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

TL;DR

It is shown that tuple counts admit a trainable unbiased risk estimator (URE) by linking the tuple-generation process to latent instance marginals and introducing simple ReLU corrections to the URE that preserve asymptotic correctness.

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.

Figures (4)

• Figure 1: Tuples of length $n{=}3$ with $m{=}1$. Horizontal axis: class prior $\pi$; vertical axis: metric value. As $\pi$ increases, TPR, Macro-$F_{1}$, and Precision remain stable or improve across datasets, while FPR decreases markedly, indicating that prior shifts predominantly influence false positives.
• Figure 2: Robustness & conditioning on MNIST($S{=}3$ seeds; solid lines = mean; shaded = $\pm\,1$ std). We report COR (accuracy), TPR, FPR, F1. (a) Prior sweep around the estimate: $\delta := \tilde{\pi}-\hat{\pi} \in [-0.30,0.30]$; COR/F1 change marginally and FPR stays near zero. (b) Tuple-count noise: with probability $p$ each tuple is perturbed by $m\mapsto m\pm1$; FPR rises roughly linearly while COR/TPR/F1 decrease monotonically. (c) Identifiability gap analysis: sweep $\Delta_{\alpha} := |\pi-\alpha|$ by varying $\pi$ over the shown range (fixing $\alpha$); metrics remain largely stable with small fluctuations and consistently low FPR, except in the immediate vicinity of $\Delta_{\alpha}\!\approx\!0$ (ill-conditioned boundary).
• Figure 3: Class prior $\pi\!=\!0.2$, tuples $(n,m)\!=\!(3,1)$. Horizontal axis: training iterations; vertical axis: accuracy. Mean $\pm$ std over five trials. The plain URE may overfit (accuracy spikes then decays), whereas the corrected variants stabilize training and improve the final accuracy (see text).
• Figure 4: Metrics and stability under identifiability gap ($\Delta_{\alpha}$) on MNIST. Horizontal axis: $\Delta_{\alpha}:=|\pi-\alpha|$, where $\pi$ is the prior used in training for that run and $\alpha=m/n$ is the tuple positive rate of the $(n,m)$ configuration. We compare URE+ABS against URE. Markers show mean over $S{=}3$ seeds; vertical bars denote 95% bootstrap CIs. The red dashed line indicates the CI–based critical gap ${\Delta_{\alpha}}_{\text{crit}}^{\text{CI}}$, defined as the smallest $\Delta_{\alpha}$ at which the 95% CI width of AP first exceeds the preset threshold $w^\star{=}0.05$. (a) Best-$F_1$ vs. $\Delta_{\alpha}$. (b) AUROC vs. $\Delta_{\alpha}$. (c) AP vs. $\Delta_{\alpha}$. (d) Stability vs. $\Delta_{\alpha}$, measured by across-seed std of AP (larger is less stable). Across the $\Delta_{\alpha}$-sweep, URE+ABS maintains high accuracy with near-zero dispersion, whereas plain URE degrades and becomes increasingly variable as the prior–tuple mismatch grows.

Theorems & Definitions (11)

• Lemma 1: Tuple density as an equally weighted mixture
• Theorem 2: Flattened marginal
• Lemma 3: Identification from $(\tilde{p}_{T},p_U)$
• Theorem 4: Unbiased risk representation for NTMP
• Lemma 5: Uniform in-tuple averaging is variance-minimizing
• Theorem 6: Identifiability with heterogeneous tuples
• Theorem 7: Generalization bound and rate
• proof : Proof sketch
• Theorem 8: Risk bias under misspecified class prior
• Theorem 9: Risk bias under random count misspecification