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arxiv: 2604.07867 · v2 · submitted 2026-04-09 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn

Recognition: unknown

Stochastic Thermodynamics for Autoregressive Generative Models: A Non-Markovian Perspective

Takahiro Sagawa
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:13 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nn
keywords autoregressive modelsentropy productionstochastic thermodynamicsnon-Markovian processestransformersgenerative modelsretrospective inferenceLLMs
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The pith

Autoregressive generative models admit an entropy production from stochastic thermodynamics that can be efficiently estimated from trajectories and decomposes exactly into per-step retrospective inference terms despite non-Markovian output.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Autoregressive models including Transformers and recurrent networks generate sequences by sampling each token from a deterministic summary of the past, which produces genuinely non-Markovian observed dynamics. The paper introduces a stochastic thermodynamics framework that defines entropy production for this entire class of architectures. This entropy production can be estimated directly from sampled trajectories without incurring exponential sampling cost. It decomposes exactly into non-negative per-step contributions expressed through retrospective inference, and each contribution further factors into an information-theoretic compression loss plus a model mismatch term. The claims are illustrated with an analysis of GPT-2 and with an exact analytical reduction in the linear Gaussian case that recovers the Kalman innovation representation.

Core claim

We develop a general theoretical framework based on stochastic thermodynamics for autoregressive generative models and introduce the entropy production, which can be efficiently estimated from sampled trajectories without exponential sampling cost, despite the non-Markovian nature of the observed dynamics. The entropy production decomposes exactly into non-negative per-step contributions in terms of retrospective inference, and each of those terms further splits into information-theoretically meaningful terms: a compression loss and a model mismatch.

What carries the argument

Entropy production defined for non-Markovian observed processes generated by autoregressive conditional distributions, together with its exact decomposition into retrospective-inference contributions.

If this is right

  • Entropy production can be computed tractably from finite trajectories in models like GPT-2.
  • The quantity decomposes into per-step non-negative terms that separate compression loss from model mismatch.
  • Token-level entropy production in GPT-2 is dominated by syntactic artifacts while sentence-level values may distinguish causally ordered from non-causal text.
  • In the linear Gaussian case the framework reduces to the Kalman innovation representation with a closed-form expression for entropy production.
  • The same decomposition applies uniformly across Transformers, RNNs, state-space models, and Mamba architectures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The decomposition supplies a quantitative handle on irreversibility that could be tracked during training to monitor how well a model captures temporal structure.
  • Sentence-level entropy production may offer a diagnostic for whether a model has learned causal versus merely statistical patterns in text.
  • The framework opens a route to comparing irreversibility across different autoregressive architectures without requiring Markovian approximations.

Load-bearing premise

The entropy production for genuinely non-Markovian observed processes generated by autoregressive models admits an efficient estimator and an exact non-negative decomposition into retrospective-inference terms without additional assumptions that fail for high-dimensional models such as Transformers.

What would settle it

A high-dimensional autoregressive model such as a large Transformer in which the proposed estimator for entropy production requires exponential sampling cost or in which the per-step retrospective-inference terms fail to remain non-negative.

Figures

Figures reproduced from arXiv: 2604.07867 by Takahiro Sagawa.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the general causal structure of our setup [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic of the causal structure for the recursive [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Distribution of the per-token stochastic entropy production for sequences of [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Per-token stochastic entropy production evaluated on GPT-2 for 30 causal texts (red) and 30 non-causal texts (blue) [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Numerical verification of the analytical entropy pro [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Convergence of the Monte Carlo estimates of the per-token entropy production for [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Convergence of the Monte Carlo mean of [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Same GPT-2 experiment as Figure 4, but using another input text set generated by an independent session of Claude [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Same GPT-2 experiment as Figure 4, but using an input text set independently generated by GPT-5.4 Pro [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Same GPT-2 experiment as Figure 4, but using an input text set independently generated by Gemini 3.1 Pro [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
read the original abstract

Autoregressive generative models -- including Transformers, recurrent neural networks, classical Kalman filters, state space models, and Mamba -- all generate sequences by sampling each output from a deterministic summary of the past, producing genuinely non-Markovian observed processes. We develop a general theoretical framework based on stochastic thermodynamics for this class of architectures and introduce the entropy production, which can be efficiently estimated from sampled trajectories without exponential sampling cost, despite the non-Markovian nature of the observed dynamics. As a proof-of-concept experiment with a large language model (LLM), we evaluate the entropy production for a pre-trained Transformer-based model, GPT-2. We find that the token-level entropy production is dominated by a syntactic artifact, while the sentence-level entropy production may yield a more interpretable signal in comparisons between causally ordered and non-causal text sets. We also demonstrate the framework in the linear Gaussian case, where the model reduces to the Kalman innovation representation and the entropy production admits an analytical expression. We further show that the entropy production decomposes exactly into non-negative per-step contributions in terms of retrospective inference, and each of those terms further splits into information-theoretically meaningful terms: a compression loss and a model mismatch. Our results establish a bridge between stochastic thermodynamics and modern generative models, and provide a starting point for quantifying irreversibility in a broad class of highly non-Markovian processes such as LLMs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript develops a stochastic thermodynamics framework for autoregressive generative models (Transformers, RNNs, state-space models, etc.) that generate non-Markovian observed sequences. It defines an entropy production for these processes that admits an efficient estimator from sampled trajectories, avoiding exponential sampling costs. The entropy production is claimed to decompose exactly into non-negative per-step contributions expressed via retrospective inference; each term further splits into a compression loss and a model mismatch. Analytical results are given for the linear-Gaussian (Kalman) case, and a proof-of-concept evaluation is performed on pre-trained GPT-2, reporting that token-level entropy production is dominated by syntactic artifacts while sentence-level values may distinguish causal from non-causal text.

Significance. If the efficient estimator and exact non-negative decomposition hold without hidden assumptions that fail for high-dimensional trained models, the work supplies a concrete bridge between non-equilibrium thermodynamics and modern sequence models. It would enable quantitative study of irreversibility, information processing, and thermodynamic-like accounting in LLMs and related architectures, with potential downstream uses in model analysis, training diagnostics, and interpretability.

major comments (3)
  1. [Theoretical framework and decomposition statements] The central claim that entropy production admits an efficient estimator and an exact non-negative decomposition into retrospective-inference terms (each splitting into compression loss and model mismatch) is load-bearing. The manuscript must supply the explicit derivation showing that the path-probability ratio reduces to quantities computable from the forward conditionals and a single backward pass or summary statistic; without this, the usual exponential cost over histories reappears for genuinely non-Markovian processes.
  2. [GPT-2 experiment] § on GPT-2 experiment: the evaluation is presented only as a proof-of-concept with no quantitative metrics, baseline comparisons, statistical controls, or error bars. This leaves open whether the reported dominance of syntactic artifacts at token level and the sentence-level distinction between causal and non-causal text are robust or artifacts of the particular sampling and aggregation choices.
  3. [Linear-Gaussian / Kalman case] Linear-Gaussian case: the claim of an analytical expression is important for validation, yet the manuscript must demonstrate that the general decomposition recovers the known Kalman-filter entropy-production formula without additional assumptions; otherwise the reduction serves only as a consistency check rather than independent support.
minor comments (1)
  1. [Notation and definitions] Notation for retrospective inference and the two information-theoretic splits should be introduced with explicit equations early in the theoretical section to avoid ambiguity when the same symbols appear in both the general and linear-Gaussian treatments.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation of our theoretical framework and experimental results. We address each major comment below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Theoretical framework and decomposition statements] The central claim that entropy production admits an efficient estimator and an exact non-negative decomposition into retrospective-inference terms (each splitting into compression loss and model mismatch) is load-bearing. The manuscript must supply the explicit derivation showing that the path-probability ratio reduces to quantities computable from the forward conditionals and a single backward pass or summary statistic; without this, the usual exponential cost over histories reappears for genuinely non-Markovian processes.

    Authors: We agree that an explicit, self-contained derivation is essential for the central claims. While the manuscript derives the entropy production and its decomposition from the path-probability ratio using the autoregressive structure, we acknowledge that the reduction to forward conditionals and retrospective inference (via a single backward pass) could be presented more transparently. In the revised manuscript we will expand the theoretical section with a detailed, step-by-step derivation that explicitly shows how the non-Markovian path ratio factors into computable per-step terms without exponential summation over histories. revision: yes

  2. Referee: [GPT-2 experiment] § on GPT-2 experiment: the evaluation is presented only as a proof-of-concept with no quantitative metrics, baseline comparisons, statistical controls, or error bars. This leaves open whether the reported dominance of syntactic artifacts at token level and the sentence-level distinction between causal and non-causal text are robust or artifacts of the particular sampling and aggregation choices.

    Authors: We accept that the GPT-2 section, presented as a proof-of-concept, would benefit from greater rigor. In the revision we will add quantitative metrics (mean entropy-production values with standard errors across multiple trajectories), baseline comparisons (e.g., against shuffled or randomly generated text), and statistical controls (significance tests for the reported distinctions). We will also document the exact sampling and aggregation procedures to allow reproducibility and assessment of robustness. revision: yes

  3. Referee: [Linear-Gaussian / Kalman case] Linear-Gaussian case: the claim of an analytical expression is important for validation, yet the manuscript must demonstrate that the general decomposition recovers the known Kalman-filter entropy-production formula without additional assumptions; otherwise the reduction serves only as a consistency check rather than independent support.

    Authors: We agree that an explicit recovery of the known Kalman-filter result strengthens the validation. In the revised manuscript we will include a dedicated subsection that applies the general decomposition directly to the linear-Gaussian (Kalman) case and verifies that it reproduces the standard entropy-production formula without extra assumptions, thereby confirming consistency with established results. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces entropy production for non-Markovian processes generated by autoregressive models and claims an efficient estimator from trajectories plus an exact non-negative decomposition into retrospective-inference terms that further split into compression loss and model mismatch. No equations or sections are provided that reduce the claimed decomposition or estimator to a tautological redefinition of the input quantities (such as the forward conditionals themselves) or to a self-citation chain whose load-bearing step is unverified. The linear-Gaussian analytical case and GPT-2 experiment are presented as independent verifications rather than forced by construction. The framework is therefore self-contained against external benchmarks from stochastic thermodynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework extends standard stochastic-thermodynamics definitions to non-Markovian autoregressive dynamics; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard assumptions of stochastic thermodynamics for defining entropy production in driven non-equilibrium systems
    The paper extends these assumptions from Markovian to non-Markovian observed processes generated by autoregressive models.

pith-pipeline@v0.9.0 · 5553 in / 1387 out tokens · 54387 ms · 2026-05-10T17:13:54.579863+00:00 · methodology

discussion (0)

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Reference graph

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