Moved trustregion docs from gtsam_experimental to gtsam
Richard Roberts
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Richard Roberts at 2011-10-10 11:30:37 -0400
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%% Saved with string encoding Unicode (UTF-8)
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@webpage{Hauser06lecture,
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Author = {Raphael Hauser},
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Date-Added = {2011-10-10 15:21:22 +0000},
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Date-Modified = {2011-10-10 15:24:31 +0000},
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Title = {Lecture Notes on Unconstrained Optimization},
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Url = {http://www.numerical.rl.ac.uk/nimg/oupartc/lectures/raphael/},
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Year = {2006},
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\begin_layout Section
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Overview of Trust-region Methods
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
For nice figures, see
|
||||
\begin_inset space ~
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset CommandInset citation
|
||||
LatexCommand cite
|
||||
key "Hauser06lecture"
|
||||
|
||||
\end_inset
|
||||
|
||||
(in our /net/hp223/borg/Literature folder).
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
We just deal here with a small subset of trust-region methods, specifically
|
||||
approximating the cost function as quadratic using Newton's method, and
|
||||
using the Dogleg method and later to include Steihaug's method.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The overall goal of a nonlinear optimization method is to iteratively find
|
||||
a local minimum of a nonlinear function
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\hat{x}=\arg\min_{x}f\left(x\right)
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\begin_inset Formula $f\left(x\right)\to\mathbb{R}$
|
||||
\end_inset
|
||||
|
||||
is a scalar function.
|
||||
In GTSAM, the variables
|
||||
\begin_inset Formula $x$
|
||||
\end_inset
|
||||
|
||||
could be manifold or Lie group elements, so in this document we only work
|
||||
with
|
||||
\emph on
|
||||
increments
|
||||
\emph default
|
||||
|
||||
\begin_inset Formula $\delta x\in\R n$
|
||||
\end_inset
|
||||
|
||||
in the tangent space.
|
||||
In this document we specifically deal with
|
||||
\emph on
|
||||
trust-region
|
||||
\emph default
|
||||
methods, which at every iteration attempt to find a good increment
|
||||
\begin_inset Formula $\norm{\delta x}\leq\Delta$
|
||||
\end_inset
|
||||
|
||||
within the
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
trust radius
|
||||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula $\Delta$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Further, most nonlinear optimization methods, including trust region methods,
|
||||
deal with an approximate problem at every iteration.
|
||||
Although there are other choices (such as quasi-Newton), the Newton's method
|
||||
approximation is, given an estimate
|
||||
\begin_inset Formula $x^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
of the variables
|
||||
\begin_inset Formula $x$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
f\left(x^{\left(k\right)}\oplus\delta x\right)\approx M^{\left(k\right)}\left(\delta x\right)=f^{\left(k\right)}+g^{\left(k\right)\t}\delta x+\frac{1}{2}\delta x^{\t}G^{\left(k\right)}\delta x\text{,}\label{eq:M-approx}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\begin_inset Formula $f^{\left(k\right)}=f\left(x^{\left(k\right)}\right)$
|
||||
\end_inset
|
||||
|
||||
is the function at
|
||||
\begin_inset Formula $x^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula $g^{\left(x\right)}=\left.\frac{\partial f}{\partial x}\right|_{x^{\left(k\right)}}$
|
||||
\end_inset
|
||||
|
||||
is its gradient, and
|
||||
\begin_inset Formula $G^{\left(k\right)}=\left.\frac{\partial^{2}f}{\partial x^{2}}\right|_{x^{\left(k\right)}}$
|
||||
\end_inset
|
||||
|
||||
is its Hessian (or an approximation of the Hessian).
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Trust-region methods adaptively adjust the trust radius
|
||||
\begin_inset Formula $\Delta$
|
||||
\end_inset
|
||||
|
||||
so that within it,
|
||||
\begin_inset Formula $M$
|
||||
\end_inset
|
||||
|
||||
is a good approximation of
|
||||
\begin_inset Formula $f$
|
||||
\end_inset
|
||||
|
||||
, and then never step beyond the trust radius in each iteration.
|
||||
When the true minimum is within the trust region, they converge quadratically
|
||||
like Newton's method.
|
||||
At each iteration
|
||||
\begin_inset Formula $k$
|
||||
\end_inset
|
||||
|
||||
, they solve the
|
||||
\emph on
|
||||
trust-region subproblem
|
||||
\emph default
|
||||
to find a proposed update
|
||||
\begin_inset Formula $\delta x$
|
||||
\end_inset
|
||||
|
||||
inside the trust radius
|
||||
\begin_inset Formula $\Delta$
|
||||
\end_inset
|
||||
|
||||
, which decreases the approximate function
|
||||
\begin_inset Formula $M^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
as much as possible.
|
||||
The proposed update is only accepted if the true function
|
||||
\begin_inset Formula $f$
|
||||
\end_inset
|
||||
|
||||
decreases as well.
|
||||
If the decrease of
|
||||
\begin_inset Formula $M$
|
||||
\end_inset
|
||||
|
||||
matches the decrease of
|
||||
\begin_inset Formula $f$
|
||||
\end_inset
|
||||
|
||||
well, the size of the trust region is increased, while if the match is
|
||||
not close the trust region size is decreased.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Minimizing Eq.
|
||||
\begin_inset space ~
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:M-approx"
|
||||
|
||||
\end_inset
|
||||
|
||||
is itself a nonlinear optimization problem, so there are various methods
|
||||
for approximating it, including Dogleg and Steihaug's method.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Adapting the Trust Region Size
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
As mentioned in the previous section, we increase the trust region size
|
||||
if the decrease in the model function
|
||||
\begin_inset Formula $M$
|
||||
\end_inset
|
||||
|
||||
matches the decrease in the true cost function
|
||||
\begin_inset Formula $S$
|
||||
\end_inset
|
||||
|
||||
very closely, and decrease it if they do not match closely.
|
||||
The closeness of this match is measured with the
|
||||
\emph on
|
||||
gain ratio
|
||||
\emph default
|
||||
,
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\rho=\frac{f\left(x\right)-f\left(x\oplus\delta x_{d}\right)}{M\left(0\right)-M\left(\delta x_{d}\right)}\text{,}
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\begin_inset Formula $\delta x_{d}$
|
||||
\end_inset
|
||||
|
||||
is the proposed dogleg step to be introduced next.
|
||||
The decrease in the model function is always non-negative, and as the decrease
|
||||
in
|
||||
\begin_inset Formula $f$
|
||||
\end_inset
|
||||
|
||||
approaches it,
|
||||
\begin_inset Formula $\rho$
|
||||
\end_inset
|
||||
|
||||
approaches
|
||||
\begin_inset Formula $1$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
If the true cost function increases,
|
||||
\begin_inset Formula $\rho$
|
||||
\end_inset
|
||||
|
||||
will be negative, and if the true cost function decreases even more than
|
||||
predicted by
|
||||
\begin_inset Formula $M$
|
||||
\end_inset
|
||||
|
||||
, then
|
||||
\begin_inset Formula $\rho$
|
||||
\end_inset
|
||||
|
||||
will be greater than
|
||||
\begin_inset Formula $1$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
A typical update rule [
|
||||
\color blue
|
||||
see where this came from in paper
|
||||
\color inherit
|
||||
] is
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\Delta\leftarrow\begin{cases}
|
||||
\max\left(\Delta,3\norm{\delta x_{d}}\right)\text{,} & \rho>0.75\\
|
||||
\Delta & 0.75>\rho>0.25\\
|
||||
\Delta/2 & 0.25>\rho
|
||||
\end{cases}
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Dogleg
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Dogleg minimizes an approximation of Eq.
|
||||
\begin_inset space ~
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:M-approx"
|
||||
|
||||
\end_inset
|
||||
|
||||
by considering three possibilities using two points - the minimizer
|
||||
\begin_inset Formula $\delta x_{u}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
of
|
||||
\begin_inset Formula $M^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
along the negative gradient direction
|
||||
\begin_inset Formula $-g^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
, and the overall Newton's method minimizer
|
||||
\begin_inset Formula $\delta x_{n}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
of
|
||||
\begin_inset Formula $M^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
When the Hessian
|
||||
\begin_inset Formula $G^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
is positive, the magnitude of
|
||||
\begin_inset Formula $\delta x_{u}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
is always less than that of
|
||||
\begin_inset Formula $\delta x_{n}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
, meaning that the Newton's method step is
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
more adventurous
|
||||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
.
|
||||
How much we step towards the Newton's method point depends on the trust
|
||||
region size:
|
||||
\end_layout
|
||||
|
||||
\begin_layout Enumerate
|
||||
If the trust region is smaller than
|
||||
\begin_inset Formula $\delta x_{u}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
, we step in the negative gradient direction but only by the trust radius.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Enumerate
|
||||
If the trust region boundary is between
|
||||
\begin_inset Formula $\delta x_{u}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $\delta x_{n}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
, we step to the linearly-interpolated point between these two points that
|
||||
intersects the trust region boundary.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Enumerate
|
||||
If the trust region boundary is larger than
|
||||
\begin_inset Formula $\delta x_{n}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
, we step to
|
||||
\begin_inset Formula $\delta x_{n}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
To find the intersection of the line between
|
||||
\begin_inset Formula $\delta x_{u}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $\delta x_{n}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
with the trust region boundary, we solve a quadratic roots problem,
|
||||
\begin_inset Formula
|
||||
\begin{align*}
|
||||
\Delta & =\norm{\left(1-\tau\right)\delta x_{u}+\tau\delta x_{n}}\\
|
||||
\Delta^{2} & =\left(1-\tau\right)^{2}\delta x_{u}^{\t}\delta x_{u}+2\tau\left(1-\tau\right)\delta x_{u}^{\t}\delta x_{n}+\tau^{2}\delta x_{n}^{\t}\delta x_{n}\\
|
||||
0 & =uu-2\tau uu+\tau^{2}uu+2\tau un-2\tau^{2}un+\tau^{2}nn-\Delta^{2}\\
|
||||
0 & =\left(uu-2un+nn\right)\tau^{2}+\left(2un-2uu\right)\tau-\Delta^{2}+uu\\
|
||||
\tau & =\frac{-\left(2un-2uu\right)\pm\sqrt{\left(2un-2uu\right)^{2}-4\left(uu-2un+nn\right)\left(uu-\Delta^{2}\right)}}{2\left(uu-un+nn\right)}
|
||||
\end{align*}
|
||||
|
||||
\end_inset
|
||||
|
||||
From this we take whichever possibility for
|
||||
\begin_inset Formula $\tau$
|
||||
\end_inset
|
||||
|
||||
such that
|
||||
\begin_inset Formula $0<\tau<1$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
To find the steepest-descent minimizer
|
||||
\begin_inset Formula $\delta x_{u}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
, we perform line search in the gradient direction on the approximate function
|
||||
|
||||
\begin_inset Formula $M$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\delta x_{u}^{\left(k\right)}=\frac{-g^{\left(k\right)\t}g^{\left(k\right)}}{g^{\left(k\right)\t}G^{\left(k\right)}g^{\left(k\right)}}g^{\left(k\right)}\label{eq:steepest-descent-point}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Thus, mathematically, we can write the dogleg update
|
||||
\begin_inset Formula $\delta x_{d}^{\left(k\right)}$
|
||||
\end_inset
|
||||
|
||||
as
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\delta x_{d}^{\left(k\right)}=\begin{cases}
|
||||
-\frac{\Delta}{\norm{g^{\left(k\right)}}}g^{\left(k\right)}\text{,} & \Delta<\norm{\delta x_{u}^{\left(k\right)}}\\
|
||||
\left(1-\tau^{\left(k\right)}\right)\delta x_{u}^{\left(k\right)}+\tau^{\left(k\right)}\delta x_{n}^{\left(k\right)}\text{,} & \norm{\delta x_{u}^{\left(k\right)}}<\Delta<\norm{\delta x_{n}^{\left(k\right)}}\\
|
||||
\delta x_{n}^{\left(k\right)}\text{,} & \norm{\delta x_{n}^{\left(k\right)}}<\Delta
|
||||
\end{cases}
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Working with
|
||||
\begin_inset Formula $M$
|
||||
\end_inset
|
||||
|
||||
as a Bayes' Net
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
When we have already eliminated a factor graph into a Bayes' Net, we have
|
||||
the same quadratic error function
|
||||
\begin_inset Formula $M^{\left(k\right)}\left(\delta x\right)$
|
||||
\end_inset
|
||||
|
||||
, but it is in a different form:
|
||||
\begin_inset Formula
|
||||
\[
|
||||
M^{\left(k\right)}\left(\delta x\right)=\frac{1}{2}\norm{Rx-d}^{2}\text{,}
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\begin_inset Formula $R$
|
||||
\end_inset
|
||||
|
||||
is an upper-triangular matrix (stored as a set of sparse block Gaussian
|
||||
conditionals in GTSAM), and
|
||||
\begin_inset Formula $d$
|
||||
\end_inset
|
||||
|
||||
is the r.h.s.
|
||||
vector.
|
||||
The gradient and Hessian of
|
||||
\begin_inset Formula $M$
|
||||
\end_inset
|
||||
|
||||
are then
|
||||
\begin_inset Formula
|
||||
\begin{align*}
|
||||
g^{\left(k\right)} & =R^{\t}\left(Rx-d\right)\\
|
||||
G^{\left(k\right)} & =R^{\t}R
|
||||
\end{align*}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
In GTSAM, because the Bayes' Net is not dense, we evaluate Eq.
|
||||
\begin_inset space ~
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:steepest-descent-point"
|
||||
|
||||
\end_inset
|
||||
|
||||
in an efficient way.
|
||||
Rewriting the denominator (leaving out the
|
||||
\begin_inset Formula $\left(k\right)$
|
||||
\end_inset
|
||||
|
||||
superscript) as
|
||||
\begin_inset Formula
|
||||
\[
|
||||
g^{\t}Gg=\sum_{i}\left(R_{i}g\right)^{\t}\left(R_{i}g\right)\text{,}
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\begin_inset Formula $i$
|
||||
\end_inset
|
||||
|
||||
indexes over the conditionals in the Bayes' Net (corresponding to blocks
|
||||
of rows of
|
||||
\begin_inset Formula $R$
|
||||
\end_inset
|
||||
|
||||
) exploits the sparse structure of the Bayes' Net, because it is easy to
|
||||
only include the variables involved in each
|
||||
\begin_inset Formula $i^{\text{th}}$
|
||||
\end_inset
|
||||
|
||||
conditional when multiplying them by the corresponding elements of
|
||||
\begin_inset Formula $g$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset CommandInset bibtex
|
||||
LatexCommand bibtex
|
||||
bibfiles "trustregion"
|
||||
options "plain"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_body
|
||||
\end_document
|
||||
Binary file not shown.
Loading…
Reference in New Issue