reNesterov: Restart Nesterov Method for Accelerating Slowly-Convergent...

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

Restart Nesterov Method for Accelerating Slowly-Convergent Fixed-Point Iterations

Description

Using restart Nesterov method O’donoghue (2015) to accelerate general fixed-point iteration problems.

Usage

reNesterov(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: projection=function(x) x, tol=1e-7, maxiter=2000, convtype="parameter", par.track=FALSE, conv.spec=NULL.

projection

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

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

O’donoghue B, Candes E (2015). Adaptive restart for accelerated gradient schemes. Foundations of Computational Mathematics, 15(3): 715–732.

Examples

## Not run: 
set.seed(54321)
prob = lasso_task(lam=1)
reNesterov(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.