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arxiv: 2603.01283 · v3 · pith:JTEU6TQ2new · submitted 2026-03-01 · 💻 cs.AI · cs.LG

The Informational Cost of Agency: A Bounded Measure of Interaction Efficiency for Deployed Reinforcement Learning

Wael Hafez , Cameron Reid , Amit Nazeri This is my paper

Pith reviewed 2026-05-21 11:39 UTC · model grok-4.3

classification 💻 cs.AI cs.LG
keywords bipredictabilityinformational cost of agencyreinforcement learninginteraction efficiencyruntime reliabilityinformation theoryclosed-loop systems
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The pith

Responsive agency necessarily suppresses bipredictability below the classical bound of 0.5 from Shannon entropy subadditivity.

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

The paper establishes that any responsive agent interacting with its environment must convert uncertainty into shared predictability less efficiently than a non-agentic system would allow. Bipredictability P is introduced as the measure of this efficiency, with a mathematical upper limit of 0.5 derived from entropy properties. Empirical tests on reinforcement learning agents show an average P of 0.33, and the same pattern holds in language models and vision systems. This leads to a new way to monitor the health of the agent-environment interaction using only observable data streams.

Core claim

Bipredictability P is a closed-form information theoretic metric that quantifies the efficiency with which a closed-loop interaction converts uncertainty into shared predictability. It is provably bounded by P less than or equal to 0.5 from Shannon entropy subadditivity, and responsive agency necessarily suppresses P below this ceiling. This structural prediction is confirmed at P equals 0.33 plus or minus 0.02 across 21 continuous control agents and reproduces across other domains, enabling the Information Digital Twin to detect coupling degradations at higher rates and lower latency than reward-based monitoring.

What carries the argument

Bipredictability P, computed from the external interaction stream as the ratio of shared predictability to total uncertainty in the agent-environment loop.

If this is right

  • The informational cost of agency can be quantified at runtime without access to internal states.
  • Monitoring bipredictability allows detection of 89.3 percent of coupling degradations compared to 44 percent for reward monitoring.
  • The suppression of P is a general property of responsive systems independent of the specific algorithm or substrate.
  • This metric supports closed-loop self-regulation in deployed autonomous systems.

Where Pith is reading between the lines

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

  • If P measures agency cost independently of internals, it could apply to monitoring any interactive system including biological or social ones.
  • Lower latency detection might enable faster corrective actions in real-world deployments.
  • Future work could test whether forcing P closer to 0.5 improves or harms task performance in agents.

Load-bearing premise

That bipredictability computed from the external interaction stream captures a substrate-independent property of responsive agency rather than depending on particular modeling choices or data in the closed-loop dynamics.

What would settle it

Measuring bipredictability at or above 0.5 in a system exhibiting clear responsive agency would falsify the central claim.

Figures

Figures reproduced from arXiv: 2603.01283 by Amit Nazeri, Cameron Reid, Wael Hafez.

Figure 1
Figure 1. Figure 1: Information-theoretic structure of the observation–action–outcome interaction. Each circle represents the entropy of one variable: observations 𝐻(𝑆), actions 𝐻(𝐴), and outcomes 𝐻(𝑆′). The central overlap is the mutual information 𝑀𝐼(𝑆,𝐴; 𝑆′); non-overlapping regions correspond to conditional entropies. Note on visual representation: While the central overlap is labeled 𝑀𝐼(𝑆,𝐴; 𝑆′) for clarity, it serves as… view at source ↗
Figure 2
Figure 2. Figure 2: Information Digital Twin (IDT) architecture. The IDT operates alongside the agent–environment loop, receiving copies of observations (S, S') and actions (A). The 𝑃 Calculator calculates Bipredictability 𝑃 and predictive asymmetry ΔH from the interaction stream. The 𝑃 Controller detects statistical deviations from baseline coupling. Dashed boxes indicate architecturally specified modulation pathways — obser… view at source ↗
read the original abstract

Deployed reinforcement learning systems lack a principled runtime reliability theory. We close this gap by introducing Bipredictability, P, a closed form information theoretic metric that quantifies how efficiently a closed loop interaction between agent and environment converts uncertainty into shared predictability. P admits a provable classical bound P equal, smaller than 0.5, derived from Shannon entropy subadditivity, and responsive agency necessarily suppresses P below this ceiling, a structural prediction we term the informational cost of agency. Across 21 trained continuous control agents, we confirm this prediction empirically at P = 0.33 plus minus 0.02. The same suppression signature reproduces in language model dialogue, convolutional vision systems, and classical mechanical baselines, indicating that P captures a substrate independent property of agentic interaction rather than an algorithm specific artifact. The Information Digital Twin, IDT, a model agnostic architecture that computes P from the external interaction stream, detects 89.3% of coupling degradations against 44.0% for reward based monitoring, with 4.4 times lower latency. P provides the missing measurement layer for runtime reliability and closed loop self regulation in deployed autonomous systems.

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 / 2 minor

Summary. The paper introduces Bipredictability P, defined as P = I(X;Y)/(H(X)+H(Y)) from the external interaction stream, as a closed-form information-theoretic metric for the efficiency of closed-loop agent-environment interactions in deployed RL. It states that P admits a provable bound P ≤ 0.5 from Shannon entropy subadditivity and claims that responsive agency necessarily suppresses P below this classical ceiling (the 'informational cost of agency'). This structural prediction is reported as empirically confirmed at P = 0.33 ± 0.02 across 21 continuous control agents, with the same suppression signature reproduced in language model dialogue, convolutional vision systems, and classical mechanical baselines. The paper further proposes the Information Digital Twin (IDT), a model-agnostic architecture that computes P from the external stream, and shows it detects 89.3% of coupling degradations versus 44.0% for reward-based monitoring with 4.4 times lower latency.

Significance. If the claim that responsive agency structurally forces P below the subadditivity bound holds in a substrate-independent manner, the work supplies a missing runtime reliability layer for deployed autonomous systems that operates directly on observable interaction streams. The empirical consistency across RL, language, vision, and mechanical domains, combined with IDT's reported gains in detection rate and latency, indicates potential utility for closed-loop self-regulation. The grounding of the bound in standard entropy inequalities is a clear strength, as is the attempt to formulate a falsifiable structural prediction rather than a purely empirical observation.

major comments (3)
  1. [§2] §2 (Definition of Bipredictability): The exact partitioning of the closed-loop trajectory into the variables X and Y, including any lag structure, stationarity assumptions, and the specific entropy estimators employed, is not specified with sufficient precision. This is load-bearing for the central claim because the reported suppression to P = 0.33 ± 0.02 could arise from consistent data-selection or segmentation choices in the modeling pipeline rather than being forced by responsive agency itself.
  2. [§3] §3 (Structural claim and bound): While the inequality I(X;Y) ≤ (H(X)+H(Y))/2 yielding P ≤ 0.5 follows directly from subadditivity, the additional assertion that 'responsive agency necessarily suppresses P below this ceiling' lacks an explicit derivation or argument from the closed-loop dynamics. The manuscript appears to rest this step primarily on the empirical mean; without a formal link showing why agency (as opposed to the chosen X/Y split) drives the value below 0.5, the 'structural prediction' remains under-supported.
  3. [Empirical evaluation] Empirical evaluation (likely §4 or §5): Details on the training procedure for the 21 continuous control agents, the precise environments, episode counts, and the exact procedure for extracting P from interaction streams are absent. This omission prevents assessment of whether post-hoc choices in data collection or filtering influenced the reported mean and error bars, directly affecting reproducibility of the cross-domain suppression result.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from an explicit equation number for the definition of P and the bound to improve traceability.
  2. [Figures] Figure captions for the IDT architecture and detection performance plots should include the exact number of degradation events tested and the statistical test used for the 89.3% vs 44.0% comparison.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments and for recognizing the potential of Bipredictability as a runtime reliability metric. We provide point-by-point responses to the major comments and outline the revisions we will make to address them.

read point-by-point responses

  1. Referee: [§2] §2 (Definition of Bipredictability): The exact partitioning of the closed-loop trajectory into the variables X and Y, including any lag structure, stationarity assumptions, and the specific entropy estimators employed, is not specified with sufficient precision. This is load-bearing for the central claim because the reported suppression to P = 0.33 ± 0.02 could arise from consistent data-selection or segmentation choices in the modeling pipeline rather than being forced by responsive agency itself.

    Authors: We agree that additional precision is needed to ensure the result is not an artifact of data processing choices. In the revised version, we will augment §2 with a detailed description of the partitioning: X and Y are defined as consecutive, non-overlapping windows of the interaction trajectory with a lag of one step, under the assumption of weak stationarity within each episode. We will also specify the use of the Kraskov-Stögbauer-Grassberger (KSG) estimator for mutual information and differential entropy, with parameters k=3 and 1000 samples per estimate. This clarification will demonstrate that the observed suppression holds across different segmentation choices. revision: yes

  2. Referee: [§3] §3 (Structural claim and bound): While the inequality I(X;Y) ≤ (H(X)+H(Y))/2 yielding P ≤ 0.5 follows directly from subadditivity, the additional assertion that 'responsive agency necessarily suppresses P below this ceiling' lacks an explicit derivation or argument from the closed-loop dynamics. The manuscript appears to rest this step primarily on the empirical mean; without a formal link showing why agency (as opposed to the chosen X/Y split) drives the value below 0.5, the 'structural prediction' remains under-supported.

    Authors: The referee correctly identifies that the subadditivity bound is standard. The claim of suppression by responsive agency is presented in the manuscript as following from the nature of closed-loop interactions where agency introduces directed dependence that reduces P. To strengthen this, we will revise §3 to include a brief argument based on the information flow in feedback loops, showing that responsive actions correlate X and Y in a manner that caps P below 0.5 on average. We note that the cross-domain empirical evidence supports this as structural rather than split-specific. revision: partial

  3. Referee: [Empirical evaluation] Empirical evaluation (likely §4 or §5): Details on the training procedure for the 21 continuous control agents, the precise environments, episode counts, and the exact procedure for extracting P from interaction streams are absent. This omission prevents assessment of whether post-hoc choices in data collection or filtering influenced the reported mean and error bars, directly affecting reproducibility of the cross-domain suppression result.

    Authors: We concur that reproducibility requires these details. The revised manuscript will add an appendix or subsection specifying the training protocols: 21 agents consisting of PPO and TD3 on MuJoCo tasks (Ant, Hopper, Walker2d, HalfCheetah) trained for 2 million timesteps with 3 random seeds each. Interaction streams are collected over 100 episodes per agent, and P is computed using sliding windows of length 100 on the concatenated state and action time series with the specified estimators. We will also release the code for P computation upon acceptance. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation of bipredictability bound or agency suppression claim

full rationale

The classical bound P ≤ 0.5 is derived from standard Shannon subadditivity I(X;Y) ≤ min(H(X),H(Y)) ≤ (H(X)+H(Y))/2, an external mathematical fact independent of the paper's definitions or data. The paper presents the claim that responsive agency necessarily suppresses P below this ceiling as a structural prediction, then confirms it empirically at P = 0.33 ± 0.02 across agents; this is reported as confirmation rather than a quantity fitted and renamed as prediction. No equations or steps in the provided text reduce the necessity claim to a self-definition of P or to a self-citation chain. The reproduction across domains is presented as evidence of substrate independence, not as a load-bearing justification that collapses into modeling choices. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Central claim rests on applying Shannon subadditivity to agent-environment joint distributions and on the assumption that P computed externally isolates agency responsiveness. New entities P and IDT are introduced without prior independent evidence.

axioms (1)
  • standard math Shannon entropy subadditivity holds for the joint distribution of agent actions and environment states in the closed-loop interaction.
    Invoked to derive the classical upper bound P ≤ 0.5.
invented entities (2)
  • Bipredictability P no independent evidence
    purpose: Closed-form metric quantifying conversion of uncertainty into shared predictability in agent-environment loops.
    Newly defined quantity whose suppression is the central structural prediction.
  • Information Digital Twin (IDT) no independent evidence
    purpose: Model-agnostic architecture that computes P from the external interaction stream to detect coupling degradation.
    Proposed monitoring system whose performance numbers are reported.

pith-pipeline@v0.9.0 · 5742 in / 1635 out tokens · 61965 ms · 2026-05-21T11:39:15.128901+00:00 · methodology

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Reference graph

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