Quantifying Memory Utilization with Effective State-Size

The need to develop a general framework for architecture analysis is becoming increasingly important, given the expanding design space of sequence models. To this end, we draw insights from classical signal processing and control theory, to develop a quantitative measure of $\mathit{\text{memory}}\mathit{\text{utilization}}$: the internal mechanisms through which a model stores past information to produce future outputs. This metric, which we call $\mathit{\text{effective}}\mathit{\text{state-size}}$ (ESS), is tailored to the fundamental class of systems with $\mathit{\text{input-invariant}}$ and $\mathit{\text{input-varying}}\mathit{\text{linear}}\mathit{\text{operators}}$, 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 $\mathit{\text{memory}}\mathit{\text{capacity}}$ (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.