Non-parametric Binary regression in metric spaces with KL loss

TL;DR

This paper introduces a non-parametric binary regression method in metric spaces using Lipschitz functions and KL loss, addressing computational efficiency and statistical challenges with novel algorithms and theoretical bounds.

Contribution

It presents a parameter-free optimization algorithm and an adaptive truncation technique to handle unbounded loss functions in non-parametric binary regression.

Findings

Efficient interior point optimization algorithm developed

Adaptive truncation approach for unbounded loss functions

Lower bounds showing truncation necessity

Abstract

We propose a non-parametric variant of binary regression, where the hypothesis is regularized to be a Lipschitz function taking a metric space to [0,1] and the loss is logarithmic. This setting presents novel computational and statistical challenges. On the computational front, we derive a novel efficient optimization algorithm based on interior point methods; an attractive feature is that it is parameter-free (i.e., does not require tuning an update step size). On the statistical front, the unbounded loss function presents a problem for classic generalization bounds, based on covering-number and Rademacher techniques. We get around this challenge via an adaptive truncation approach, and also present a lower bound indicating that the truncation is, in some sense, necessary.