As the space of causal sequence modeling architectures continues to grow, the need to develop a general framework for their analysis becomes increasingly important. With this aim, we draw insights from classical signal processing and control theory, to develop a quantitative measure of memory utilization: the internal mechanisms through which a model stores past information to produce future outputs. This metric, which we call effective state-size (ESS), is tailored to the fundamental class of systems with input-invariant and input-varying linear operators, encompassing a variety of computational units such as variants of attention, convolutions, and recurrences. Unlike prior work on memory utilization, which either relies on raw operator visualizations (e.g. attention maps), or simply the total memory capacity (i.e. cache size) of a model, our metrics provide highly interpretable and actionable measurements. In particular, we show how ESS can be leveraged to improve initialization strategies, inform novel regularizers and advance the performance-efficiency frontier through model distillation. Furthermore, we demonstrate that the effect of context delimiters (such as end-of-speech tokens) on ESS highlights cross-architectural differences in how large language models utilize their available memory to recall information. Overall, we find that ESS provides valuable insights into the dynamics that dictate memory utilization, enabling the design of more efficient and effective sequence models.

As models that process sequences (like text or speech) become more complex, it’s increasingly important to understand how they “remember” and "utilize" past information to generate accurate future outputs. This research introduces a tool derived from signal processing and control theory called Effective State-Size (ESS) that measures how efficiently a model uses its internal memory.Unlike older methods that just look at visual patterns or how much memory a model has, ESS offers a more theoretically grounded way to analyze memory use. We show that ESS can help in several ways: designing better model starting points (initializations), creating new training techniques (regularizers), and making models faster and more efficient through distillation. We also find that ESS reveals how different models respond to context cues (like end-of-sentence tokens), giving insight into their memory utilization patterns.