Technical Summary: MoRe: Modular Representations for Principled Continual Representation Learning on Sequential Data

1. Problem Definition

The paper addresses the challenge of Continual Learning (CL), specifically the problem of catastrophic forgetting when models adapt to new data sequentially. The authors argue that existing CL methods (both supervised and unsupervised) largely rely on heuristic strategies—such as parameter constraints, replay, distillation, or architectural isolation—to decide what to preserve and what to update. These approaches often treat the "what to update" decision as a task-level or parameter-level problem rather than a representational one.

The core issue identified is that tasks require distinct yet structured representations that can be selectively updated without disrupting existing knowledge. However, current methods lack a principled criterion grounded in the data's intrinsic structure to determine which parts of a representation should be reused versus replaced. The paper posits that for sequential data, time-delayed dependencies provide a natural signal to uncover the intrinsic organization of data, revealing how fundamental representations give rise to specific ones.

2. Methodology: MoRe Framework

The authors propose MoRe (Modular Representations), a framework for principled continual representation learning. Unlike methods that allocate modularity at the architectural level (e.g., separate networks per task), MoRe identifies modularity within the representation itself.

2.1 Generative Model and Assumptions

MoRe assumes the data is generated by a hierarchical process where latent variables $Z_t$ are organized into $d$ layers ($Z^1_t, \dots, Z^d_t$):

• Hierarchy: Higher layers encode fundamental, domain-agnostic knowledge (e.g., physical principles), while lower layers capture specialized, domain-specific information.
• Unidirectional Time-Delayed Dependencies: Cross-layer effects flow unidirectionally from fundamental to specific layers with a time delay (e.g., $Z^i_t$ depends on $Z^j_{t-\tau}$ where $j < i$). Instantaneous cross-layer dependencies are forbidden.
• Intra-layer Dependencies: Dependencies within the same layer are unrestricted.

2.2 Theoretical Guarantees (Identifiability)

A central contribution is the proof that this specific generative structure allows for the identifiability of the latent process:

• Theorem 1 (Latent Variable Identifiability): Under conditions of smooth density and sufficient variability, instantaneous relations ensure that latent variables can only be mixed within the same layer, not across layers.
• Theorem 2 (Hierarchical Order Identifiability): The unidirectional, time-delayed cross-layer effects constrain the admissible permutations of layers. The hierarchy is identifiable up to a "compatible order" (an order that does not reverse true time-delayed effects).
• Corollary 1: Combining these, the full latent process is identifiable up to within-layer indeterminacies and compatible hierarchical order. This transforms modules from heuristic constructs into well-defined objects, enabling principled decisions on reuse and expansion.

2.3 Adaptation Algorithm

MoRe employs a two-phase adaptation procedure for new datasets $D_k$:

1. Fitting and Assessment (Inference-Only):

   • New data is projected into the existing coordinate system using the previous encoder $f_{k-1}$.
   • Self-Consistency Scoring: Each existing module is scored ($a_j$) based on whether its assumed generative structure (temporal autocorrelation and conditional independence from other modules' residuals) is preserved in the new data.
   • Residual Analysis: The reconstruction residual in observation space is analyzed to identify candidate new coordinates ($V_{new}$) orthogonal to the existing basis, surfacing missing structure.

2. Selective Updating:

   • Gating Mechanism: Each module (existing and candidate new) is associated with a Hard Concrete gate ($g_j$). The self-consistency score $a_j$ initializes the gate prior (high $a_j$ $\rightarrow$ positive logit; low $a_j$ $\rightarrow$ negative logit).
   • Alignment: A learnable alignment matrix $T$ (initialized to identity) allows for small coordinate adjustments to reused modules without full retraining.
   • Objective Function: The loss combines data fit (change-of-variables likelihood), an alignment regularizer ($\|T - I\|_F^2$), and sparsity penalties on gates ($L_0$) and new transition blocks ($J_{new}$).
   • Commitment: At the end of training, gates are binarized. Modules with $g_j=0$ are deactivated (but retained in the library), and new modules with $g_j=1$ are added.

3. Key Contributions

1. MoRe Framework: A method for decomposing representations into a hierarchy of fundamental and specific modules that reflects the data-generating process rather than task boundaries.
2. Identifiability Guarantees: Theoretical proofs showing that under cross-layer time-delayed dependencies, the hierarchical modular structure is identifiable, providing a principled basis for deciding which modules to reuse or expand.
3. Adaptation Procedure: A two-phase algorithm that interprets new data through existing modules, scores them for consistency, and selectively aligns, retains, or expands them using learnable gates.
4. Empirical Validation: Demonstrations on synthetic benchmarks and real-world Large Language Model (LLM) activation data showing interpretable hierarchical structure and improved plasticity-stability trade-offs.

4. Experimental Results

4.1 Synthetic Benchmarks

• Identifiability (S-RQ1): MoRe successfully recovered latent factors across varying dimensions (up to 128) and depths (up to 4 layers) in both linear and nonlinear settings, achieving high Mean Correlation Coefficients (MCC > 0.9).
• Plasticity-Stability (S-RQ2): In transition scenarios involving new factors or missing factors, MoRe achieved near-zero or negative forgetting (improving old-factor recovery) while maintaining plasticity comparable to fine-tuning. Ablation studies confirmed the alignment matrix $T$ is critical for protecting old representations.
• Factor Allocation (S-RQ3): MoRe correctly routed factors to reused or new modules. In "hard" selective cases where old and new factors partially overlapped, gate accuracy dropped to 0.5, reflecting inherent ambiguity, but required sparsity penalties on both old and new gates to function correctly.

4.2 Real-World LLM Activations

• Hierarchical Emergence (RW-RQ1): Applied to activations from Pythia-1.4B and Gemma-2-9B on datasets like ArXiv, AMPS, and Wikipedia, MoRe uncovered a "broad-to-specific" structure.
  • Concept Concentration: On the AMPS dataset (math curriculum), lower layers ($L_0$) showed higher concept concentration (specificity to difficulty tiers) than higher layers, aligning with the curriculum hierarchy.
  • Cross-Layer Links: Extracted B-matrix links showed transitions from fundamental concepts (e.g., arithmetic) to specific ones (e.g., linear algebra) in AMPS, and from abstract math to applied domains in ArXiv.
• Label Efficiency (RW-RQ2): Linear probes trained on frozen MoRe representations outperformed PCA baselines in few-shot settings. For example, MoRe achieved equivalent performance to PCA with 256 dimensions using only 64 dimensions from a single layer and half the labeled data.

5. Significance and Claims

The paper claims that the central challenge of continual learning is representational. By grounding the adaptation process in the identifiable structure of the data (specifically, time-delayed cross-layer dependencies), MoRe offers a principled alternative to heuristic task-based or parameter-based isolation.

• Principled Foundation: The framework suggests that continual adaptation should not be driven by task labels but by the intrinsic organization of the data, allowing for the reuse of fundamental building blocks while expanding specific components.
• Interpretability: The learned hierarchy is interpretable, reflecting the transition from general principles to specific applications, as evidenced by the alignment with human-defined hierarchies in real-world datasets.
• Efficiency: The method demonstrates that rich representations, once learned, allow for label-efficient downstream adaptation, requiring only a few samples to assemble representations for new tasks.

The authors remain modest regarding limitations, noting that the current implementation uses linear instantiations and that the theoretical assumptions (unidirectional delays, sparsity) may only be approximately satisfied in practice. They view the work as a foundational step toward a complete continual learning system, leaving long-horizon stability and non-linear scaling for future research.