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A general goal of interactive learning is to investigate broad ways of leveraging human feedback, and understand the benefits of learning from potentially complex feedback. We study a special case of linear regression with access to comparisons between pairs of samples. Learning from such queries is motivated by several important applications, where obtaining comparisons can be much easier than direct labels, and/or when comparisons can be more reliable. We develop an interactive algorithm that utilizes both labels and comparisons to obtain a linear estimator, and show that it only requires a very small amount of direct labels to achieve low error. We also give minimax lower bounds for the problem, showing that our algorithm is optimal up to log factors. Finally, experiments show that our algorithm outperforms label-only algorithms when labels are scarce, and it can be practical for real world applications 

In supervised learning, we leverage a labeled dataset to design methods for function estimation. In many practical situations, we are able to obtain alternative feedback, possibly at a low cost. A broad goal is to understand the usefulness of, 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 consider ordinal 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 datasets and 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.