A PAC-Bayesian Analysis of Distance-Based Classifiers: Why Nearest-Neighbour works!
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Abstract We present PAC-Bayesian bounds for the generalisation error of the K-nearest-neighbour classifier (K-NN). This is achieved by casting the K-NN classifier into a kernel space framework in the limit of vanishing kernel bandwidth. We establish a relation between prior measures over the coefficients in the kernel expansion and the induced measure on the weight vectors in kernel space. Defining a sparse prior over the coefficients allows the application of a PAC-Bayesian folk theorem that leads to a generalisation bound that is a function of the number of redundant training examples: those that can be left out without changing the solution. The presented bound requires to quantify a prior belief in the sparseness of the solution and is evaluated after learning when the actual redundancy level is known. Even for small sample size (m 100) the bound gives non-trivial results when both the expected sparseness and the actual redundancy are high.
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