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

In $[]$: 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, $y_{{?}_t\text{r}\text{a}\text{i}\text{n}\text{,}}$ tr$\_$test, $y_{-}$test, $10)$

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