daarem: Damped Anderson Acceleration with Restarts and...

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

Damped Anderson Acceleration with Restarts and Epsilon-Montonicity for Accelerating Slowly-Convergent, Monotone Fixed-Point Iterations

Description

An ‘off-the-shelf’ acceleration scheme for accelerating the convergence of any smooth, monotone, slowly-converging fixed-point iteration. It can be used to accelerate the convergence of a wide variety of montone iterations including, for example, expectation-maximization (EM) algorithms and majorization-minimization (MM) algorithms. This is an modified version of the original daarem in daarem package, including projection and user defined convergence function.

Usage

daarem(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: order=10, mon.tol=0.01, cycl.mon.tol=0.0, alpha=1.2, kappa=25, projection=function(x) x, tol=1e-7, maxiter=2000, convtype="parameter", par.track=FALSE, conv.spec=NULL.

order

An integer >= 1 denoting the order of the DAAREM acceleration scheme. Default is 10.

mon.tol

A nonnegative scalar that determines whether the montonicity condition is violated. The monotonicity condition is violated whenver L(x[k+1]) > L(x[k]) + mon.tol. Such violations determine how much damping is to be applied on subsequent steps of the algorithm. Default value of mon.tol is 1.e-02.

cycl.mon.tol

A nonegative scalar that determines whether a montonicity condition is violated after the end of the cycle. This cycle-level monotonicity condition is violated whenver L(x[end cycle]) > L(x[start cycle]) + cycl.mon.tol. Here, x[start cycle] refers to the value of x at the beginning of the current cycle while x[end cycle] refers to the value of x at the end of the current cycle. Such violations also determine how much damping is to be applied on subsequent steps of the algorithm.

kappa

A nonnegative parameter which determines the “half-life” of relative damping and how quickly relative damping tends to one. In the absence of monotonicity violations, the relative damping factor is <= 1/2 for the first kappa iterations, and it is then greater than 1/2 for all subsequent iterations. The relative damping factor is the ratio between the norm of the unconstrained coefficients in Anderson acceleration and the norm of the damped coefficients. In the absence of any monotonicity violations, the relative damping factor in iteration k is 1/(1 + a^{(kappa - k)}).

alpha

A parameter > 1 that determines the initial relative damping factor and how quickly the relative damping factor tends to one. The initial relative damping factor is 1/(1 + a^{kappa}). In the absence of any monotonicity violations, the relative damping factor in iteration k is 1/(1 + a^{(kappa - k)}).

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

Henderson, N.C. and Varadhan, R. (2019) Damped Anderson acceleration with restarts and monotonicity control for accelerating EM and EM-like algorithms, Journal of Computational and Graphical Statistics, Vol. 28(4), 834-846.

Examples

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