ICML2025 Image Generation representative generation language generation in the limit group closure dimension fairness in generative models mode collapse diversity

Representative Language Generation¶

Conference: ICML2025
arXiv: 2505.21819
Code: None (purely theoretical work)
Area: Language Generation Theory / Algorithmic Fairness
Keywords: representative generation, language generation in the limit, group closure dimension, fairness in generative models, mode collapse, diversity

TL;DR¶

Proposes a theoretical framework of "representative generation", which requires the outputs of generative models to proportionally represent various groups of interest in the training data, and introduces "group closure dimension" as a key combinatorial quantity to characterize generatability.

Background & Motivation¶

In recent years, generative models (LLMs, diffusion models, etc.) have achieved great success, but issues of insufficient diversity and bias have become increasingly prominent:

Mode collapse: Generative models tend to overrepresent certain groups while ignoring minority groups.
Social bias: Language models exhibit systematic biases in gender, race, etc., in their generated content.
Theoretical gap: Kleinberg & Mullainathan (2024) proposed the "generation in the limit" framework, and Li et al. (2024) further formalized it, but neither considered fairness/diversity constraints.

The core motivation of this work is to incorporate representativeness constraints into the generative theory framework, requiring the output distribution of the generative model to proportionally reflect the share of each group in the training data, thereby studying the feasibility boundaries of fair generation at a theoretical level.

Method¶

Basic Setup¶

Let \(\mathcal{X}\) be a countable sample space, and \(\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}\) be a hypothesis class. The support of each \(h \in \mathcal{H}\) is defined as:

\[\text{supp}(h) := \{x \in \mathcal{X} : h(x) = 1\}\]

Introduce a set of groups of interest \(\mathcal{A} \subseteq 2^{\mathcal{X}}\) over \(\mathcal{X}\), which is typically a countable partition of \(\mathcal{X}\), \(\mathcal{A} = \{A_1, A_2, \ldots\}\).

Definition of Representative Generation¶

Given a tolerance parameter \(\alpha > 0\), a generator \(\mathcal{G}\) achieves \(\alpha\)-representative generation if for all time steps \(t\):

\[\|\mathcal{G}(x_{1:t})|_{\mathcal{A}} - \overline{x_{1:t}}|_{\mathcal{A}}\|_{\infty} \leq \alpha\]

where \(\overline{x_{1:t}}|_{\mathcal{A}}(i)\) is the proportion of samples in the input sequence belonging to group \(A_i\), and \(\mathcal{G}(x_{1:t})|_{\mathcal{A}}(i)\) is the probability mass of the generator's output distribution on group \(A_i\).

Group Closure Dimension¶

The core contribution of this work is the introduction of the Group Closure Dimension \(\text{GC}_\alpha(\mathcal{H}, \mathcal{A})\):

For a given set of \(d\) distinct samples \(x_1, \ldots, x_d\), the closure \(\langle x_1, \ldots, x_d \rangle_{\mathcal{H}}\) is defined as the intersection of the supports of all hypotheses consistent with these samples. The group closure dimension is the maximum number of samples \(d\) satisfying certain "blocked group" conditions.

When \(\text{GC}_\alpha(\mathcal{H}, \mathcal{A}) < \infty\), \(\alpha\)-representative uniform generation is feasible.
When \(\text{GC}_\alpha(\mathcal{H}, \mathcal{A}) = \infty\), \(\alpha\)-representative uniform generation is infeasible.

Representative Generation in the Limit¶

For generation in the limit:

Information-theoretic level: For countably infinite hypothesis classes and group sets, representative generation in the limit is feasible under certain conditions.
Computational level: When using only membership queries, representative generation in the limit is non-computable—which stands in stark contrast to the positive results for standard generation in the limit in Kleinberg et al. (2024).

Key Differences with Standard Generation¶

Property	Standard Generation	Representative Generation
Uniform Generation	Characterized by closure dimension	Characterized by group closure dimension
Generation in the Limit (Information-Theoretic)	Feasible for countable classes	Feasible for countable classes under conditions
Generation in the Limit (Computational)	Computable with membership queries	Non-computable with membership queries

Key Experimental Results¶

This work is a purely theoretical work with no experimental data. The main results are presented in the form of theorems:

Theorem	Content	Significance
Theorem (Characterization of Uniform Generation)	\(\alpha\)-representative uniform generation \(\Leftrightarrow\) \(\text{GC}_\alpha(\mathcal{H}, \mathcal{A}) < \infty\)	Complete characterization, necessary and sufficient condition
Theorem (Non-uniform Generation)	Similar characterizations extended to non-uniform cases	More general setup
Theorem (Positive Result for Generation in the Limit)	Countable classes are representatively generatible in the limit under specific conditions	Information-theoretic feasibility
Theorem (Negative Result for Generation in the Limit)	Representative generation in the limit is non-computable using only membership queries	Computational barrier

Highlights & Insights¶

Group closure dimension as an exact characterization: Similar to how VC dimension functions in learning theory, the group closure dimension precisely characterizes the feasibility boundary of representative generation.
Fairness introduces an inherent computational cost: While standard generation in the limit can be achieved using membership queries, adding representativeness constraints makes it non-computable, revealing a deep tension between fairness and computability.
Connecting generation theory and fairness theory: Bridges fairness concepts such as multicalibration and outcome indistinguishability with generation theory.
Formalizing the mode collapse problem: Representativeness constraints are essentially a theoretical characterization of mode collapse, providing a theoretical foundation for practical mitigation.

Limitations & Future Work¶

Purely theoretical framework: All results are established under formal theoretical settings and lack experimental validation with actual generative models (such as GPT or diffusion models).
Countable space assumption: In practical applications, the sample space is typically continuous (e.g., image space), which limits the applicability of the countable assumption.
Strong partition assumption: It assumes that the groups form a partition of \(\mathcal{X}\), whereas groups often overlap in practice.
Limitations of the membership query model: Real-world generative models do much more than membership queries, and the practical significance of the negative results requires further discussion.
No efficiency analysis: Even in cases where it is information-theoretically feasible, the sample or time complexity of the algorithms is not analyzed.

Kleinberg & Mullainathan (2024): Pioneering framework for language generation in the limit, which this work directly extends.
Li, Raman & Tewari (2024): Formalizes generation from a learning theory perspective, introducing the closure dimension.
Gold (1967): Classic work on language identification in the limit, serving as the theoretical foundation for this work.
Hébert-Johnson et al. (2018): Multicalibration framework, which inspired the design of the representativeness constraints.
Bender et al. (2021): The "Stochastic Parrots" paper, pointing out the practical harms of lack of diversity in LLMs.

Rating¶

Novelty: ⭐⭐⭐⭐⭐ — First to introduce representativeness/fairness constraints into formal language generation theory; group closure dimension is a completely new concept.
Experimental Thoroughness: ⭐⭐ — Purely theoretical work without experiments, but the proofs of the theorems are rigorous and complete.
Writing Quality: ⭐⭐⭐⭐ — Rigorous formalization and clear structure, though the entry barrier is high for non-theory readers.
Value: ⭐⭐⭐⭐ — Provides a solid theoretical foundation for fair generation and reveals the inherent conflict between fairness and computability.