3.1 1D case

First, let’s start with a simple case. Suppose we need to find \(x\) that mimize \[f(x)=1+(x-3)^{2}\] by using the gradient decent which described in the algorithm

gradient_descent = function(step_size,initial_guess){
  x = initial_guess
  d_fx=10

  while(abs(d_fx)>0.01){
    d_fx = 2*(x-3)
    x = x-step_size*d_fx

  }

  return(x)
}

Now, let’s visulize the result by checking if we find the minimum point or not. You should be able to see the red dots(the found solution) appeared at the bottom of the function curve

x = gradient_descent(0.01,0)
x

## [1] 2.995136

##Check
t= seq(0,5,by = 0.01)
y = 1+(t-3)^2
plot(t,y)
points(x,1+(x-3)^2,col='red',pch=23,lwd=6)

Yes! We made it. Now, let’s try the case in 2D.

3.2 2D case

Suppose we need to find \(x = [x_1,x_2]\) that mimize

\[f\left(x_{1}, x_{2}\right)=1+\left(x_{1}-3\right)^{2}+\left(x_{2}-2\right)^{2}\]

by using the gradient decent which described in the algorithm

gradient_descent = function(step_size,initial_guess){
  x = initial_guess
  d_fx= rep(10,length(initial_guess))
  N =10000
  trace <- vector("list", N)
  i = 1
  while(max(abs(d_fx))>0.01){
    d_fx[1] = 2*(x[1]-3)
    d_fx[2] = 2*(x[2]-2)
    x = x-step_size*d_fx
    trace[[i]] <- x
    i <- i + 1

  }
  result = list("answer"=x,"trace"=Filter(Negate(is.null), trace) )
  return(result)
}
library(data.table)
x = gradient_descent(0.01,c(-10,-10))
trace_loc = transpose(as.data.frame(x$trace))
colnames(trace_loc )=c("t1","t2")
trace_loc$z = 1+(trace_loc$t1-3)^2+(trace_loc$t2-2)^2
trace_loc$episode = seq(1,length(trace_loc$z),1)

3.3 Using plotly to make an animation

Plotly’s R graphing library makes interactive, publication-quality graphs. For this 2D case, we can make an animation to visualize the trajectory of the decent by using Plotly.

##Check
library("plotly")
library(reshape2)
library(rlist)
t1= seq(-10,10,by = 0.1)
t2= seq(-10,10,by =0.1)
z =outer(1+(t1-3)^2,(t2-2)^2,`+`)
rownames(z) <- seq(-10,10,by =0.1)
colnames(z) <-seq(-10,10,by =0.1)
matrix=melt(z)
data=setNames(matrix, c("t1", "t2", "VALUE"))

fig = plot_ly(x = ~data$t1, y = ~data$t2, z = ~data$VALUE,type = 'scatter3d',mode ='lines',
              line = list(color = '#1f77b4', width = 1.5))
fig <- fig %>% add_trace( x = ~trace_loc$t1, y = ~trace_loc$t2, z = ~trace_loc$z,
                           type = 'scatter3d',mode = 'lines',
                           line = list(color = 'yellow', width = 10))
fig <- fig %>% add_markers( x = ~trace_loc$t1, y = ~trace_loc$t2, z = ~trace_loc$z,
                          frame = ~trace_loc$episode, type = 'scatter3d',mode = 'markers',
                          showlegend = F,color = 'green')


fig