A study of logistic classifier: uniform consistency in finite-dimensional linear spaces

Abstract

Let X be a random variable taking values in a finite dimensional linear space and Y ∈ {0, 1} its associated label. We study the case, where conditional distribution p(x) = P(Y = 1 | X = x) depends on x through some linear form θx. We show that in this case, under a mild assumption on the distribution µ of X, a maximum-likelihood estimator pˆ, as well as the induced class of logistic classifiers, are uniformly (w.r.t. p) consistent.