$2\mathrm{.}1\mathrm{.}31\mathrm{.}2$ 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.

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

In $[]$: $1$ 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$\_$ test, $\{$:$y_{-}$test,$10)$

# print $(f^{\text{B}\text{e}\text{s}\text{t}}K$ : best$\_k$ with error rate: $\{$best$\_$error$\_$rate$\}")$