Minimax Rate Optimal Adaptive Nearest Neighbor Classification and Regression
This work addresses a theoretical gap in kNN methods for statistical learning, improving performance in scenarios with unbounded feature distributions, though it is incremental as it builds on existing minimax optimality results.
The authors tackled the suboptimal convergence rate of standard k-nearest neighbor (kNN) methods for classification and regression when feature distributions have unbounded support, proposing an adaptive kNN method that selects different k for different samples without requiring precise knowledge of the underlying distribution, and showed it matches the minimax lower bound, significantly outperforming the standard method.
k Nearest Neighbor (kNN) method is a simple and popular statistical method for classification and regression. For both classification and regression problems, existing works have shown that, if the distribution of the feature vector has bounded support and the probability density function is bounded away from zero in its support, the convergence rate of the standard kNN method, in which k is the same for all test samples, is minimax optimal. On the contrary, if the distribution has unbounded support, we show that there is a gap between the convergence rate achieved by the standard kNN method and the minimax bound. To close this gap, we propose an adaptive kNN method, in which different k is selected for different samples. Our selection rule does not require precise knowledge of the underlying distribution of features. The new proposed method significantly outperforms the standard one. We characterize the convergence rate of the proposed adaptive method, and show that it matches the minimax lower bound.