Horizon-Constrained Rashomon Sets for Chaotic Forecasting
Pith reviewed 2026-05-10 09:40 UTC · model grok-4.3
The pith
In chaotic systems, the effective Rashomon set of near-optimal models contracts exponentially with prediction lead time at the maximum Lyapunov exponent rate.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We bridge predictive multiplicity and chaotic dynamics by introducing horizon-constrained Rashomon sets that characterize how model equivalence evolves with prediction horizon. We prove that the effective Rashomon set contracts exponentially with lead time at a rate determined by the maximum Lyapunov exponent. Lyapunov-weighted metrics provide tighter bounds on predictive disagreement. Decision-aligned selection algorithms choose among near-optimal models based on downstream utility rather than forecast accuracy alone.
What carries the argument
Horizon-constrained Rashomon sets, which adjust the standard definition of near-equivalent models to account for exponential divergence of predictions driven by the maximum Lyapunov exponent of the underlying system.
If this is right
- The number of models that remain predictively equivalent drops exponentially as the forecast horizon increases.
- Lyapunov-weighted metrics yield stricter and more accurate limits on model disagreement than unweighted alternatives.
- Selecting models by downstream decision utility rather than raw accuracy improves outcomes in chaotic forecasting tasks.
- The framework applies directly to both synthetic chaotic attractors and real-world series such as traffic and weather data.
Where Pith is reading between the lines
- Shorter forecast horizons may retain more model choices, while longer ones force reliance on fewer, more carefully vetted models.
- The approach could be adapted to quantify uncertainty in non-chaotic but noisy systems by replacing Lyapunov rates with other divergence measures.
- Safety-critical applications might use pre-computed Lyapunov estimates to decide the maximum reliable horizon before multiplicity collapses.
- Ensemble methods in machine learning could incorporate dynamical-systems diagnostics to set horizon-specific equivalence thresholds.
Load-bearing premise
The dynamical systems under study have a well-defined positive maximum Lyapunov exponent, and the initial ensemble of models begins with predictions similar enough that divergence dominates other sources of difference.
What would settle it
Observe an ensemble of models on a known chaotic system such as Lorenz-96 and measure whether the growth rate of forecast disagreements exactly matches the known maximum Lyapunov exponent; if the effective Rashomon set fails to contract at that rate, the central claim is false.
Figures
read the original abstract
Predictive multiplicity and chaotic dynamics represent two fundamental challenges in machine learning that have evolved independently despite their conceptual connections. We bridge this gap by introducing horizon-constrained Rashomon sets, a theoretical framework that characterizes how model multiplicity evolves with prediction horizon in chaotic systems. Unlike static prediction tasks where the Rashomon set remains fixed, chaos induces exponential divergence among initially similar models, fundamentally transforming the nature of predictive equivalence. We prove that the effective Rashomon set contracts exponentially with lead time at a rate determined by the maximum Lyapunov exponent and introduce Lyapunov-weighted metrics that provide tighter bounds on predictive disagreement. Leveraging these insights, we develop decision-aligned selection algorithms that choose among near-optimal models based on downstream utility rather than forecast accuracy alone. Extensive experiments on synthetic chaotic systems (Lorenz-96, Kuramoto-Sivashinsky) and real-world applications (wind power, traffic, weather) demonstrate that our framework improves decision quality by 18-34\% while maintaining competitive predictive performance. This work establishes the first rigorous connection between chaos theory and predictive multiplicity, providing principled guidance for deploying machine learning in safety-critical chaotic domains.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces horizon-constrained Rashomon sets to model how predictive multiplicity evolves under chaotic dynamics. It claims a proof that the effective Rashomon set contracts exponentially with lead time at a rate set by the maximum Lyapunov exponent, introduces Lyapunov-weighted metrics for tighter disagreement bounds, and proposes decision-aligned model selection. Experiments on Lorenz-96, Kuramoto-Sivashinsky, wind power, traffic, and weather data report 18-34% gains in decision quality while preserving predictive accuracy.
Significance. If the central derivation holds under the stated assumptions, the work forges a rigorous link between chaos theory and Rashomon-set analysis that is absent from prior literature. The Lyapunov-weighted metrics and decision-aligned selection offer concrete tools for safety-critical forecasting domains. The multi-system experimental suite (synthetic plus real) provides a useful empirical testbed, though the reported gains require clearer baselines and error bars to be fully convincing.
major comments (3)
- [§3] §3 (proof of exponential contraction): The derivation that the effective Rashomon set contracts as exp(−λt) with λ the maximum Lyapunov exponent assumes that initial ensemble forecast differences lie inside the linear regime of the tangent map and that chaotic divergence dominates over model-specific biases. No explicit verification is provided that the L2 distance between short-horizon predictions of the independently trained networks is ≪ attractor diameter on Lorenz-96 or Kuramoto-Sivashinsky; without this check the theoretical rate cannot be guaranteed to govern the observed multiplicity decay.
- [§4.2, §5] §4.2 and §5 (Lyapunov-weighted metrics and experiments): The contraction rate is defined using the maximum Lyapunov exponent estimated from the same chaotic trajectories used to train and evaluate the models. This introduces a data-dependent quantity into the metric, so the claimed “parameter-free” character of the horizon-constrained set is not fully realized; the paper should clarify whether the rate is computed in a truly out-of-sample manner or post-hoc.
- [Table 2, §5.3] Table 2 / §5.3 (decision-quality results): The 18-34% improvement is stated without reporting the exact baselines, number of independent runs, or statistical significance tests. Because the central claim rests on these quantitative gains, the absence of these details prevents assessment of whether the improvement is robust or merely an artifact of a particular baseline choice.
minor comments (3)
- [§2] Notation for the horizon-constrained Rashomon set R_h is introduced without an explicit equation reference in the main text; adding Eq. (X) would improve readability.
- [Figure 3] Figure 3 (multiplicity decay curves) lacks error bars or shading for variability across random seeds; this makes visual assessment of the exponential fit difficult.
- [Related Work] The abstract claims “the first rigorous connection,” yet the related-work section omits several recent papers on chaotic time-series forecasting with ensemble methods; a brief citation update would strengthen the novelty statement.
Simulated Author's Rebuttal
Thank you for the detailed and constructive feedback on our manuscript. We have carefully considered each of the major comments and provide point-by-point responses below, indicating the revisions we plan to implement.
read point-by-point responses
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Referee: [§3] §3 (proof of exponential contraction): The derivation that the effective Rashomon set contracts as exp(−λt) with λ the maximum Lyapunov exponent assumes that initial ensemble forecast differences lie inside the linear regime of the tangent map and that chaotic divergence dominates over model-specific biases. No explicit verification is provided that the L2 distance between short-horizon predictions of the independently trained networks is ≪ attractor diameter on Lorenz-96 or Kuramoto-Sivashinsky; without this check the theoretical rate cannot be guaranteed to govern the observed multiplicity decay.
Authors: We agree that an explicit verification of the linear regime assumption strengthens the theoretical claims. In the revised manuscript, we will include in Section 3 and the supplementary material a quantitative check demonstrating that the L2 distances between short-horizon predictions from independently trained networks are substantially smaller than the attractor diameter for both Lorenz-96 and Kuramoto-Sivashinsky systems. This will confirm that the exponential contraction governed by the maximum Lyapunov exponent applies under the experimental conditions. revision: yes
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Referee: [§4.2, §5] §4.2 and §5 (Lyapunov-weighted metrics and experiments): The contraction rate is defined using the maximum Lyapunov exponent estimated from the same chaotic trajectories used to train and evaluate the models. This introduces a data-dependent quantity into the metric, so the claimed “parameter-free” character of the horizon-constrained set is not fully realized; the paper should clarify whether the rate is computed in a truly out-of-sample manner or post-hoc.
Authors: We thank the referee for pointing this out. Upon review, the maximum Lyapunov exponent is estimated from trajectories held out from the training and evaluation sets to ensure it is computed in an out-of-sample fashion. However, we will revise Section 4.2 to explicitly describe this procedure and clarify that while the horizon-constrained set incorporates a system-specific but fixed parameter (the LE), the metric remains free of model-specific hyperparameters. We will also address the 'parameter-free' phrasing if it overstates the case. revision: partial
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Referee: [Table 2, §5.3] Table 2 / §5.3 (decision-quality results): The 18-34% improvement is stated without reporting the exact baselines, number of independent runs, or statistical significance tests. Because the central claim rests on these quantitative gains, the absence of these details prevents assessment of whether the improvement is robust or merely an artifact of a particular baseline choice.
Authors: We acknowledge the need for greater transparency in the experimental results. In the revised manuscript, we will update Table 2 and Section 5.3 to specify the exact baseline models used for comparison, report the number of independent runs (with error bars), and include statistical significance tests (e.g., p-values from appropriate tests) to substantiate the 18-34% improvements in decision quality. revision: yes
Circularity Check
No circularity: theoretical contraction follows from independent definition of Lyapunov exponent
full rationale
The paper's central claim is a proof that the effective Rashomon set contracts at a rate governed by the system's maximum Lyapunov exponent, which is a pre-existing property of the underlying chaotic dynamical system (Lorenz-96, Kuramoto-Sivashinsky) rather than a quantity fitted from the Rashomon set or model ensemble itself. No equation in the provided abstract or framing reduces the claimed contraction to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain; the Lyapunov exponent is invoked as an external fact from chaos theory. Experiments may estimate λ from trajectories for numerical illustration, but this does not make the theoretical derivation circular by construction. The derivation remains self-contained against external benchmarks from dynamical systems.
Axiom & Free-Parameter Ledger
free parameters (1)
- maximum Lyapunov exponent
axioms (1)
- domain assumption Chaotic dynamical systems possess a well-defined positive maximum Lyapunov exponent that governs exponential divergence of nearby trajectories.
invented entities (2)
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horizon-constrained Rashomon set
no independent evidence
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Lyapunov-weighted metrics
no independent evidence
Reference graph
Works this paper leans on
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[1]
Empirical Validation of Rashomon Set Contraction A central prediction of Theorem 1 is that horizon- constrained Rashomon sets contract exponentially with lead time at a rate governed by the maximum Lyapunov exponent. To validate this theoretical claim empirically, we construct Rashomon sets across multiple horizons for the Lorenz-96 system under varying c...
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[2]
Decision Quality Improvements Across Domains and Horizons While Rashomon set contraction characterizes the evolu- tion of model multiplicity, the practical value of our frame- work depends on whether decision-aligned selection trans- lates into tangible utility improvements. Figure 2 presents a comprehensive analysis of decision quality across three real-...
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[3]
Cross-Horizon Model Agreement and Predictive Ambiguity Understanding how model predictions remain correlated or decorrelate across horizons provides insight into the structure of chaos-induced multiplicity. Figure 3 quantifies these agree- ment patterns through pairwise correlation analysis and ambi- guity growth curves. The correlation structure revealed...
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[4]
Figure 4 examines the consequences of includ- ing diverse model families in the candidate pool
Architectural Diversity and Model Family Effects The hypothesis classHfrom which we construct Rashomon sets need not be restricted to variants of a single architecture. Figure 4 examines the consequences of includ- ing diverse model families in the candidate pool. Figure 4 establishes that expanding the hypothesis class to include architecturally diverse ...
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[5]
Figure 5 characterizes this tradeoff across horizons and tolerance values
Sensitivity Analysis: Tolerance Parameter and Set Size The tolerance parameterε k governs Rashomon set member- ship and therefore critically influences both the multiplicity available for selection and the computational cost of enumer- ation. Figure 5 characterizes this tradeoff across horizons and tolerance values. The tolerance sensitivity analysis in F...
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[6]
Figure 6 synthesizes these relationships
Chaos Strength Modulation and Framework Performance To systematically assess how our framework responds to varying dynamical regimes, we modulate the chaos strength in Lorenz-96 by adjusting the forcing parameterFand mea- sure resulting changes in predictability, multiplicity, and util- ity gains. Figure 6 synthesizes these relationships. The systematic r...
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[7]
Figure 7 validates these predictions against empirical mea- surements
Theoretical Validation: Contraction Rates, Horizons, and Sample Complexity Our theoretical framework makes specific quantitative pre- dictions regarding contraction rates (Theorem 1), effective horizons (Definition 2), and sample complexity (Theorem 3). Figure 7 validates these predictions against empirical mea- surements. The comprehensive theoretical va...
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[8]
Table II decomposes total runtime into constituent operations
Computational Performance Analysis Understanding the computational costs associated with each component of our framework guides optimization efforts and informs deployment decisions. Table II decomposes total runtime into constituent operations. The computational profile in Table II identifies Rashomon set construction as the primary bottleneck, suggestin...
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[9]
Implementation Specifications for Reservoir Computing To ensure reproducibility and facilitate extension of our work, we document complete implementation details for the reservoir computing architecture that forms the core of our ex- perimental evaluation. The reservoir computer implements a recurrent dynamical system governed by the update equation: rt =...
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[10]
Lyapunov Exponent Estimation Methodology Accurate estimation of the maximum Lyapunov exponent λmax constitutes a critical component of our framework, as this quantity governs both theoretical predictions and practical al- gorithmic parameters. We employ the Rosenstein algorithm, a widely-adopted method designed specifically for noisy, finite- length time ...
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[11]
Decision Optimization Procedures Given a forecasting modelhand its predic- tions ˆxh t+k, we must solve for the optimal action a∗(h) =argmax a∈A E[u(ˆxh t+k ,a)]that maximizes expected utility. The appropriate optimization method depends on the structure of both the action spaceAand the utility function u. For applications with differentiable utility func...
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[12]
Table III summarizes results across the full range from weakly chaotic to strongly chaotic regimes
Extended Empirical Results: Chaos Strength Sensitivity To comprehensively characterize how framework perfor- mance depends on underlying system properties, we conduct a systematic sensitivity analysis varying the chaos strength in Lorenz-96 via the forcing parameterF. Table III summarizes results across the full range from weakly chaotic to strongly chaot...
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[13]
Table IV examines this possibility through system- atic transfer experiments
Cross-Domain Transfer Learning Analysis A natural question concerns whether knowledge extracted from one chaotic domain can transfer to improve performance in another. Table IV examines this possibility through system- atic transfer experiments. The transfer learning results in Table IV have important practical implications for rapid deployment of our fra...
discussion (0)
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