parabolic_em: Parabolic-EM Method for Accelerating Slowly-Convergent...

In bhtang127/AccelBenchmark: Benchmark Various Accelerating Methods for Fixed Point Iterations

Parabolic-EM Method for Accelerating Slowly-Convergent Fixed-Point Iterations

Description

Using parabolic-EM method Berlinet (2009) to accelerate general fixed-point iteration problems.

Usage

parabolic_em(par, fixptfn, objfn, ..., control = list())

Arguments

par	Vector for initial parameters
fixptfn	Fixed point updating function
objfn	Objective function
...	Other arguments required by fixptfn and objfn
control	A list containing parameters controlling the algorithm

Details

The task it to minimize objfn. Default values of control are: warmup=5, h=0.1, a=1.5, maxtry=Inf, version="geometric", objfn.inc=0, projection=function(x) x, tol=1e-7, maxiter=2000, convtype="parameter", par.track=FALSE, conv.spec=NULL.

warmup

An integer variable indicating the number of fixptfn to be evaluated before starting this algorithm. Default is 5.

a

A positive real number in the line search of geometric method. Default is 1.5.

h

A positive real number indicating the step size in the line search step. Default is 0.1.

maxtry

An integer variable indicating maximum number of try when searching for optimal step length in ever iteration. Default is Inf.

version

A string indicating the method used in searching the step length t. t = 1 + h \times a^i for "geometric" and t = 1 + i \times h for "arithmetic". Default is "geometric".

projection

A function projecting the parameter after each iteration. Default is identity function f(x) = x.

objfn.inc

A non-negative scalar that dictates the degree of non-montonicity. Default is 0. Set objfn.inc = 0 to obtain monotone convergence. Setting objfn.inc = Inf gives a non-monotone scheme. In-between values result in partially-monotone convergence.

tol

A small, positive scalar that determines when iterations should be terminated, see convtype for details. Default is 1e-7

maxiter

An integer denoting the maximum limit on the number of evaluations of fixptfn. Default is 2000.

convtype

A string indicating the convergence criteria. If it is "parameter", the algorithm will termenate when L2 norm of parameters difference x_{new} - x_{old} < tol. If it is "objfn", the algorithm will terminate when the absolute difference of objective function |L_{new} - L_{old}| < tol. If it is "user" or conv.spec is not NULL. Then the convergence is guided by the user defined function conv.spec. Default is "parameter".

par.track

An bool value indicating whether to track parameters along the algorithm. TRUE for tracking and FALSE for not. Default is FALSE

conv.spec

A function for user specified convergence criteria. When using "parameter" or "objfn" option in convtype, this should be NULL. The function should have the form f(old_parameter, new_parameter, old_objective, new_objective, tolerance) and return 1 if convergent, 0 if not. Defalut is NULL.

Value

A list of results

par	Parameter values, x* that are the fixed-point of fixptfn F such that x*=F(x*) if convergence is successful.
value.objfn	The objective function value at termination.
fpevals	Number of times the fixed-point function fixptfn was evaluated.
objfevals	Number of times the objective function objfn was evaluated.
iter	Numbers of iteration used at termination. (for different algorithms, multiple fixed point iteration might be evaluated in one iteration)
convergence	An integer code indicating whether the algorithm converges. 1 for convergence and 0 denote failure.
objfn.track	An array tracking objective function values along the algorithm
par.track	A matrix tracking parameters along the algorithm, where each row is an array of parameters at some iteration. If not tracking paramters, this will be NULL

References

Berlinet A, Roland C (2009). Parabolic acceleration of the EM algorithm. Statistics and Computing, 19(1): 35–47.

Examples

## Not run: 
set.seed(54321)
prob = lasso_task(lam=1)
parabolic_em(prob$initfn(), prob$fixptfn, prob$objfn, X=prob$X, y=prob$y)

## End(Not run)

bhtang127/AccelBenchmark documentation built on May 30, 2022, 2:21 a.m.