# Simplicity bias, algorithmic probability, and the random logistic map

arXiv: https://arxiv.org/abs/2401.00593 · HTML: https://mlsanity.com/paper/2401.00593/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 70 references that are in the map.
- First posted on arXiv 31 Dec 2023.

## Builds on (most cited first, 15 of 15)

- [Deep learning generalizes because the parameter-function map is biased towards simple functions](https://mlsanity.com/paper/1805.08522/) (2018)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Simplicity Bias in Transformers and their Ability to Learn Sparse Boolean Functions](https://mlsanity.com/paper/2211.12316/) (2022)
- [A Fine-Grained Spectral Perspective on Neural Networks](https://mlsanity.com/paper/1907.10599/) (2019)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Deep neural networks have an inbuilt Occam's razor](https://mlsanity.com/paper/2304.06670/) (2023)
- [Neural networks are a priori biased towards Boolean functions with low entropy](https://mlsanity.com/paper/1909.11522/) (2019)
- [Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem](https://mlsanity.com/paper/2012.00338/) (2020)
- [Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds](https://mlsanity.com/paper/1904.08622/) (2019)
- [A Note on Kernel Methods for Multiscale Systems with Critical Transitions](https://mlsanity.com/paper/1804.09415/) (2018)
- [Data-driven prediction of a multi-scale Lorenz 96 chaotic system using deep learning methods: Reservoir computing, ANN, and RNN-LSTM](https://mlsanity.com/paper/1906.08829/) (2019)
- [Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks](https://mlsanity.com/paper/1306.0322/) (2013)
- [A Dilemma for Solomonoff Prediction](https://mlsanity.com/paper/2206.06473/) (2022)
- [Coding-theorem Like Behaviour and Emergence of the Universal Distribution from Resource-bounded Algorithmic Probability](https://mlsanity.com/paper/1711.01711/) (2017)
- [Multiclass classification utilising an estimated algorithmic probability prior](https://mlsanity.com/paper/2212.07426/) (2022)

## Abstract (arXiv)

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 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. We discover that simplicity bias is observable in the random logistic map for specific ranges of μ 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 μ=3.83, 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.