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Quantifying Memory Utilization with Effective State-Size

Rom N. Parnichkun, Neehal Tumma, Armin W. Thomas, Alessandro Moro, Qi An, Taiji Suzuki, Atsushi Yamashita, Michael Poli, Stefano Massaroli

TL;DR

Problem: understanding memory utilization beyond capacity in sequence models with linear and input-varying operators. Approach: Introduce effective state-size (ESS) defined as $\text{ESS}_i = \operatorname{rank}(H_i)$ with $H_i = T_{i:, :i-1}$, and develop tolerance-ESS and entropy-ESS variants; unify linear systems and LIVs under a single operator view. Contributions: theoretical links between ESS and minimal recurrent realization; empirical validation across memory-intensive tasks; demonstrations of initialization, regularization, distillation, and state modulation in language models. Significance: ESS provides a principled, interpretable metric to guide memory-aware design and optimization in diverse sequence-model architectures.

Abstract

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 \textit{memory utilization}: the internal mechanisms through which a model stores past information to produce future outputs. This metric, which we call \textbf{\textit{effective state-size}} (ESS), is tailored to the fundamental class of systems with \textit{input-invariant} and \textit{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 \textit{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.

Quantifying Memory Utilization with Effective State-Size

TL;DR

Problem: understanding memory utilization beyond capacity in sequence models with linear and input-varying operators. Approach: Introduce effective state-size (ESS) defined as with , and develop tolerance-ESS and entropy-ESS variants; unify linear systems and LIVs under a single operator view. Contributions: theoretical links between ESS and minimal recurrent realization; empirical validation across memory-intensive tasks; demonstrations of initialization, regularization, distillation, and state modulation in language models. Significance: ESS provides a principled, interpretable metric to guide memory-aware design and optimization in diverse sequence-model architectures.

Abstract

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 \textit{memory utilization}: the internal mechanisms through which a model stores past information to produce future outputs. This metric, which we call \textbf{\textit{effective state-size}} (ESS), is tailored to the fundamental class of systems with \textit{input-invariant} and \textit{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 \textit{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.
Paper Structure (72 sections, 2 theorems, 18 equations, 51 figures, 3 tables)

This paper contains 72 sections, 2 theorems, 18 equations, 51 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Classifying sequence models.
  4. Quantifying memory utilization in sequence models.
  5. Theory
  6. A unified representation of sequence models.
  7. Operator-recurrence duality and the minimal recurrent realization of linear systems.
  8. Interpreting effective state-size.
  9. Memory capacity in LIVs.
  10. Computing Effective State-Size
  11. Tolerance-ESS.
  12. Entropy-ESS.
  13. Computational complexity of ESS.
  14. Aggregation of ESS across model and data dimensions.
  15. Empirical Validation of Effective State-Size
  16. ...and 57 more sections

Key Result

Theorem 3.1

Given any causal input-invariant operator $T$, there exist infinite variations of linear recurrences in the form of Equation eq:ssm that realize an equivalent input-output operator.

Figures (51)

  • Figure 1: An overview of the effective state-size metric and its various downstream applications.
  • Figure 2: Scatter plots of accuracy vs ESS/kv across featurizers. Within each featurizer plot, all task-model configurations from the sweep corresponding to each featurizer are shown.
  • Figure 3: Correlation between ESS and accuracy over the course of model training bucketed by TSS and kv.
  • Figure 4: ESS/kv vs TSS/kv as a proxy for model performance as measured by correlation.
  • Figure 5: ESS-distillation loss (activation) correlation.
  • ...and 46 more figures

Theorems & Definitions (4)

  • Theorem 3.1
  • Theorem 3.2
  • proof
  • proof