$1\mathrm{.}2$ Finding the Optimal Value of K for KNN

The choice of $\text{'}$K$\text{'}$ in K$-$Nearest Neighbors $($KNN$)$ significantly affects the model's ability to generalize well from the training data to unseen data. This task focuses on identifying the optimal $\text{'}$K$\text{'}$ that achieves a balance between overfitting and underfitting.

Objective:

Implement a function to find and return the optimal $\text{'}$K$\text{'}$ for a KNN model, evaluated on given training and testing$/$validation data.

Requirements:

The function should be named find$\_$best$\_$k $.$

Parameters:

x$\_$train : A $2$D array of the training features.

y$\_$train : A $1$D array of the training labels.

x$\_$test : A $2$D array of the testing$/$validation features.

y$\_$test : A $1$D array of the testing$/$validation labels.

k$\_$max: An integer representing the maximum value of $\text{'}K\text{'}$ to be considered in the search for the optimal $\text{'}K\text{'}.$

Return:

The function should return two values:

best$\_$k: An integer representing the optimal number of neighbors based on the evaluation.

best$\_$error$\_$rate: A float representing the lowest error rate achieved with the optimal $\text{'}$K$\text{'}.$