Semi-supervised Active Regression

TL;DR

This paper introduces a semi-supervised active learning framework for linear regression that minimizes label queries by leveraging partially labeled data, achieving near-optimal bounds based on an instance-dependent parameter called reduced rank.

Contribution

The paper formalizes semi-supervised active regression, introduces the reduced rank parameter, and provides an efficient algorithm with optimal query complexity bounds for ridge and kernel ridge regression.

Findings

Proposes an algorithm with query complexity O(R_X/ε)

Establishes matching lower bounds for active ridge regression

Improves bounds for ridge and kernel ridge regression cases

Abstract

Labelled data often comes at a high cost as it may require recruiting human labelers or running costly experiments. At the same time, in many practical scenarios, one already has access to a partially labelled, potentially biased dataset that can help with the learning task at hand. Motivated by such settings, we formally initiate a study of $semi-supervised$ $active$ $learning$ through the frame of linear regression. In this setting, the learner has access to a dataset $X \in \mathbb{R}^{(n_1+n_2) \times d}$ which is composed of $n_1$ unlabelled examples that an algorithm can actively query, and $n_2$ examples labelled a-priori. Concretely, denoting the true labels by $Y \in \mathbb{R}^{n_1 + n_2}$, the learner's objective is to find $\widehat{\beta} \in \mathbb{R}^d$ such that, \begin{equation} \| X \widehat{\beta} - Y \|_2^2 \le (1 + \epsilon) \min_{\beta \in \mathbb{R}^d} \| X \beta - Y \|_2^2 \end{equation} while making as few additional label queries as possible. In order to bound the label queries, we introduce an instance dependent parameter called the reduced rank, denoted by $R_X$, and propose an efficient algorithm with query complexity $O(R_X/\epsilon)$. This result directly implies improved upper bounds for two important special cases: (i) active ridge regression, and (ii) active kernel ridge regression, where the reduced-rank equates to the statistical dimension, $sd_\lambda$ and effective dimension, $d_\lambda$ of the problem respectively, where $\lambda \ge 0$ denotes the regularization parameter. For active ridge regression we also prove a matching lower bound of $O(sd_\lambda / \epsilon)$ on the query complexity of any algorithm. This subsumes prior work that only considered the unregularized case, i.e., $\lambda = 0$.