Understanding LoRA as Knowledge Memory: An Empirical Analysis

Large Language Models (LLMs) must continuously learn and update knowledge to remain effective in dynamic real-world environments. While Low-Rank Adaptation (LoRA) is widely used for such memory updates, existing studies mainly rely on qualitative downstream evaluations, leaving the quantitative capacity limits and underlying dynamics of exact parametric memory largely unexplored. To bridge this gap, we employ LoRA as a controlled memory capacity probe within the latent space to systematically quantify exact parametric memory. We introduce the Parametric Memory Law, a robust power law linking loss reduction Delta L to effective parameters and sequence length. At the token level, fine-grained analysis reveals a deterministic phase transition, demonstrating that a prediction probability of p > 0.5 constitutes a sufficient condition for verbatim recall under greedy decoding. Driven by these insights, we introduce MemFT, a threshold-guided optimization strategy that dynamically redistributes the training budget toward sub-threshold tokens. Empirical evaluations demonstrate that MemFT can enhance memory fidelity and efficiency. Code will be released at https://github.com/zjunlp/ParametricMemoryLaw.

Large language models (LLMs) encode knowledge in parametric weights, making it costly to update or extend without retraining. Retrieval-augmented generation (RAG) mitigates this limitation by appending retrieved text to the input, but operates purely through context expansion, where external knowledge competes as tokens within the attention mechanism. As a result, its influence is indirect and often unstable, particularly in long context and multi hop reasoning scenarios. We propose Knowledge Capsules, structured nonparametric memory units that represent normalized relational knowledge and can be constructed directly from document corpora using a frozen base model. Instead of injecting knowledge as text, we introduce an External Key Value Injection (KVI) framework that compiles capsules into attention-compatible key value representations, enabling external knowledge to directly participate in the model's attention computation. By shifting knowledge integration from context-level augmentation to memory level interaction, the proposed framework consistently outperforms RAG and GraphRAG across multiple QA benchmarks, with improved stability and accuracy in long context and multi hop reasoning, while requiring no parameter updates.

Efficiently updating Large Language Models (LLMs) with new or evolving factual knowledge remains a central challenge, as even parameter-efficient adaptation can erode previously acquired reasoning abilities. This tension reflects a plasticity-stability dilemma: models must incorporate new knowledge while preserving skill-critical representations. In this work, we study this trade-off through the spectral structure of multilayer perceptron weight matrices. We show, both theoretically and empirically, that information essential for reasoning is not localized only in dominant singular directions, but is instead distributed across the singular spectrum. Motivated by this observation, we introduce PALoRA, a two-stage framework for knowledge injection with reduced interference. PALoRA first trains a Singular Value Fine-Tuning (SVF) expert on a reasoning dataset and uses its learned singular scaling vector as a frozen geometric probe to identify components that are critical for the target skill. It then performs factual knowledge injection with Low-Rank Adaptation (LoRA) under a structural orthogonality constraint, ensuring that updates avoid the identified skill-relevant subspace. Across Llama 3.1 8B and Mistral 7B, and across mathematical, coding, and scientific reasoning benchmarks, PALoRA preserves on average 95% of the SVF expert's reasoning performance while maintaining competitive factual recall. It consistently improves skill retention over prior spectral Parameter-Efficient Fine-Tuning (PEFT) methods while adding less than 0.006% parameter overhead.