Introduction to Gradient Descent

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Write a function that finds the minimum value of a vector using gradient descent.

The function accepts $1$ input vector $($the data within which we are finding the minimum value$),$ and has $2$ output arguments: the minimum value and the number of times the algorithm had to run to get there.

function $[$minimum$,$ number$\_$of$\_$steps$]=$ gradient$\_$descent$\_$intro$($input$\_$vector$)$

The initial step size for the decent should be half the length of the input$\_$vector and the step$\_$size should be halved at each step until it reaches $1,$ where it will stay until the code concludes.

Start at the first index in the vector.

if the derivative is positive, go left. if it's negative, go right. make sure your index does not go out of bounds. the minimum index is $1,$ the maximum index is length$($input$\_$vector$)$

keep track of the last $2$ values of the function and stop the algorithm once a value reappears.

EXAMPLE

If the sequence of values is $-2\mathrm{.}8,-2\mathrm{.}9,-2\mathrm{.}999999,-3\mathrm{.}000000,-2\mathrm{.}8$

then the sequence of "minimums" should be something like

$-2\mathrm{.}8$

$-2\mathrm{.}9$

$-3\mathrm{.}000000$

$-2\mathrm{.}999999$

$-3\mathrm{.}000000$

then the function should return that $-3\mathrm{.}000000$ is the minimum and that it took $5$ steps to find it$.$

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