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.

$2\mathrm{.}1\mathrm{.}3\mathrm{.}1$ 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$\text{'}.$

def find$\_$best$\_$k$($X$\_$train, y$\_$train, X$\_$test, y$\_$test, k$\_$max$)$:

$"""$

Finds the best value of K for KNN based on the given training and testing$/$validation data.

Parameters:

$-$ X$\_$train: Training data features.

$-$ y$\_$train: Training data labels.

$-$ X$\_$test: Testing$/$validation data features.

$-$ y$\_$test: Testing$/$validation data labels.

$-$ k$\_$max: The maximum value of K to consider.

Returns:

$-$ best$\_$k: The optimal value of K that results in the lowest error rate.

$-$ best$\_$error$\_$rate: The lowest error rate corresponding to the best K$.$

$"""$

return best$\_$k$,$ best$\_$error$\_$rate

# Usage example:

# best$\_$k$,$ best$\_$error$\_$rate $=$ find$\_$best$\_$k$($X$\_$train, y$\_$train, X$\_$test, y$\_$test, $10)$

# print$($f$"$Best K: $\{$best$\_$k$\}$ with error rate: $\{$best$\_$error$\_$rate$\}")$