On the Sample Complexity of Active Learning with Membership Queries

Efficient learning of user preferences is crucial for many modern decision making systems but typically requires costly labeled data. Active learning reduces this cost, yet standard methods are computationally expensive due to pool-based evaluation. Further, most methods assume all query feedback is equally reliable, ignoring that pairwise queries between nearly identical or entirely dissimilar items yield ambiguous, low-confidence responses. To address the issue of feedback reliability, we introduce a novel confidence aware response model that explicitly accounts for these ambiguous comparisons. To overcome the computational bottleneck of pool-based evaluation, we propose an active query synthesis framework, Info-Synth that generates optimal queries by maximizing a mutual information-based objective within a continuous space. Moreover, we propose two strategies, Pair M-dist and Pair Opt-dist, that extend Info-Synth to select effective queries even when restricted to finite query pools. We demonstrate our framework's versatility and performance across synthetic preference learning, constrained text summary datasets, and subjective, continuous-space controller gain tuning for a simulated mobile robot.

We study a \emph{max-risk} objective for active learning in a multi-group mean estimation $d$-armed bandits: a learner adaptively allocates a budget of $T$ samples across $d$ groups to minimize the worst-case uncertainty index $\max_{k\in[d]}σ_k^2/n_k$, where $σ_k$ is the standard deviation of the distribution of arm $d$, and $n_k$ is the number of times arm $d$ is sampled. We develop a local minimax framework and prove the first general lower bound for this objective, valid for any finite-variance hypothesis class. The bound separates difficulty into three orthogonal factors: a \emph{budget} term, a \emph{heteroscedasticity} index measuring how unevenly the uncertainty is spread across arms, and a model-dependent complexity measure, the \emph{Variance Local Curvature} ($\mathrm{VLC}$), which captures how much information a local change of variance creates inside the hypothesis class. For smooth classes, the $\mathrm{VLC}$ is a reparametrization of a variance--Fisher information, with closed-form values for common families. Benchmarking against the strongest available upper bound shows near-optimality up to logarithmic factors in broad regimes, and pinpoints a systematic gap in highly heterogeneous instances. Our proof introduces two key ingredients: a loss-induced $\ell_1$ geometry on the decision space, and a representation-based instance generator that reduces hard-instance construction to an explicit random matrix calculation.

We study the task of agnostically learning general (as opposed to homogeneous) ReLUs under the Gaussian distribution with respect to the squared loss. In the passive learning setting, recent work gave a computationally efficient algorithm that uses $poly(d,1/ε)$ labeled examples and outputs a hypothesis with error $O(opt)+ε$, where $opt$ is the squared loss of the best fit ReLU. Here we focus on the interactive setting, where the learner has some form of query access to the labels of unlabeled examples. Our main result is the first computationally efficient learner that uses $d polylog(1/ε)+\tilde{O}(\min\{1/p, 1/ε\})$ black-box label queries, where $p$ is the bias of the target function, and achieves error $O(opt)+ε$. We complement our algorithmic result by showing that its query complexity bound is qualitatively near-optimal, even ignoring computational constraints. Finally, we establish that query access is essentially necessary to improve on the label complexity of passive learning. Specifically, for pool-based active learning, any active learner requires $\tildeΩ(d/ε)$ labels, unless it draws a super-polynomial number of unlabeled examples.