A Competitive Algorithm for Agnostic Active Learning

We address the problem of active logistic regression in the realizable setting. It is well known that active learning can require exponentially fewer label queries compared to passive learning, in some cases using $\log \frac{1}{\eps}$ rather than $\poly(1/\eps)$ labels to get error $\eps$ larger than the optimum. We present the first algorithm that is polynomially competitive with the optimal algorithm on every input instance, up to a factor $d^2$ and factors polylogarithmic in the error and domain size. In particular, on a simple family of logistic threshold instances with a uniform distribution, our algorithm uses $\polylog(1/\eps)$ labels, whereas every passive learner requires $Ω(1/\eps^2)$ labels. Our algorithm is based on sampling and can be extended to learn more general class of functions. We further support our theoretical results with experiments demonstrating performance gains for logistic regression compared to existing active learning algorithms.

This work revisits a fundamental question in active learning: how powerful is the ability to synthesize arbitrary queries? Compared to pool-based active learning, where the learner only selects queries from a given unlabeled pool, it has been observed that this seemingly mild change in query ability may dramatically alter the difficulty of statistical learning. In particular some hypothesis classes, such as two dimensional halfspaces, permit only polynomial error decay in the number of samples when the learner must draw its labels from the pool, yet become exponentially learnable once synthesized queries are allowed. This striking gap suggests that membership query synthesis induces a fundamentally different mode of learning, one that is not adequately captured by existing active learning theory and calls for new analytical tools to characterize its sample complexity for general hypothesis classes. Motivated by this phenomenon, we develop several sufficient conditions, present intriguing examples, and propose a conjectural perspective toward understanding which hypothesis classes admit efficient learning through synthesized queries.

Active learning reduces labeling costs by selecting samples that maximize information gain. A dominant framework, Query-by-Committee (QBC), typically relies on perturbation-based diversity by inducing model disagreement through random feature subsetting or data blinding. While this approximates one notion of epistemic uncertainty, it sacrifices direct characterization of the plausible hypothesis space. We propose the complementary approach: Rashomon Ensembled Active Learning (REAL) which constructs a committee by exhaustively enumerating the Rashomon Set of all near-optimal models. To address functional redundancy within this set, we adopt a PAC-Bayesian framework using a Gibbs posterior to weight committee members by their empirical risk. Leveraging recent algorithmic advances, we exactly enumerate this set for the class of sparse decision trees. Across synthetic and established active learning baselines, REAL outperforms randomized ensembles, particularly in moderately noisy environments where it strategically leverages expanded model multiplicity to achieve faster convergence.