Compression Through Time: An Interplay of Separability, Stationarity and Capacity

Recurrent models process long contexts efficiently by compressing their history into a fixed-size state, but modern architectures typically do so in a single causal pass over the sequence. Each input must therefore be compressed before the model knows how it will later be used, forcing a limited state to compromise across possible future demands. We introduce dynamic compression, which allows a recurrent model to selectively revisit past tokens and revise its fixed-size state through additional recurrent updates. The model need not preserve every part of the history at uniformly high fidelity in its recurrent state, because lower-fidelity information can be revisited from the retained raw sequence when it becomes relevant. We study this in a controlled setting where the model first learns multiple functions in-context and, later in the same sequence, encounters a series of few-shot tasks that each require it to identify and reuse one of those functions. A single-pass model must preserve every function at sufficient fidelity for any future task, whereas selective re-scanning allows the model to revisit and refine only the function currently needed. We find that dynamic compression substantially reduces the recurrent state required for accurate reuse and scales more favorably as the number of stored functions grows. These results demonstrate a computation--memory tradeoff in which recurrent models can spend more computation revisiting their history to make more effective use of a fixed-size state.

Recursive computation repeatedly compresses or reuses intermediate states, creating a simple tension: information that must remain useful across longer recursive paths is also exposed to more opportunities for loss before reaching the final prediction. Existing reconstruction or local-prediction objectives provide tractable supervision, but do not ensure that the retained information remains sufficient for subsequent recursive computation. We identify local predictive sufficiency with recursive predictive closure: controlling local predictive deficiencies at individual interfaces controls the resulting discrepancy at the root. We then turn this principle into a tractable training procedure. Starting from a variational characterization, we derive finite predictive tests and an empirical predictive deficiency that measures predictive value retained across compression. Its predictive sensitivities define margin-relaxed half-space constraints on parameter updates, and we project the host optimizer's proposed update onto their intersection only when predictive preservation would otherwise be violated. Across temporal graphs, language memory, vision-language-action control, and recursive self-improvement, the method matches or improves the corresponding host models under matched compression budgets, with larger gains under heavier recursive or memory demands, while better preserving predictive information across successive transformations. Crucially, the same task-agnostic predictive-preservation principle is instantiated across all four settings through host-compatible interventions while keeping the endpoint task, backbone, and evaluation protocol fixed. These results establish predictive preservation at recursive interfaces as a general training principle for recursive compression.

Despite the increasing prevalence of large language models (LLMs), we still have a limited understanding of how their representational spaces are structured. This limits our ability to interpret how and what they learn or relate them to learning in humans. We argue LLMs are best seen as an instance of lossy compression, where over training they learn by retaining only information in their training data relevant to their objective(s). We show pre-training results in models that are optimally compressed for next-sequence prediction, approaching the Information Bottleneck bound on compression. Across an array of open weights models, each compresses differently, likely due to differences in the data and training recipes used. However even across different families of LLMs the optimality of a model's compression, and the information present in it, can predict downstream performance on across a wide array of benchmarks, letting us directly link representational structure to actionable insights about model performance. In the general case the work presented here offers a unified Information-Theoretic framing for how these models learn that is deployable at scale.