Learning from N-Tuple Data with M Positive Instances: Unbiased Risk Estimation and Theoretical Guarantees

TL;DR

This paper introduces an unbiased risk estimator for learning from n-tuple data with only positive counts, providing theoretical guarantees and demonstrating improved performance over existing weak supervision methods.

Contribution

The authors develop a novel unbiased risk estimator for NTMP supervision, extending it to variable tuple sizes and counts, with proven consistency and generalization bounds.

Findings

Outperforms baseline weak supervision methods on NTMP tasks

Remains robust under class imbalance and diverse tuple configurations

Provides theoretical guarantees including consistency and generalization bounds

Abstract

Weakly supervised learning often operates with coarse aggregate signals rather than instance labels. We study a setting where each training example is an $n$-tuple containing exactly m positives, while only the count m per tuple is observed. This NTMP (N-tuple with M positives) supervision arises in, e.g., image classification with region proposals and multi-instance measurements. We show that tuple counts admit a trainable unbiased risk estimator (URE) by linking the tuple-generation process to latent instance marginals. Starting from fixed (n,m), we derive a closed-form URE and extend it to variable tuple sizes, variable counts, and their combination. Identification holds whenever the effective mixing rate is separated from the class prior. We establish generalization bounds via Rademacher complexity and prove statistical consistency with standard rates under mild regularity assumptions. To improve finite-sample stability, we introduce simple ReLU corrections to the URE that preserve asymptotic correctness. Across benchmarks converted to NTMP tasks, the approach consistently outperforms representative weak-supervision baselines and yields favorable precision-recall and F1 trade-offs. It remains robust under class-prior imbalance and across diverse tuple configurations, demonstrating that count-only supervision can be exploited effectively through a theoretically grounded and practically stable objective.