Simplicity bias, algorithmic probability, and the random logistic map

Creators: Hamzi, Boumediene; Dingle, Kamaludin

Abstract

Simplicity bias is an intriguing phenomenon prevalent in various input–output maps, characterized by a preference for simpler, more regular, or symmetric outputs. Notably, these maps typically feature high-probability outputs with simple patterns, whereas complex patterns are exponentially less probable. This bias has been extensively examined and attributed to principles derived from algorithmic information theory and algorithmic probability. In a significant advancement, it has been demonstrated that the renowned logistic map , a staple in dynamical systems theory, and other one-dimensional maps exhibit simplicity bias when conceptualized as input–output systems. Building upon this work, our research delves into the manifestations of simplicity bias within the random logistic map, specifically focusing on scenarios involving additive noise. This investigation is driven by the overarching goal of formulating a comprehensive theory for the prediction and analysis of time series.

Our primary contributions are multifaceted. We discover that simplicity bias is observable in the random logistic map for specific ranges of and noise magnitudes. Additionally, we find that this bias persists even with the introduction of small measurement noise, though it diminishes as noise levels increase. Our studies also revisit the phenomenon of noise-induced chaos, particularly when , revealing its characteristics through complexity-probability plots. Intriguingly, we employ the logistic map to illustrate a paradoxical aspect of data analysis: more data adhering to a consistent trend can occasionally lead to reduced confidence in extrapolation predictions, challenging conventional wisdom.

We propose that adopting a probability-complexity perspective in analyzing dynamical systems could significantly enrich statistical learning theories related to series prediction and analysis. This approach not only facilitates a deeper understanding of simplicity bias and its implications but also paves the way for novel methodologies in forecasting complex systems behavior, especially in scenarios dominated by uncertainty and stochasticity.

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Acknowledgement

BH thanks Prof. Jeroen Lamb (Imperial College London) for useful discussions about random dynamical systems that inspired the work in this paper. BH acknowledge support from the Jet Propulsion Laboratory, USA, California Institute of Technology, USA , under a contract with the National Aeronautics and Space Administration and from Beyond Limits (Learning Optimal Models) through CAST (The Caltech Center for Autonomous Systems and Technologies). KD thanks Muhammad Alaskandarani for useful discussions and work on the early parts of this work. This work has been partially supported by the Gulf University for Science and Technology, including by project code: ISG Case 9.

Contributions

Boumediene Hamzi: Methodology, Investigation, Conceptualization. Kamaludin Dingle: Methodology, Investigation, Conceptualization.

Data Availability

No data was used for the research described in the article.

Conflict of Interest

The authors declare that the work presented in this paper is original and has not been published elsewhere in any form or language (partially or in full), except in abstract format for conferences. This work is the result of our research conducted primarily at the Gulf University for Science and Technology and has been partially supported by project code: ISG Case 9. We have no conflicts of interest to disclose and affirm that all sources used have been appropriately credited following ethical research practices. Furthermore, we agree to the terms of submission and publication in the journal to which we are submitting, and we confirm that all co-authors have approved the manuscript for submission.

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