Write a python program for Randomizing the Leader: Consider the problem of online learning on the simplex

${}_d$ where $d=1000$; At round $t,$ you predict $p_t$ and receive a vector $z_t$ and suffer a loss of

$p_t^T{z}_t.$ Assume the adversary picks the vector $z_t$ as the $t-$th row in the dataset Dataset$\_$Z$.$

Implement FTRL with quadratic and entropic regularization for this problem and plot the

regret over time.

Now, consider the following algorithm which first picks and fixes a random $1000$ dimensional

vector $R$ sampled uniformly from $[0,\frac{1}{}{]}^d$ and uses the following rule for prediction

$p_{t+1}=argmi{n}_{\text{p}\text{i}\text{n}{}_d}{\displaystyle \underset{i=1}{\overset{t}{}}}(p^T(z_i+R))$

How would you choose $$ for this problem? For the value chosen, plot the regret bound

for this algorithm as well. How does the regret bound compare with the previous two

algorithms for this problem?