animation (version 2.4.1)

grad.desc: Gradient Descent Algorithm for the 2D case

Description

This function provids a visual illustration for the process of minimizing a real-valued function through Gradient Descent Algorithm.

Usage

grad.desc(FUN = function(x, y) x^2 + 2 * y^2, rg = c(-3, -3, 3, 3), init = c(-3,  3), gamma = 0.05, tol = 0.001, gr = NULL, len = 50, interact = FALSE,  col.contour = "red", col.arrow = "blue", main)

Arguments

FUN

a bivariate objective function to be minimized (variable names do not have to be x and y); if the gradient argument gr is NULL, deriv will be used to calculate the gradient, in which case we should not put braces around the function body of FUN (e.g. the default function is function(x, y) x^2 + 2 * y^2)

rg

ranges for independent variables to plot contours; in a c(x0, y0, x1, y1) form

init

starting values

gamma

size of a step

tol

tolerance to stop the iterations, i.e. the minimum difference between $F(x[i])$ and $F(x[i+1])$

gr

the gradient of FUN; it should be a bivariate function to calculate the gradient (not the negative gradient!) of FUN at a point $(x,y)$, e.g. function(x, y) 2 * x + 4 * y. If it is NULL, R will use deriv to calculate the gradient

len

desired length of the independent sequences (to compute z values for contours)

interact

logical; whether choose the starting values by clicking on the contour plot directly?

col.contour, col.arrow

colors for the contour lines and arrows respectively (default to be red and blue)

main

the title of the plot; if missing, it will be derived from FUN

Value

A list containing A list containing

Details

Gradient descent is an optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or the approximate gradient) of the function at the current point. If instead one takes steps proportional to the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent.

The arrows are indicating the result of iterations and the process of minimization; they will go to a local minimum in the end if the maximum number of iterations ani.options('nmax') has not been reached.

References

http://vis.supstat.com/2013/03/gradient-descent-algorithm-with-r/

See Also

deriv, persp, contour, optim

Examples

## default example
oopt = ani.options(interval = 0.3, nmax = ifelse(interactive(), 50, 2))
xx = grad.desc()
xx$par  # solution
xx$persp(col = "lightblue", phi = 30)  # perspective plot

## define more complex functions; a little time-consuming
f1 = function(x, y) x^2 + 3 * sin(y)
xx = grad.desc(f1, pi * c(-2, -2, 2, 2), c(-2 * pi, 2))
xx$persp(col = "lightblue", theta = 30, phi = 30)

## need to provide the gradient when deriv() cannot handle the function
grad.desc(FUN = function(x1, x2) {
    x0 = cos(x2)
    x1^2 + x0
}, gr = function(x1, x2) {
    c(2 * x1, -sin(x2))
}, rg = c(-3, -1, 3, 5), init = c(-3, 0.5), main = expression(x[1]^2 + cos(x[2])))

## or a even more complicated function
ani.options(interval = 0, nmax = ifelse(interactive(), 200, 2))
f2 = function(x, y) sin(1/2 * x^2 - 1/4 * y^2 + 3) * cos(2 * x + 1 - exp(y))
xx = grad.desc(f2, c(-2, -2, 2, 2), c(-1, 0.5), gamma = 0.1, tol = 1e-04)

## click your mouse to select a start point
if (interactive()) {
    xx = grad.desc(f2, c(-2, -2, 2, 2), interact = TRUE, tol = 1e-04)
    xx$persp(col = "lightblue", theta = 30, phi = 30)
}

## HTML animation pages
saveHTML({
    ani.options(interval = 0.3)
    grad.desc()
}, img.name = "grad.desc", htmlfile = "grad.desc.html", ani.height = 500, 
    ani.width = 500, title = "Demonstration of the Gradient Descent Algorithm", 
    description = "The arrows will take you to the optimum step by step.")

ani.options(oopt)