Probing Classifiers › Control Tasks

The most important methodological innovation in probing research is the control task, introduced by Hewitt and Lisserman (2019). Control tasks address a fundamental question: if a probe achieves 90% accuracy, does that tell us something interesting about the model, or does it just tell us that logistic regression is good at this task regardless of what representations you give it?

To understand why this question is sharp, consider a 768-dimensional representation space with only 80 training examples and 6 classes. The probe has $768 \times 6 = 4{,}608$ parameters plus biases, nearly 60 parameters per training example. In this regime, the probe has enough capacity to perfectly separate any 80 points in 768 dimensions by random chance. If the probe can memorize arbitrary labels with high accuracy, then high accuracy on real labels tells you nothing specific about what BERT learned. The control task is designed to measure this memorization capacity directly.

The idea behind a random control task is elegant. Assign random labels to the training examples, drawn from the same label distribution as the real task but with no semantic connection to the input. Then train the probe on these random labels using the model's representations. Measure the accuracy.

If the probe achieves high accuracy even with random labels, it means either the representations are so high-dimensional that the probe can memorize the training set (overfitting), or the probe architecture is too expressive and can extract arbitrary patterns from the representations.

If accuracy on random labels is near chance (i.e., the probe can't overfit), but accuracy on real labels is high, you have stronger evidence that the probe is extracting meaningful information from the representations.

The control task thus defines a baseline. The selectivity metric captures how much of the probe's accuracy comes from linguistic signal rather than from the probe's capacity to memorize:

$$\text{selectivity} = \text{accuracy}_{\text{task}} - \text{accuracy}_{\text{control}}$$

where:

• $\text{accuracy}\_{\text{task}}$: probe accuracy trained on real linguistic labels (e.g., POS tags)
• $\text{accuracy}\_{\text{control}}$: probe accuracy trained on randomly assigned labels drawn from the same label distribution

A high selectivity score means the probe succeeds specifically because of what the real labels capture about the input, not because of general memorization capacity. If the probe could memorize arbitrary labels equally well, the two accuracies would be similar and selectivity would approach zero.

The selectivity score has an intuitive interpretation in information-theoretic terms. The control accuracy measures how much the representations help with any classification task of this type (because they span a rich 768-dimensional space that a linear classifier can overfit). The real task accuracy measures how much they help with the specific linguistic task. The difference is specifically what those linguistic labels capture above and beyond what any labels would capture.

A related control is label remapping, where the actual labels are deterministically shuffled by a fixed permutation. Unlike random assignment, this preserves the label distribution but destroys the semantic relationship between input and label. Checking that the probe achieves lower accuracy under remapping than under real labels confirms the probe is learning something beyond label frequency statistics.

Label remapping is particularly useful when the label distribution is skewed. If nouns make up 40% of tokens and the probe achieves 60% accuracy on a random control (by predicting "noun" everywhere), the selectivity metric might be misleading without careful accounting for the majority class.

Another control is to train the probe on representations from a randomly initialized (untrained) model. If a probe achieves comparable accuracy using random model representations as using pretrained model representations, it suggests the task is trivially easy from input features, not that the model has learned anything meaningful.

This is particularly important for tasks like named entity type prediction, where the surface form of words (capitalization, character patterns) already carries strong signal. A randomly initialized model's embedding layer might encode capitalization information in a simple lookup-table way that is sufficient to solve the task. Comparing pretrained model accuracy against random model accuracy tells you how much the pretraining contributed beyond what character-level features can provide.

Understanding control tasks geometrically is illuminating. The representation space for BERT-base is 768-dimensional. A logistic regression probe defines a hyperplane in this space. In 768 dimensions, you can always find a hyperplane that separates a small number of points, regardless of their labels. This is a consequence of the fact that $n$ points in general position in $d$-dimensional space are linearly separable as long as $n \leq d + 1$.

With 80 training examples and 768 dimensions, we are nowhere near the regime where linear separability fails. The probe can memorize any labeling. The control task explicitly measures this memorization capacity. The selectivity gap tells you: beyond the general separability of points in high-dimensional space, how much does the specific geometry of BERT's representations (shaped by pretraining on English text) help predict linguistic labels?