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Nonparametric Regression with Comparisons: Escaping the Curse of Dimensionality with Ordinal Information
Yichong Xu · Hariank Muthakana · Sivaraman Balakrishnan · Aarti Singh · Artur Dubrawski

Wed Jul 11 02:00 AM -- 02:20 AM (PDT) @ A6

In supervised learning, we leverage a labeled dataset to design methodsfor function estimation.In many practical situations, we are able to obtainalternative feedback, possibly at a low cost. A broad goal is to understand the usefulnessof, and to design algorithms to exploit, this alternative feedback.We focus on a semi-supervised setting where we obtain additional ordinal (or comparison) information for potentially unlabeled samples. We considerordinal feedback of varying qualities where we have either a perfect ordering of the samples, a noisy ordering of the samples or noisy pairwise comparisons between the samples.We provide a precise quantification of the usefulness of these types of ordinal feedback in non-parametric regression, showing that in many cases it is possible to accurately estimate an underlying function with a very small labeled set, effectively escaping the curse of dimensionality.We develop an algorithm called Ranking-Regression (RR) and analyze its accuracy as a function of size of the labeled and unlabeled datasetsand various noise parameters.We also present lower bounds, that establish fundamental limits for the task and show that RR is optimal in a variety of settings. Finally, we present experiments that show the efficacy of RR and investigate its robustness to various sources of noise and model-misspecification.

Author Information

Yichong Xu (Carnegie Mellon University)
Hariank Muthakana (Carnegie Mellon University)
Sivaraman Balakrishnan (Carnegie Mellon University)
Aarti Singh (Carnegie Mellon University)
Artur Dubrawski (CMU)

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