763 lines
15 KiB
Plaintext
763 lines
15 KiB
Plaintext
#LyX 2.0 created this file. For more info see http://www.lyx.org/
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\end_inset
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\end_layout
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\begin_layout Section
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Overview of Trust-region Methods
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\end_layout
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\begin_layout Standard
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For nice figures, see
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\begin_inset space ~
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\end_inset
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\begin_inset CommandInset citation
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LatexCommand cite
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key "Hauser06lecture"
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\end_inset
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(in our /net/hp223/borg/Literature folder).
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\end_layout
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\begin_layout Standard
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We just deal here with a small subset of trust-region methods, specifically
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approximating the cost function as quadratic using Newton's method, and
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using the Dogleg method and later to include Steihaug's method.
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\end_layout
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\begin_layout Standard
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The overall goal of a nonlinear optimization method is to iteratively find
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a local minimum of a nonlinear function
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\begin_inset Formula
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\[
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\hat{x}=\arg\min_{x}f\left(x\right)
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\]
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\end_inset
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where
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\begin_inset Formula $f\left(x\right)\to\mathbb{R}$
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\end_inset
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is a scalar function.
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In GTSAM, the variables
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\begin_inset Formula $x$
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\end_inset
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could be manifold or Lie group elements, so in this document we only work
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with
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\emph on
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increments
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\emph default
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\begin_inset Formula $\delta x\in\R n$
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\end_inset
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in the tangent space.
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In this document we specifically deal with
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\emph on
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trust-region
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\emph default
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methods, which at every iteration attempt to find a good increment
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\begin_inset Formula $\norm{\delta x}\leq\Delta$
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\end_inset
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within the
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\begin_inset Quotes eld
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\end_inset
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trust radius
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\begin_inset Quotes erd
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\end_inset
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\begin_inset Formula $\Delta$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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Further, most nonlinear optimization methods, including trust region methods,
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deal with an approximate problem at every iteration.
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Although there are other choices (such as quasi-Newton), the Newton's method
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approximation is, given an estimate
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\begin_inset Formula $x^{\left(k\right)}$
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\end_inset
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of the variables
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\begin_inset Formula $x$
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\end_inset
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,
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\begin_inset Formula
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\begin{equation}
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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}
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\end{equation}
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\end_inset
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where
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\begin_inset Formula $f^{\left(k\right)}=f\left(x^{\left(k\right)}\right)$
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\end_inset
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is the function at
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\begin_inset Formula $x^{\left(k\right)}$
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\end_inset
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,
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\begin_inset Formula $g^{\left(x\right)}=\left.\frac{\partial f}{\partial x}\right|_{x^{\left(k\right)}}$
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\end_inset
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is its gradient, and
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\begin_inset Formula $G^{\left(k\right)}=\left.\frac{\partial^{2}f}{\partial x^{2}}\right|_{x^{\left(k\right)}}$
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\end_inset
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is its Hessian (or an approximation of the Hessian).
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\end_layout
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\begin_layout Standard
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Trust-region methods adaptively adjust the trust radius
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\begin_inset Formula $\Delta$
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\end_inset
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so that within it,
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\begin_inset Formula $M$
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\end_inset
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is a good approximation of
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\begin_inset Formula $f$
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\end_inset
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, and then never step beyond the trust radius in each iteration.
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When the true minimum is within the trust region, they converge quadratically
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like Newton's method.
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At each iteration
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\begin_inset Formula $k$
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\end_inset
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, they solve the
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\emph on
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trust-region subproblem
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\emph default
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to find a proposed update
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\begin_inset Formula $\delta x$
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\end_inset
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inside the trust radius
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\begin_inset Formula $\Delta$
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\end_inset
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, which decreases the approximate function
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\begin_inset Formula $M^{\left(k\right)}$
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\end_inset
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as much as possible.
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The proposed update is only accepted if the true function
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\begin_inset Formula $f$
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\end_inset
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decreases as well.
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If the decrease of
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\begin_inset Formula $M$
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\end_inset
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matches the decrease of
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\begin_inset Formula $f$
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\end_inset
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well, the size of the trust region is increased, while if the match is
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not close the trust region size is decreased.
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\end_layout
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\begin_layout Standard
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Minimizing Eq.
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\begin_inset space ~
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:M-approx"
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\end_inset
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is itself a nonlinear optimization problem, so there are various methods
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for approximating it, including Dogleg and Steihaug's method.
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\end_layout
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\begin_layout Section
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Adapting the Trust Region Size
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\end_layout
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\begin_layout Standard
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As mentioned in the previous section, we increase the trust region size
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if the decrease in the model function
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\begin_inset Formula $M$
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\end_inset
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matches the decrease in the true cost function
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\begin_inset Formula $S$
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\end_inset
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very closely, and decrease it if they do not match closely.
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The closeness of this match is measured with the
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\emph on
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gain ratio
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\emph default
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,
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\begin_inset Formula
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\[
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\rho=\frac{f\left(x\right)-f\left(x\oplus\delta x_{d}\right)}{M\left(0\right)-M\left(\delta x_{d}\right)}\text{,}
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\]
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\end_inset
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where
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\begin_inset Formula $\delta x_{d}$
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\end_inset
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is the proposed dogleg step to be introduced next.
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The decrease in the model function is always non-negative, and as the decrease
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in
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\begin_inset Formula $f$
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\end_inset
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approaches it,
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\begin_inset Formula $\rho$
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\end_inset
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approaches
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\begin_inset Formula $1$
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\end_inset
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.
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If the true cost function increases,
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\begin_inset Formula $\rho$
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\end_inset
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will be negative, and if the true cost function decreases even more than
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predicted by
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\begin_inset Formula $M$
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\end_inset
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, then
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\begin_inset Formula $\rho$
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\end_inset
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will be greater than
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\begin_inset Formula $1$
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\end_inset
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.
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A typical update rule [
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\color blue
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see where this came from in paper
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\color inherit
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] is
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\begin_inset Formula
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\[
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\Delta\leftarrow\begin{cases}
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\max\left(\Delta,3\norm{\delta x_{d}}\right)\text{,} & \rho>0.75\\
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\Delta & 0.75>\rho>0.25\\
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\Delta/2 & 0.25>\rho
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\end{cases}
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\]
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\end_inset
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\end_layout
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\begin_layout Section
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Dogleg
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\end_layout
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\begin_layout Standard
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Dogleg minimizes an approximation of Eq.
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\begin_inset space ~
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:M-approx"
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\end_inset
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by considering three possibilities using two points - the minimizer
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\begin_inset Formula $\delta x_{u}^{\left(k\right)}$
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\end_inset
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of
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\begin_inset Formula $M^{\left(k\right)}$
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\end_inset
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along the negative gradient direction
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\begin_inset Formula $-g^{\left(k\right)}$
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\end_inset
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, and the overall Newton's method minimizer
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\begin_inset Formula $\delta x_{n}^{\left(k\right)}$
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\end_inset
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of
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\begin_inset Formula $M^{\left(k\right)}$
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\end_inset
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.
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When the Hessian
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\begin_inset Formula $G^{\left(k\right)}$
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\end_inset
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is positive, the magnitude of
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\begin_inset Formula $\delta x_{u}^{\left(k\right)}$
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\end_inset
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is always less than that of
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\begin_inset Formula $\delta x_{n}^{\left(k\right)}$
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\end_inset
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, meaning that the Newton's method step is
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\begin_inset Quotes eld
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\end_inset
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more adventurous
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\begin_inset Quotes erd
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\end_inset
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.
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How much we step towards the Newton's method point depends on the trust
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region size:
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\end_layout
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\begin_layout Enumerate
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If the trust region is smaller than
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\begin_inset Formula $\delta x_{u}^{\left(k\right)}$
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\end_inset
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, we step in the negative gradient direction but only by the trust radius.
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\end_layout
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\begin_layout Enumerate
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If the trust region boundary is between
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\begin_inset Formula $\delta x_{u}^{\left(k\right)}$
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\end_inset
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and
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\begin_inset Formula $\delta x_{n}^{\left(k\right)}$
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\end_inset
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, we step to the linearly-interpolated point between these two points that
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intersects the trust region boundary.
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\end_layout
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\begin_layout Enumerate
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If the trust region boundary is larger than
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\begin_inset Formula $\delta x_{n}^{\left(k\right)}$
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\end_inset
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, we step to
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\begin_inset Formula $\delta x_{n}^{\left(k\right)}$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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To find the intersection of the line between
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\begin_inset Formula $\delta x_{u}^{\left(k\right)}$
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\end_inset
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and
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\begin_inset Formula $\delta x_{n}^{\left(k\right)}$
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\end_inset
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with the trust region boundary, we solve a quadratic roots problem,
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\begin_inset Formula
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\begin{align*}
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\Delta & =\norm{\left(1-\tau\right)\delta x_{u}+\tau\delta x_{n}}\\
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\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}\\
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0 & =uu-2\tau uu+\tau^{2}uu+2\tau un-2\tau^{2}un+\tau^{2}nn-\Delta^{2}\\
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0 & =\left(uu-2un+nn\right)\tau^{2}+\left(2un-2uu\right)\tau-\Delta^{2}+uu\\
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\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)}
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\end{align*}
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\end_inset
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From this we take whichever possibility for
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\begin_inset Formula $\tau$
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\end_inset
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such that
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\begin_inset Formula $0<\tau<1$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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To find the steepest-descent minimizer
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\begin_inset Formula $\delta x_{u}^{\left(k\right)}$
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\end_inset
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, we perform line search in the gradient direction on the approximate function
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\begin_inset Formula $M$
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\end_inset
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,
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\begin_inset Formula
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\begin{equation}
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\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}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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Thus, mathematically, we can write the dogleg update
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\begin_inset Formula $\delta x_{d}^{\left(k\right)}$
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\end_inset
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|
as
|
|
\begin_inset Formula
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\[
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\delta x_{d}^{\left(k\right)}=\begin{cases}
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-\frac{\Delta}{\norm{g^{\left(k\right)}}}g^{\left(k\right)}\text{,} & \Delta<\norm{\delta x_{u}^{\left(k\right)}}\\
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\left(1-\tau^{\left(k\right)}\right)\delta x_{u}^{\left(k\right)}+\tau^{\left(k\right)}\delta x_{n}^{\left(k\right)}\text{,} & \norm{\delta x_{u}^{\left(k\right)}}<\Delta<\norm{\delta x_{n}^{\left(k\right)}}\\
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\delta x_{n}^{\left(k\right)}\text{,} & \norm{\delta x_{n}^{\left(k\right)}}<\Delta
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\end{cases}
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\]
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\end_inset
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\end_layout
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\begin_layout Section
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Working with
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\begin_inset Formula $M$
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\end_inset
|
|
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|
as a Bayes' Net
|
|
\end_layout
|
|
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\begin_layout Standard
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|
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
|
|
\[
|
|
M^{\left(k\right)}\left(\delta x\right)=\frac{1}{2}\norm{Rx-d}^{2}\text{,}
|
|
\]
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|
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|
\end_inset
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|
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|
where
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|
\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
|
|
|
|
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|
\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
|