In k-nearest neighbors, what impact does increasing k have on the variance of the model?

Increasing k in the k-nearest neighbors (KNN) algorithm has the effect of decreasing the variance of the model. This is because a larger k means that the model considers a greater number of neighbors to make predictions, leading to a more generalized decision boundary. With more neighbors influencing the output, the impact of individual data points is reduced, which tends to smooth out the model's predictions and make them less sensitive to noise in the training data.

By averaging over more neighbors, the likelihood of overfitting to particular nuances in the dataset decreases. For instance, in situations where the training data is sparse or contains outliers, a small k could lead to a high variance model that captures random fluctuations rather than the underlying trend. As k increases, the model tends to focus more on the overall distribution of the data rather than the peculiarities of individual samples, thus improving generalization and stability.

The relationship between k and variance is a critical consideration in the tuning of KNN, where the goal is often to balance bias and variance to achieve optimal predictive performance.