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Information-Theoretic Probing with Minimum Description Length

Elena Voita, Ivan Titov

arXiv:2003.12298v1cs.CL

TL;DR

Standard probe accuracy does not reliably reflect differences between representations or between genuine and random tasks. The paper replaces accuracy with minimum description length, using variational and online coding to measure both predictive quality and the effort required to achieve it. These methods agree in their results and are more informative and stable than standard probes.

  • Problem

    Probe accuracy can fail to distinguish pretrained from randomly initialized representations and genuine linguistic labels from random control tasks.

  • Method

    Information-theoretic probing measures the minimum description length of labels given representations, including model-transmission cost, using variational and online coding.

  • Results

    Variational and online coding agree in their results, and MDL probing is more informative and stable than standard probes.

  • Takeaways & Limitations

    MDL naturally characterizes probe quality together with the effort needed to achieve it without manual search for probe settings.

  • Takeaways & Limitations

    The work refers to theoretical codelengths of associated encodings rather than practical implementations of transmission algorithms.

Abstract

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To measure how well pretrained representations encode some linguistic property, it is common to use accuracy of a probe, i.e. a classifier trained to predict the property from the representations. Despite widespread adoption of probes, differences in their accuracy fail to adequately reflect differences in representations. For example, they do not substantially favour pretrained representations over randomly initialized ones. Analogously, their accuracy can be similar when probing for genuine linguistic labels and probing for random synthetic tasks. To see reasonable differences in accuracy with respect to these random baselines, previous work had to constrain either the amount of probe training data or its model size. Instead, we propose an alternative to the standard probes, information-theoretic probing with minimum description length (MDL). With MDL probing, training a probe to predict labels is recast as teaching it to effectively transmit the data. Therefore, the measure of interest changes from probe accuracy to the description length of labels given representations. In addition to probe quality, the description length evaluates "the amount of effort" needed to achieve the quality. This amount of effort characterizes either (i) size of a probing model, or (ii) the amount of data needed to achieve the high quality. We consider two methods for estimating MDL which can be easily implemented on top of the standard probing pipelines: variational coding and online coding. We show that these methods agree in results and are more informative and stable than the standard probes.

1 Introduction

Standard probe accuracy often fails to distinguish pretrained representations from random baselines or genuine linguistic tasks from random controls. The paper proposes MDL probing, which measures label description length while accounting for the effort required to achieve probe quality.

  • Motivation: Standard probe accuracy can be similar for pretrained and randomly initialized representations, or for genuine labels and random control tasks.Previous work made these differences clearer by restricting probe data or model size.
  • Information-theoretic probing: MDL probing measures the minimum number of bits needed to transmit labels given representations, including the cost of transmitting the model.This combines compressed data length with model-transmission cost.
  • Interpretation: Codelength captures both probe quality and the effort needed to achieve it, such as probe complexity or the amount of data needed to reveal structure.Clearer representation–label structure can be communicated with a simpler model or less data.
  • MDL estimators: Variational coding includes probe-weight transmission cost, whereas online coding measures learning from progressively larger data portions.Both methods estimate MDL, with variational coding using a variational loss and online coding transmitting data in sequence.
  • Empirical effectiveness: MDL architectures distinguish linguistic and random control tasks despite similar accuracies, using smaller models for linguistic tasks and larger models for controls.These architectures emerge from MDL optimization rather than manual search.
  • Contributions: The paper reports that MDL probing is more informative and stable than standard probes.The proposed approach is designed to measure MDL on top of standard probe-training pipelines.

2 Information-Theoretic Viewpoint

The paper reframes probing as transmitting labels given representations, measuring minimum description length rather than accuracy alone. Total codelength combines predictive fit with the effort required to communicate or learn the probing model, estimated using variational and online codes.

  • Information-theoretic probing: MDL probing measures the minimum number of bits needed to transmit labels given representations, rather than probe accuracy alone.Because labels are transmitted using a model, the total codelength includes both compressed data and model transmission costs.
  • Information-theoretic probing: Total codelength reflects both probe quality and the effort required to achieve it, including model simplicity and the amount of data needed to reveal structure.Representations with clearer label-related structure can be explained using a simpler model or less training data.
  • Information-theoretic interpretation: The compression gain is bounded by mutual information, but effective representations also yield simple models predicting labels from inputs.High mutual information is necessary for compression, whereas model codelength distinguishes representations that require simpler versus more complex predictive rules.
  • Two MDL estimators: Variational coding explicitly transmits model parameters before encoding labels, while online coding encodes successive label blocks using models learned from earlier data.Variational coding incorporates model cost directly; online coding avoids directly transmitting model weights.
  • Two MDL estimators: Online codelength rewards probes that perform well with few examples and is related to the area under the learning curve.Early-learned regularities allow later data blocks to be transmitted more efficiently.
  • Connection to previous probing: The variational and online codes agree in their model-cost trends, linking efficient learning from little data and efficient prediction with a small architecture to the strength of data regularity.Earlier constraints on training-data amount or probe size can therefore be understood as indirect ways of accounting for effort.

3 Description Length and Control Tasks

The experiments compare MDL-based description length with probe accuracy on linguistic and random control tasks. MDL distinguishes the tasks, reflects model and data effort, and remains more stable across compression methods, settings, and random seeds.

  • Experimental setup: PoS probes use ELMo representations and compare linguistic labels with randomly sampled word-type control labels.The control task samples each word type’s PoS tag independently from the empirical linguistic distribution.
  • Description-length results: Codelengths for control tasks are at least twice larger than for linguistic tasks, while both compression methods show similar trends.The linguistic task favors the first layer; control-task codelength increases above the embedding layer.
  • Description-length results: For LAYER 0, control-task accuracy is higher but its code is twice longer than the linguistic task’s code.MDL captures the greater cost required to achieve the control task’s accuracy.
  • Description-length results: The embedding-layer codelength for the linguistic task is approximately twice that of the first layer.The passage reports the same trend across several other tasks later in the paper.
  • Description-length results: Linguistic tasks induce smaller probe models, with 33–75 neurons in the second and third layers, whereas control tasks induce denser models.Variational coding produces these architectures during training rather than through manual architecture selection.
  • Stability and reliability: MDL is stable across probe settings and random seeds, unlike accuracy, which can change rankings and even reverse linguistic–control comparisons.For one seed, LAYER 1 and LAYER 2 accuracies are 97.48 and 97.31; for another, they are 97.38 and 97.48.

4 Description Length and Random Models

The experiments compare trained and randomly initialized ELMo representations using MDL across edge probing tasks. Codelength consistently distinguishes contextualized representations from lexical and random representations, while also revealing how representations evolve across layers.

  • Trained vs random: Randomly initialized representations perform surprisingly strongly with accuracy, but MDL shows large codelength differences between trained and randomly initialized representations.This comparison addresses the limitation that accuracy alone does not reflect what representations encode.
  • Experimental setup: The evaluation covers PoS tagging, constituency and dependency labeling, named entity recognition, semantic role labeling, coreference resolution, and relation classification.The experiments use edge probing tasks with representations from ELMo trained on the Billion Word Benchmark.
  • Experimental setup: The probing setup uses span representations formed by projection and self-attention pooling, then concatenates them before a two-layer MLP and softmax.The architecture accesses contextual vectors and one or two integer spans, depending on the task.
  • LAYER 0 vs contextual: Contextualized representations have codelengths about twice smaller than embedding-layer representations for most tasks, including in randomly initialized models.The result is attributed to greater mutual information between labels and contextualized representations than between labels and context-agnostic embeddings.
  • Layer evolution: For randomly initialized models, MDL is identical across layers for all tasks, whereas trained-model layers differ in codelength.The trained model's Layer 2 is worse than Layer 1, while random layers do not evolve.

5 Related work

Related work positions probing classifiers as the standard tool for linking neural representations to linguistic properties, while highlighting limitations in interpreting their behavior. Other approaches analyze learning curves, model mechanisms, neurons, or predictions through templates.

  • Probing classifiers: Probing classifiers are commonly used to associate neural network representations with linguistic properties, and Belinkov and Glass provide a comprehensive survey.The passage also cites work arguing that diagnostic classifiers are unsuitable for understanding learning dynamics.
  • Learning curves: Learning curves have been used to evaluate how quickly models learn tasks and whether performance reflects pretrained representations or fine-tuning.The cited examples include Yogatama et al. and Talmor et al.
  • Other analysis methods: Other model-analysis methods inspect attention weights or individual neurons, and evaluate predictions with manually defined templates.These approaches examine mechanisms or sensitivity to specific grammatical errors.

6 Conclusions

The paper proposes information-theoretic probing based on MDL to evaluate representations through both predictive quality and the effort required to achieve it. It reports that MDL probing is easy to add to standard pipelines and yields more informative and stable results than standard probes.

  • Contribution: Information-theoretic probing measures the minimum description length of labels given representations.The proposal reframes probing as an information-theoretic measurement rather than standard accuracy evaluation.
  • Contribution: MDL characterizes probe quality together with the effort needed to achieve it, including the complexity of the probing model or the amount of data required.The paper presents this as a theoretically justified account without manual search for settings.
  • Results: MDL probing results are more informative and stable than those of standard probes.The conclusion reports this comparison at the level of probing results rather than a single task or metric.

A.1 Settings

The appendix reports experimental results for linguistic and control tasks using codelength and compression measures. Results are organized in paired comparisons, with probe hidden-layer size included as an additional setting.

  • Results format: Table 8 reports results in linguistic-task/control-task pairs.This layout compares the two task types directly within each reported result.
  • Results format: Codelength in Table 8 is measured in kbits using variational or online coding.The table identifies the coding method associated with each codelength.
  • Results format: The variable h denotes the probe hidden-layer size in Table 8.The table includes hidden-layer size as part of the reported configuration.

A.2 Random seeds: control task

Figure 5 presents control-task results across five random seeds under the default MLP-2 setting with h = 1000.

  • Figure 5 reports control-task results for 5 random seeds.
  • The figure uses the default MLP-2 setting.
  • The default setting specifies h = 1000.

B Description Length and Random Models

The appendix reports pruned architectures of trained variational probes across seven linguistic tasks, documenting each task's starting probe architecture.

  • Table 9 reports a pruned trained variational probe for Part of Speech, starting from 1024-256-256.
  • Tables 10–12 report pruned trained variational probes for constituent labeling, dependency labeling, and named entity recognition.The starting probes are 1024-256-256 for constituent labeling, (1024+1024)-512-256 for dependency labeling, and 1024-256-256 for named entity recognition.
  • Tables 13–15 report pruned trained variational probes for semantic role labeling, coreference resolution, and relation classification.Each starts from (1024+1024)-512-256.
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