Minimax Optimal Algorithms with Fixed-$k$-Nearest Neighbors

TL;DR

This work tackles minimax-optimal learning using fixed-$k$ nearest neighbors in a distributed setting by splitting data into $M$ groups and aggregating the $k$-NN information. It introduces M-split ($k,M$) rules for classification, regression, and density estimation, plus distance-based selective refinements that remove logarithmic gaps and achieve near-optimal or exact minimax rates under standard regularity (doubling, Hölder, margin) conditions. A key contribution is showing that fixed-$k$ rules across many splits can match the statistical power of growing-$k$ rules while delivering substantial computational and storage benefits, especially in large-scale or distributed environments. The paper also provides empirical validation on synthetic and real data, demonstrating competitive performance and practical speedups, and discusses variants and connections to prior ensemble methods like BigNN and denoising approaches. Overall, the results offer a principled, scalable path to minimax-optimal NN-based learning in distributed systems, with clear guidance on parameter choices (notably fixing $k$ and tuning $M$).

Abstract

This paper presents how to perform minimax optimal classification, regression, and density estimation based on fixed-$k$ nearest neighbor (NN) searches. We consider a distributed learning scenario, in which a massive dataset is split into smaller groups, where the $k$-NNs are found for a query point with respect to each subset of data. We propose \emph{optimal} rules to aggregate the fixed-$k$-NN information for classification, regression, and density estimation that achieve minimax optimal rates for the respective problems. We show that the distributed algorithm with a fixed $k$ over a sufficiently large number of groups attains a minimax optimal error rate up to a multiplicative logarithmic factor under some regularity conditions. Roughly speaking, distributed $k$-NN rules with $M$ groups has a performance comparable to the standard $Θ(kM)$-NN rules even for fixed $k$.

Figures (5)

• Figure 1: Maximum allowed ratio $\frac{L}{M}$ indicated by our theory for different $k$'s, when $k$ is kept fixed. This plot summarizes the information in Fig. \ref{['fig:good_gamma_tau']} in Appendix.
• Figure 2: Summary of the excess risks of the NN classifiers for the mixture of two Gaussians experiments in Section \ref{['sec:exp_synthetic']}.
• Figure 3: Validation error profiles from 10-fold cross validation. Here, as expected, the optimal $M$ chosen for $(1,M)$-NN rules is in the same order of the optimal $k$ for the standard $k$-NN rules.
• Figure 4: Sample size vs. mean squared error plots for $K$-NN and $(k,K/k)$-NN density estimation rules for $k\in\{1,2,3,4,5\}$ over columns for mixture of Gaussian distributions of dimension $d\in \{1,2,3,4,5\}$. Here, $K$ was chosen as $\Theta(N^{\frac{2\sigma}{d+2\sigma}})$, where $\sigma=2$ for mixture of Gaussians was plugged in.
• Figure 1.1: Visualization of (a) the allowed pairs of $(\gamma,\tau)$ for each fixed $k\ge 1$ and (b) the corresponding selection factors. The maximum values of the selection factors for each $k$ in (b) are summarized in Fig. \ref{['fig:maximum_allowed_factor']}.

Theorems & Definitions (48)

• Theorem 1: Cover--Hart1967
• Lemma 2: Cover--Hart1967
• Theorem 3: Loftsgaarden--Quesenberry1965
• Proposition 4
• Theorem 5: Regression
• Theorem 6: Classification
• Remark 7: Reduction to regression
• Theorem 8: Qiao--Duan--Cheng2019, rephrased
• Corollary 9: Regression
• Corollary 10: Classification