The ZOOM Minimization Package

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The ZOOM Minimization Package

• David SachsMark FischlerFermi National
  Accelerator Laboratory

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Minimization Package

• Object-oriented C library for minimization
• Minuit capabilities, and more
• Suitable for use
• In stand-alone programs
• As part of Root applications
• In other frameworks
• The OO design allows the Minimizer to play well
  with others
• As central part of a fitter provided as examples
  will be
• an unbinned maximum likelihood fitter
• a binned chi-squared fitter,

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Benefits of OO Design

• User convenience
• Domain concepts reflected more directly in code
• Natural to encapsulate function to be minimized
• Avoidance of global variables allows
• multiple simultaneous minimizations
• use in multi-threaded applications
• Extensibility
• Separation of concepts allows for independent
  replacement
• Benefits from any revision of code
• Find features that should have been there
• Sometimes you uncover flaws yes, even in Minuit

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Brief Survey of C Minimizers

• Other C packages are available, but do not use
  modern C
• and so lack the OO advantages
• Other Projects include
• The MINUIT Project SEAL/MathLib
• Hand-translation of Minuit (James and Winkler)
• ROOT minimization/fitting
• Hand-smoothed automated translation of Minuit

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An Example of Use
double f(const stdvectorltdoublegt x) return
x0x0 // some function of // x0
thru x4 int main() using namespace
Minimization const int Ndimensions 5
Problem m(UserFunction(Ndimensions, f))
m.minimize() cout ltlt m.currentPoint() ltlt endl
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Directions of Extensibility

• class Terminator
• When do we stop improving the minimum?
• class Domain
• How do we restrict the allowed values of the
  parameters?
• class Algorithm
• How do we refine existing minimization methods?
• How do we add new minimization techniques?

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Terminators

• A terminator is an object that can be asked
  whether or not minimization is complete
• it must be an instance of a class derived from
  class Terminator
• Several standard terminators are provided
• These can be combined by ORs and ANDs
• Terminators typically make use of the state of
  the minimization
• Users can easily create their own terminator
  classes

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An Example Terminator
class NCalls public Terminator int
n public explicit NCalls(int limit)n(limit)
... TerminationType finished (const
ProblemState p) return
p.functionCallCount gt n ? TTStop
TTContinue
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Domains

• A domain is an object that maps internal
  coordinates to external coordinates and vice
  versa
• It must be an instance of a class derived from
  Domain
• Custom domain classes must provide
• Mapping from external M-space to interior
  unrestricted N-space, and vice versa
• Gradient calculation
• One domain is provided RectilinearDomain
• independent -?, ?, a,b, -?, b, a, ?
  intervals
• maps for a,b intervals match those in Minuit

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Some Possible Custom Domains

• Orthogonal domain, but with sigmoid mappings
  instead of arcsine mappings
• Superior for some cases, worse for others
• Multiple probability space
• All parameters must be non-negative
• Parameters must sum to 1
• Interior (or surface) of N-Sphere
• Lots of possible mappings
• But watch out if minimum lies very near a mapping
  singularity

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Algorithms

• An algorithm is an object that encapsulates a
  minimization technique
• It must be an instance of a class derived from
  Algorithm
• Custom algorithm classes can be written by
  experts.
• Two algorithms are provided
• Migrad
• Simplex
• Other algorithms will be provided
• Seek and others completing Minuit collection
• Further algorithm implementation possibilities
• Fumili
• biConjGradStab
• LEAMAX

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Adding a New Algorithm

• We might wish to add a Simplex-like algorithm but
  with
• improved estimation of the distance to the
  minimum (edm)
• strategies to avoid false convergence
• The mathematics of finding the best strategy is
  still hard
• OO design makes adding the algorithm
  straightforward, once the proper strategy is
  discovered

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The Simplex Algorithm

• The algorithm
• N1 points in N-space form a simplex
• Repeatedly reflect the worst point about the
  centroid of the remaining points
• Sometimes shrink if that improves function
• Sometimes expand if reflected point is best point
• When estimated distance to minimum is small, stop
• Problem The edm estimate formed by simplex is
  largely fantasy.

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Discovery of a Flaw in Minuit

• Simplex can converge to a non-minimum
• Even on well-behaved (quadratic) functions!
• When starting a large distance from the true
  minimum
• How does this happen?
• Simplex becomes thin, aligned with an
  equipotential
• There is a good direction to expand off that line
  but
• simplex shrinks before it finds that direction
  and
• triggers convergence criterion based on the false
  edm
• If the edm were big at that point, we would at
  least know something was wrong
• The fix is still not trivial but knowing a
  meaningful edm is a start

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Apparent Convergence
MINUIT
Minimization
The simplex algorithm is stopped when it
indicates it is near the function minimum.
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Convergence Failure
MINUIT
Minimization
MINUITs simplex indicates convergence while
still far from it. The error is fixed in
Minimization
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Conclusion

• Minimization package available from
• http//cepa.fnal.gov/aps/minimization.shtml
• Standard build mechanism (configure/make)
• Modern, Object Oriented C
• Suitable for some multi-threaded operation
• Designed to be extensible to new
• Termination criteria
• Domains (parameter restrictions)
• Algorithms
• Corrects a flaw in Minuit
• In the Simplex algorithm

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The ZOOM Minimization Package

• David SachsMark FischlerFermi National
  Accelerator Laboratory