Corpus-Level Constraints in ML

• Corpus-level constraints are global rules defined over entire datasets that ensure fairness, statistical balance, and logical consistency in model predictions.
• They are enforced using techniques like Lagrangian relaxation and logical query evaluation, which adjust local scores to meet aggregate conditions.
• Applications span bias mitigation in structured prediction, ontology validation in knowledge graphs, and improved evaluation metrics for complex tasks.

Corpus-level constraints constitute a class of global conditions or properties imposed over an entire dataset, corpus, or collection of documents, rather than at the level of a single sample, sentence, or property. Their formal specification and enforcement enables models, evaluation metrics, and databases to maintain global statistical properties, restrict aggregate behavior, or support advanced queries requiring evidence integration across a repository far exceeding the scope of local, instance-specific rules. This paradigm spans structured prediction in vision and NLP, semantic knowledge base validation, and advanced corpus-level question answering—each domain exhibiting both theoretical and practical innovations for definition, specification, and enforcement of such constraints.

1. Formal Definition and Mathematical Foundations

Corpus-level constraints are typically formulated as inequalities, equalities, or logical statements involving globally aggregated statistics over the output or structure of a model or database:

• In structured prediction, corpus-level constraints encapsulate conditions such as demographic ratios or statistical relationships across an entire prediction batch. For instance, in multi-label classification or visual semantic role labeling (vSRL), let $y^i \in Y^i \subseteq \{0,1\}^K$ encode the $K$ binary predictions for the $i$th instance. To constrain gender bias with respect to an object or verb $o$, define $b(o, g) = c(o, g) / \sum_{g'} c(o, g')$ for count $c(o, g)$ in the corpus. The desired test-time constraint enforces $$b^*(o, \text{man}) - \gamma \leq \tilde{b}(o, \text{man}) \leq b^*(o, \text{man}) + \gamma$$ for all $o$, where $b^*$ derives from the training set and $\gamma$ is a margin (Zhao et al., 2017).
• In large-scale QA or knowledge base analysis, a corpus-level constraint is any query whose answer is a function of aggregation over the entire document set or database $C = \{d_1, \ldots, d_N\}$, i.e., $a = A(\{g(d_i, q)\}_{i = 1}^N)$, requiring evidence or computations spanning the corpus rather than reducible to any individual $d_i$ (Lu et al., 21 Jan 2026).
• In semantic knowledge bases, logical frameworks such as eMAPL on MARS encode corpus-level constraints as first-order or counting-quantified formulas over multi-attributed relational structures, enabling specification of global ontological or property relationships, e.g., class disjointness, value uniqueness, or ontological non-circularity (Martin et al., 2020).

These constraints generalize local restrictions by capturing dependencies, balance, or logical relationships that only make sense at aggregate scale.

2. Applications across Machine Learning and Knowledge Representation

2.1. Bias Calibration and Fairness in Structured Prediction

RBA (“Reducing Bias Amplification”) incorporates corpus-level constraints in structured vision models to calibrate output distributions, mitigating bias amplification prevalent in social or demographic labels. CRFs and neural architectures are modified at inference time with a Lagrangian relaxation scheme: a dual objective introduces global constraints as penalty terms, adjusting per-instance local scores to satisfy aggregate corpus-level constraints while retaining baseline recognition accuracy (Zhao et al., 2017). This approach led to $\sim$47.5% and $\sim$40.5% reduction in bias amplification on multilabel classification and vSRL, respectively, with negligible performance tradeoff.

2.2. Global Querying and Semantic Validation in Knowledge Graphs

The MARS/eMAPL framework underlies powerful corpus-level validation of Wikidata and similar knowledge repositories, encoding constraints such as class union/disjointness, value-type restrictions, cardinality, and ontology acyclicity as first-order logical conditions with set and counting extensions. Violations are found by evaluating negative conditions (counterexamples) as queries over the entire corpus, ensuring that global structure and semantics are preserved and errors or inconsistencies are precisely located (Martin et al., 2020).

2.3. Corpus-level Reasoning in Question Answering

CorpusQA formalizes the requirement that some QA tasks require global integration and aggregation across a repository. For example, identifying entities matching a statistical threshold, comparing records across documents, or computing multi-stage aggregate statistics are all defined as corpus-level constraints. Models and benchmarks explicitly test for the ability to handle such queries, revealing that standard retrieval-augmented models fail dramatically as evidence becomes more dispersed, and that only architectures supporting full-corpus holistic synthesis retain meaningful performance (Lu et al., 21 Jan 2026).

3. Enforcement and Inference Methods

3.1. Lagrangian Relaxation for Structured Models

Enforcement in structured prediction proceeds by augmenting inference with dual variables corresponding to corpus-level linear or nonlinear constraints. Given $A$ (constraint matrix) and $b$ (RHS vector), optimization solves

$$\max_{\{y^i\}} \sum_i f_\theta(y^i, i) \quad \text{s.t.} \quad A \sum_{i=1}^N y^i - b \leq 0$$

via coordinate ascent on the Lagrangian:

• For fixed dual variables $\lambda$, solve per-instance inference with modified potentials.
• Update $\lambda$ via projected subgradient ascent using batch statistics until constraints are (nearly) satisfied (Zhao et al., 2017).

This method operates as a “wrapper” requiring only adjustment to local scores, is generally convergent in $50$–$100$ iterations, and supports arbitrary corpus-level linear constraints.

3.2. Logical Query Evaluation in Knowledge Bases

In eMAPL/MARS, constraint satisfaction or violation reduces to efficient evaluation of first-order (plus counting/set atoms) queries over a global database. Compilation to SPARQL or SQL is tractable, and data complexity remains polynomial time, allowing practical application even at knowledge base scale (Martin et al., 2020).

3.3. Hierarchical or Memory-Augmented Reasoning for QA

CorpusQA’s agentic architectures maintain an external memory buffer to iteratively summarize, compare, and aggregate information across corpus partitions, supporting dynamic composition of global aggregation functions ($A$) over local extractors ($g$). This paradigm is essential, as LLM performance for such queries exhibits exponential decay with input length unless advanced memory-augmented mechanisms are applied (Lu et al., 21 Jan 2026).

4. Impact on Evaluation Metrics and Statistical Aggregation

A common use of corpus-level aggregation is in MT system evaluation, where metrics such as BLEU and chrF operate at the corpus scale. However, research exposes critical mathematical and empirical differences between “corpus-level” aggregation (ratio of sums) and sentence-level aggregation (mean of sentence-level scores):

• Corpus-level BLEU/chrF is a length-weighted average, causing long sentences to dominate and yielding poor correlation with human judgments (Pearson $\rho \sim 0.3$–$0.4$).
• Sentence-level aggregation (SLA) ignores length weighting, averaging per-sentence scores and aligning much more closely with human evaluation ($\rho \sim 0.7$–$0.8$), providing direct variance estimates and eliminating the need for expensive bootstrap procedures.
• Corpus-level aggregation is empirically unstable, with low correlation to human and bootstrapped scores as test sets grow.
• The strong recommendation from current research is to abandon corpus-level aggregation of lexical metrics in favor of SLA for robust, interpretable system evaluation (Cavalin et al., 2024).

5. Corpus-Level Constraints in Knowledge Representation: MARS/eMAPL Examples

The expressivity of corpus-level constraints in MARS/eMAPL enables a rich ontology and functional constraint vocabulary, including:

Constraint Type	eMAPL Formula Example (Sketch)	Semantic Purpose
Union of Classes	$\forall u,Q,i$. union_of$(u,Q)\wedge$instance_of$(i,u)\rightarrow\exists c\in Q$ instance_of$(i,c)$	Enforce class composition relationships
Disjointness	$$\forall c_1,c_2.\;\text{disjoint\_with}(c_1, c_2)\rightarrow\neg\exists i.$$ instance_of$(i,c_1)\wedge$instance_of$(i,c_2)$	Prevent overlapping class membership
No-Value Global Ban	$$\forall p,s.\;\text{no\_value}(p,s)\rightarrow\neg\exists o.\;p(s,o)$$	Ban assignment of value to property corpus-wide
Ontology Non-Circularity	$\forall i_1,i_2.\;$ instance_of$(i_1,i_2)\rightarrow\neg$subclass_of$(i_1,i_2)$	Prevent instance–subclass cycles

By compiling such formulas into executable queries over the global knowledge graph, one can check and enforce far-reaching constraints on ontological structure and property distribution, with polynomial-time complexity in data size (Martin et al., 2020).

6. Practical Limitations and Emerging Directions

• Practical deployment requires batch-wise or full-corpus processing to guarantee reliable statistical estimation of empirical ratios or aggregate properties (Zhao et al., 2017).
• Enforcement sensitivity to hyperparameters (e.g., step size in dual updates) necessitates careful calibration for stability.
• As context lengths and corpus sizes scale into the multi-million token regime, even human performance collapses, and only memory-augmented architectural innovations remain viable (Lu et al., 21 Jan 2026).
• There is a critical need for hybrid symbolic-neural approaches integrating program synthesis, NL2SQL, and statistical aggregation to support compositional reasoning over unstructured corpora.

Corpus-level constraints are foundational in ensuring fairness, logical consistency, and trustworthy evaluation in modern machine learning and semantic technologies. They drive methodological advances in large-scale inference, robust evaluation metric design, and principled knowledge base validation. Ongoing work focuses on scalable agentic architectures, logical query languages, and training objectives aligned with global reasoning correctness (Zhao et al., 2017, Cavalin et al., 2024, Martin et al., 2020, Lu et al., 21 Jan 2026).