The Problem Setup

In classical binary classification, you have labeled positive and negative examples. But what if you only have positives and a pool of unlabeled data — where some unlabeled points are actually positives but you don’t know which?

This is Positive-Unlabeled (PU) learning, and it comes up more often than you might think:

• Identifying fraudulent transactions (confirmed fraud vs. unknown)
• Drug discovery (known active compounds vs. untested molecules)
• Preference learning from implicit feedback (clicked vs. not-clicked-but-maybe-relevant)

The Key Insight: Risk Reformulation

Let $p(x)$ be the positive class density and $u(x)$ the unlabeled density. The unlabeled set is a mixture:

\[u(x) = \pi \cdot p(x) + (1 - \pi) \cdot n(x)\]

where $\pi = P(Y=1)$ is the class prior and $n(x)$ is the negative density.

The standard binary risk can be rewritten as:

\[R(f) = \pi \, \mathbb{E}_p[\ell(f(x), +1)] + (1-\pi) \, \mathbb{E}_n[\ell(f(x), -1)]\]

Since we can estimate $\mathbb{E}_n[\ell]$ from the unlabeled set (after subtracting the positive contribution), PU learning becomes tractable without ever observing negative labels directly.

Non-Negative Risk Estimator

A practical challenge: the estimated risk can go negative during training, causing overfitting. The nnPU estimator (Kiryo et al., 2017) clips the negative part:

\[\hat{R}_{nnPU}(f) = \pi \, \hat{\mathbb{E}}_p[\ell_+] + \max\!\left(0,\, \hat{\mathbb{E}}_u[\ell_-] - \pi \, \hat{\mathbb{E}}_p[\ell_-]\right)\]

This simple fix dramatically stabilizes training and is now the default in most PU implementations.

Connection to My Research

In our work on LLM preference auditing, we model the RLHF annotation process as a PU problem: gold-standard pairwise comparisons form the positive pool, and a large collection of model-generated preferences forms the unlabeled pool. OT then bridges the two, yielding a principled confidence score for each unlabeled preference annotation.

When to Use PU Learning

PU learning is the right tool when:

1. Collecting negative labels is expensive or impossible
2. Your unlabeled data is a realistic mixture of both classes
3. You have at least a rough estimate of the class prior $\pi$

A common pitfall: treating unlabeled as negative. This introduces systematic label noise that biases the classifier toward over-predicting negatives. PU methods avoid this by design.

Libraries

• pulearn (Python): sklearn-compatible PU classifiers
• pu-learning on GitHub: nnPU and unbiased PU implementations in PyTorch