The Entire Quantile Path of a Risk-Agnostic SVM Classifier

A quantile binary classifier uses the rule: Classify x as +1 if P (Y = 1|X = x) ≥ τ , and as −1 otherwise, for a fixed quantile parameter τ ∈ [0, 1]. It has been shown that Support Vector Machines (SVMs) in the limit are quantile classifiers with τ = 12 . In this paper, we show that by using asymmetric cost of misclassification SVMs can be appropriately extended to recover, in the limit, the quantile binary classifier for any τ . We then present a principled algorithm to solve the extended SVM classifier for all values of τ simultaneously. This has two implications: First, one can recover the entire conditional distribution P (Y = 1|X = x) = τ for τ ∈ [0, 1]. Second, we can build a risk-agnostic SVM classifier where the cost of misclassification need not be known apriori. Preliminary numerical experiments show the effectiveness of the proposed algorithm.