Verifying the Selected Completely at Random Assumption in Positive-Unlabeled Learning

TL;DR

This work tackles the validity of the SCAR assumption in Positive-Unlabeled (PU) learning by proposing a statistically principled, computationally efficient test. The method first estimates a putative positive set from PU data, then generates artificial SCAR labels to approximate the null distribution of a chosen test statistic, enabling a p-value-based decision between SCAR and SAR. The authors establish theoretical guarantees for type I error control and consistency under idealized conditions, and demonstrate through extensive experiments that the test detects SAR deviations while maintaining nominal error rates across diverse datasets. Practically, the approach serves as a pre-processing step to select appropriate PU algorithms (SCAR-friendly or SAR-aware) and is supported by robust empirical evidence favoring the KS-based statistic for reliable performance.

Abstract

The goal of positive-unlabeled (PU) learning is to train a binary classifier on the basis of training data containing positive and unlabeled instances, where unlabeled observations can belong either to the positive class or to the negative class. Modeling PU data requires certain assumptions on the labeling mechanism that describes which positive observations are assigned a label. The simplest assumption, considered in early works, is SCAR (Selected Completely at Random Assumption), according to which the propensity score function, defined as the probability of assigning a label to a positive observation, is constant. On the other hand, a much more realistic assumption is SAR (Selected at Random), which states that the propensity function solely depends on the observed feature vector. SCAR-based algorithms are much simpler and computationally much faster compared to SAR-based algorithms, which usually require challenging estimation of the propensity score. In this work, we propose a relatively simple and computationally fast test that can be used to determine whether the observed data meet the SCAR assumption. Our test is based on generating artificial labels conforming to the SCAR case, which in turn allows to mimic the distribution of the test statistic under the null hypothesis of SCAR. We justify our method theoretically. In experiments, we demonstrate that the test successfully detects various deviations from SCAR scenario and at the same time it is possible to effectively control the type I error. The proposed test can be recommended as a pre-processing step to decide which final PU algorithm to choose in cases when nature of labeling mechanism is not known.

Figures (4)

• Figure 1: Visualization of SCAR and SAR settings. Under the SCAR assumption, the probability of labeling positive observations does not depend on the feature vector while under SAR this probability depends on the features.
• Figure 2: The visualization shows how in Algorithm \ref{['Alg1']} artificial labels $\widetilde{S}$ matching the SCAR assumption are generated.
• Figure 3: Probability of rejecting $H_0$ with respect to sample size $n$ for artificial data sets 1 and 2, labeling strategy S1 and for $c=0.5$. Value $g=0$ corresponds to $H_0$ and $g>0$ to $H_1$.
• Figure 4: Probability of rejecting $H_0$ with respect to parameter $g$ for selected tabular and image datasets, for $c=0.5$. Value $g=0$ corresponds to $H_0$ (SCAR), whereas $g>0$ to $H_1$ (SAR).

Theorems & Definitions (7)

• Theorem 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3