Compositional Generalization in AI

• Compositional generalization is the ability of systems to recombine known primitives via systematic rules, echoing key aspects of human cognition.
• It drives advances in AI by challenging models to handle novel combinations, as demonstrated in benchmarks like SCAN, COGS, and GQA-CCG.
• Emerging approaches, including modular architectures and neuro-symbolic models, are enhancing robustness on compositional tasks.

Compositional generalization is the capacity of a learning system to recognize, produce, or reason about novel combinations of previously encountered foundational elements—such as words, concepts, visual parts, or operations—by systematically recombining them according to predictable rules. While this property is a cornerstone of human cognition, enabling robust extrapolation in language, vision, and reasoning, it remains a persistent challenge for many contemporary machine learning systems, particularly deep neural networks. The research field has evolved rapidly in recent years, developing theoretical frameworks, empirical benchmarks, specialized neural architectures, and objective evaluation criteria for testing and advancing compositional generalization.

1. Fundamental Principles and Formal Definitions

Compositional generalization is broadly characterized by two core principles: the existence of well-defined components (primitives or factors) and the presence of a systematic compositional mechanism that governs their combination. In the context of dataset design and theoretical analysis, this is formalized as follows:

• Well-Defined Concepts: Each data sample is associated unambiguously with a specific combination of concepts, and supports of different concept-combinations are non-overlapping.
• Compositional Rule: There exists a measurable bijection or transformation $T$ mapping the distribution of one concept-combination onto another, such that $P_{a_1, b_1}(z) = P_{a_2, b_2}(T(z))$ for all concept pairs $(a_1, b_1), (a_2, b_2)$, with $T$ acting as the identity when the concepts are unchanged (Fu et al., 20 May 2024).

This formalism enables a task-agnostic definition of compositional generalization and exposes its universality across language, vision, and control tasks.

2. Empirical Manifestations and Benchmarks

Early empirical studies revealed gaps between human and neural network performance on compositional generalization. Controlled splits—such as those in SCAN, COGS, and newly curated datasets—test the system’s ability to handle combinations not seen during training by holding out specific compositions (e.g., “jump twice and run” when “jump” and “twice” were never paired in training) (Loula et al., 2018).

Notable benchmarks include:

• SCAN: A synthetic command-to-action mapping benchmark to probe compositionality in sequence-to-sequence models (Loula et al., 2018).
• COGS: A semantic parsing dataset designed to isolate structural generalization—novel syntactic configurations of seen lexical items (Weißenhorn et al., 2022).
• GQA-CCG: A vision-and-language dataset measuring consistency of generalization across phrase/phrase, phrase/word, and word/word compositional levels (Li et al., 18 Dec 2024).
• Object Library: An environment formalizing compositional generalization in object-centric and reinforcement learning settings, operationalizing behavioral equivariance under object permutations (Zhao et al., 2022).

Performance on compositional splits remains a stringent test; standard models often excel in-distribution but dramatically underperform when required to extrapolate to unseen compositions.

3. Architectures and Training Strategies for Compositionality

Numerous neural architectures and training regimes have been developed to improve compositional generalization:

Modular and Neuro-Symbolic Approaches

• Compositional Program Generator (CPG): Assigns distinct semantic modules to each grammar rule in a context-free grammar and composes them recursively, achieving perfect generalization on SCAN and COGS with $1000\times$ fewer examples than Transformers (Klinger et al., 2023).
• Tree Stack Memory Units (Tree-SMU): Augments recursive network nodes with differentiable stacks, capturing long-range dependencies and preserving ordering for better generalization in novel mathematical expressions (Arabshahi et al., 2019).
• Hierarchical Reinforcement Learning with Analytical Expressions: Decomposes input into hierarchical expressions, using learned symbolic modules for each composition, leading to 100% accuracy on compositional splits (Liu et al., 2020).

Training Distribution and Representation Design

• Dual Representation and Entropy Regularization: Uses parallel primitive and functional representations, with entropy reduction to help the model focus only on necessary information for prediction. This approach outperforms baselines and even human learners in few-shot setups (Li et al., 2019).
• Careful Data Design: Introducing more “example primitives” in the training set (i.e., increasing the diversity of composed contexts for basic elements) dramatically improves standard sequence-to-sequence models' compositional generalization, challenging prior assessments about their weaknesses (Patel et al., 2022).
• Meta-Learning and Curriculum: Progressive learning—optimizing models on compositional samples of increasing complexity and weighting samples via meta-weight-nets—yields consistent generalization across multiple compositionality levels (Li et al., 18 Dec 2024).

Task Encoding and Representation Structure

• Disentangled and Abstract Representations: Theoretical and empirical results support the importance of architectural constraints that enforce conditional independence between components, modular decoder designs, and regularization to control representation entropy (Li, 2021, Wiedemer et al., 2023).
• Emergent Language Bottlenecks vs. Disentanglement: Unsupervised models with emergent language bottlenecks deliver stronger compositional generalization than traditional disentanglement objectives, which may yield poor downstream generalization even when compositionality metrics are high (Xu et al., 2022).

4. Theoretical Advances and Generalization Bounds

A growing body of work establishes the mathematical underpinnings of compositional generalization:

• Compositional Representation Theory: Function families are modeled as $f(z) = C(\varphi_1(z_1), ..., \varphi_K(z_K))$, where $\varphi_k$ generate intermediates and $C$ composes them (Wiedemer et al., 2023).
• Conditions for Generalization: Two support conditions are identified (Wiedemer et al., 2023): (a) compositional support—each component's marginal support must be covered in training; (b) sufficient support—the training data must be “rich” enough in local structure to permit reconstruction via the composition function's Jacobian.
• No Free Lunch Theorem: Demonstrates the impossibility of universal, task-agnostic compositional generalization—algorithmic success depends on alignment between the model’s inductive biases and the compositional rules of the data-generating process (Fu et al., 20 May 2024).
• Generalization Bound: Provides a bound relating generalization error on out-of-distribution composite tasks to the mutual information between the learned function and the compositional mechanism. Explicitly, for $L$-bounded error function,

$$\mathbb{E}\left[\mathrm{err}(P_U, f) - \mathrm{err}(D_n, f)\right] \leq \mathrm{Gen}_{IID} + \kappa_n L \Phi(I(f_S; T \mid P_S^{(T)})) + \mathcal{O}(\epsilon)$$

where $I(f_S; T \mid P_S^{(T)})$ is the conditional mutual information, reinforcing that less entanglement with the compositional rule improves generalization (Fu et al., 20 May 2024).

• Scalability: MLPs with ReLU activation can approximate a wide class of compositional task families to arbitrary precision using a linear number of neurons in the number of task modules, not in the size of the task space ($\mathcal{O}(M)$ versus $\mathcal{O}(M^K)$) (Redhardt et al., 9 Jul 2025).

5. Evaluation, Metrics, and Consistency

Assessing compositional generalization extends beyond in-distribution accuracy:

• Out-of-Distribution Splits: Benchmarks emphasize splits where all individual components are observed in training but their combinations are held out for testing (Weißenhorn et al., 2022, Li et al., 18 Dec 2024).
• Consistency Across Levels: A robust model should generalize to both complex and derived simpler compositions (phrase-phrase, phrase-word, word-word), maintaining accurate reasoning at all levels (Li et al., 18 Dec 2024).
• Interpretability: Decoding the underlying task constituents from hidden representations post-training serves both as a measure and a predictor of a model’s compositional generalization—in successful models, constituents become linearly decodable from hidden activations (Redhardt et al., 9 Jul 2025).
• Correlation with Metrics: Conventional compositionality and disentanglement metrics (e.g., SAP, IRM, DCI, MIG) do not always correlate with downstream generalization performance, suggesting the need to rely on compositional-split generalization scores and decodability measures (Xu et al., 2022).

6. Challenges, Limitations, and Ongoing Directions

Despite substantial progress, several persistent challenges remain:

• Coverage and Sufficient Support: Real-world settings may lack the exhaustive component-wise coverage assumed in theoretical work, leading to failures when compositional support is weak (Wiedemer et al., 2023).
• Generative Effects and Non-Independence: Many tasks exhibit nonlinear interactions (“generative effects”) between components, for which independent rule-based approaches can fail; new methodologies are required for such cases (Fu et al., 20 May 2024).
• Scaling Laws and Practicality: While scaling data/model size enables compositional generalization in synthetic settings, resource constraints and ambiguous task specifications limit direct applicability to large-scale natural tasks (Redhardt et al., 9 Jul 2025).

Research continues along multiple axes: integrating symbolic and neural computation, refining curriculum and meta-learning frameworks for complex composition, developing principled evaluation methods, and adopting theoretical insights to real-world data with generative or interactive dependencies.

7. Practical and Cross-Domain Impact

Compositional generalization is relevant for many applied domains:

• Natural Language Processing: Enables robust semantic parsing, machine translation, and question answering that can handle previously unseen word and structure combinations (Weißenhorn et al., 2022, Oren et al., 2020).
• Vision and Multimodal Reasoning: Supports the generation and interpretation of image captions, object-centric scene understanding, and consistent visual question answering across multiple compositional levels (Nikolaus et al., 2019, Zhao et al., 2022, Li et al., 18 Dec 2024).
• Robotics and Control: Facilitates transfer from primitive behaviors to complex instructions in grounded environments or modular world models (Kuo et al., 2020, Zhao et al., 2022).
• Mathematical and Logical Reasoning: Underpins advances in solving compositional math word problems and synthetically constructed reasoning tasks (Lan et al., 2022, Arabshahi et al., 2019).
• Representation Learning: Guides design of bottleneck structures (including emergent LLMs) to encourage more generalizable, modular internal representations (Xu et al., 2022, Ito et al., 2022).

In summary, compositional generalization is both a central theoretical problem and a cross-cutting practical objective in AI research. Advances in formal theory, curriculum design, modular architectures, and systematic evaluation are jointly shaping the path towards models that approach the flexible, principled recombinatory abilities of human cognition.