximally discriminate between the
experienced trajectories. More precisely, a constant term can be subtracted from (6.23)
as long as all S(" i) remain positive – this constant term 3 cancels in (6.22). Thus, for a
given number of roll-outs, we compute the exponential term in (6.22) as
exp
$
−
1
%
S(" i)
%
= exp
$
−h
S(" i) − min S(" i)
max S(" i) − min S(" i)
%
(6.27)
2The use of the kernel weights in the basis functions (6.3) for the purpose of time averaging has shown
better performance with respect to other weighting approaches, across all of our experiments. Therefore
this is the weighting that we suggest. Users may develop other weighting schemes as more suitable to
their needs.
3In fact, the term inside the exponent results by adding h min S(" i)
max S(" i)−min S(" i) , which cancels in (6.22),
to the term − hS(" i)
max S(" i)−min S(" i) which is equal to −1
"S(" i).
191
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 205 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | Essentially, (6.22) computes a discrete probability at time ti of each trajectory roll-out with the help of the cost (6.23). For every time step of the trajectory, a parameter update is computed in (6.24) based on a probability weighted average over trajectories. The parameter updates at every time step are finally averaged in (6.25). Note that we chose a weighted average by giving every parameter update a weight2 according to the time steps left in the trajectory and the activation of the kernel in (6.3). This average can be interpreted as using a function approximator with only a constant (offset) parameter vector to approximate the time dependent parameters. Giving early points in the trajectory a higher weight is useful since their parameters affect a large time horizon and thus higher trajectory costs. Other function approximation (or averaging) schemes could be used to arrive at a final parameter update – we preferred this simple approach as it gave very good learning results. The final parameter update is )(new) = )(old) + *). The parameter % regulates the sensitivity of the exponentiated cost and can auto-matically be optimized for every time step i to maximally discriminate between the experienced trajectories. More precisely, a constant term can be subtracted from (6.23) as long as all S(" i) remain positive – this constant term 3 cancels in (6.22). Thus, for a given number of roll-outs, we compute the exponential term in (6.22) as exp $ − 1 % S(" i) % = exp $ −h S(" i) − min S(" i) max S(" i) − min S(" i) % (6.27) 2The use of the kernel weights in the basis functions (6.3) for the purpose of time averaging has shown better performance with respect to other weighting approaches, across all of our experiments. Therefore this is the weighting that we suggest. Users may develop other weighting schemes as more suitable to their needs. 3In fact, the term inside the exponent results by adding h min S(" i) max S(" i)−min S(" i) , which cancels in (6.22), to the term − hS(" i) max S(" i)−min S(" i) which is equal to −1 "S(" i). 191 |
