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Register a new userAn adaptive multiclass nearest neighbor classifier
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We consider a problem of multiclass classification, where the training sample $S_n = {(X_i, Y_i)}_{i=1}^n$ is generated from the model $mathbb P(Y = m | X = x) = eta_m(x)$, $1 leq m leq M$, and $eta_1(x), dots, eta_M(x)$ are unknown $alpha$-Holder continuous functions.Given a test point $X$, our goal is to predict its label. A widely used $mathsf k$-nearest-neighbors classifier constructs estimates of $eta_1(X), dots, eta_M(X)$ and uses a plug-in rule for the prediction. However, it requires a proper choice of the smoothing parameter $mathsf k$, which may become tricky in some situations. In our solution, we fix several integers $n_1, dots, n_K$, compute corresponding $n_k$-nearest-neighbor estimates for each $m$ and each $n_k$ and apply an aggregation procedure. We study an algorithm, which constructs a convex combination of these estimates such that the aggregated estimate behaves approximately as well as an oracle choice. We also provide a non-asymptotic analysis of the procedure, prove its adaptation to the unknown smoothness parameter $alpha$ and to the margin and establish rates of convergence under mild assumptions.
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When data is of an extraordinarily large size or physically stored in different locations, the distributed nearest neighbor (NN) classifier is an attractive tool for classification. We propose a novel distributed adaptive NN classifier for which the number of nearest neighbors is a tuning parameter stochastically chosen by a data-driven criterion. An early stopping rule is proposed when searching for the optimal tuning parameter, which not only speeds up the computation but also improves the finite sample performance of the proposed Algorithm. Convergence rate of excess risk of the distributed adaptive NN classifier is investigated under various sub-sample size compositions. In particular, we show that when the sub-sample sizes are sufficiently large, the proposed classifier achieves the nearly optimal convergence rate. Effectiveness of the proposed approach is demonstrated through simulation studies as well as an empirical application to a real-world dataset.
kNN-MT, recently proposed by Khandelwal et al. (2020a), successfully combines pre-trained neural machine translation (NMT) model with token-level k-nearest-neighbor (kNN) retrieval to improve the translation accuracy. However, the traditional kNN algorithm used in kNN-MT simply retrieves a same number of nearest neighbors for each target token, which may cause prediction errors when the retrieved neighbors include noises. In this paper, we propose Adaptive kNN-MT to dynamically determine the number of k for each target token. We achieve this by introducing a light-weight Meta-k Network, which can be efficiently trained with only a few training samples. On four benchmark machine translation datasets, we demonstrate that the proposed method is able to effectively filter out the noises in retrieval results and significantly outperforms the vanilla kNN-MT model. Even more noteworthy is that the Meta-k Network learned on one domain could be directly applied to other domains and obtain consistent improvements, illustrating the generality of our method. Our implementation is open-sourced at https://github.com/zhengxxn/adaptive-knn-mt.
Previously in 2014, we proposed the Nearest Descent (ND) method, capable of generating an efficient Graph, called the in-tree (IT). Due to some beautiful and effective features, this IT structure proves well suited for data clustering. Although there exist some redundant edges in IT, they usually have salient features and thus it is not hard to remove them. Subsequently, in order to prevent the seemingly redundant edges from occurring, we proposed the Nearest Neighbor Descent (NND) by adding the Neighborhood constraint on ND. Consequently, clusters automatically emerged, without the additional requirement of removing the redundant edges. However, NND proved still not perfect, since it brought in a new yet worse problem, the over-partitioning problem. Now, in this paper, we propose a method, called the Hierarchical Nearest Neighbor Descent (H-NND), which overcomes the over-partitioning problem of NND via using the hierarchical strategy. Specifically, H-NND uses ND to effectively merge the over-segmented sub-graphs or clusters that NND produces. Like ND, H-NND also generates the IT structure, in which the redundant edges once again appear. This seemingly comes back to the situation that ND faces. However, compared with ND, the redundant edges in the IT structure generated by H-NND generally become more salient, thus being much easier and more reliable to be identified even by the simplest edge-removing method which takes the edge length as the only measure. In other words, the IT structure constructed by H-NND becomes more fitted for data clustering. We prove this on several clustering datasets of varying shapes, dimensions and attributes. Besides, compared with ND, H-NND generally takes less computation time to construct the IT data structure for the input data.
Both the median-based classifier and the quantile-based classifier are useful for discriminating high-dimensional data with heavy-tailed or skewed inputs. But these methods are restricted as they assign equal weight to each variable in an unregularized way. The ensemble quantile classifier is a more flexible regularized classifier that provides better performance with high-dimensional data, asymmetric data or when there are many irrelevant extraneous inputs. The improved performance is demonstrated by a simulation study as well as an application to text categorization. It is proven that the estimated parameters of the ensemble quantile classifier consistently estimate the minimal population loss under suitable general model assumptions. It is also shown that the ensemble quantile classifier is Bayes optimal under suitable assumptions with asymmetric Laplace distribution inputs.
Information-theoretic quantities, such as conditional entropy and mutual information, are critical data summaries for quantifying uncertainty. Current widely used approaches for computing such quantities rely on nearest neighbor methods and exhibit both strong performance and theoretical guarantees in certain simple scenarios. However, existing approaches fail in high-dimensional settings and when different features are measured on different scales.We propose decision forest-based adaptive nearest neighbor estimators and show that they are able to effectively estimate posterior probabilities, conditional entropies, and mutual information even in the aforementioned settings.We provide an extensive study of efficacy for classification and posterior probability estimation, and prove certain forest-based approaches to be consistent estimators of the true posteriors and derived information-theoretic quantities under certain assumptions. In a real-world connectome application, we quantify the uncertainty about neuron type given various cellular features in the Drosophila larva mushroom body, a key challenge for modern neuroscience.
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