On Fibonacci Ensembles: An Alternative Approach to Ensemble Learning Inspired by the Timeless Architecture of the Golden Ratio

TL;DR

This work introduces Fibonacci Ensembles, a principled framework for ensemble learning inspired by the Fibonacci sequence and the golden ratio. It develops two formulations: a Fibonacci-weighted aggregation with orthogonalization to achieve Rao–Blackwell variance reduction, and a second-order Fibonacci recursive flow that injects memory into the ensemble, both analyzed within a rigorous statistical learning framework. The authors establish a unifying theory linking variance reduction, expressive expansion, spectral stability, and generalization bounds, and demonstrate the approach on one-dimensional regression tasks with random Fourier feature and polynomial bases, revealing regime-dependent advantages over traditional averaging. The paper also outlines a broader general weighting theory and discusses extensions to high-dimensional problems, tree-based learners, and stacked generalization, positioning Fibonacci ensembles as a new, harmonically structured axis for designing and understanding ensemble methods.

Abstract

Nature rarely reveals her secrets bluntly, yet in the Fibonacci sequence she grants us a glimpse of her quiet architecture of growth, harmony, and recursive stability \citep{Koshy2001Fibonacci, Livio2002GoldenRatio}. From spiral galaxies to the unfolding of leaves, this humble sequence reflects a universal grammar of balance. In this work, we introduce \emph{Fibonacci Ensembles}, a mathematically principled yet philosophically inspired framework for ensemble learning that complements and extends classical aggregation schemes such as bagging, boosting, and random forests \citep{Breiman1996Bagging, Breiman2001RandomForests, Friedman2001GBM, Zhou2012Ensemble, HastieTibshiraniFriedman2009ESL}. Two intertwined formulations unfold: (1) the use of normalized Fibonacci weights -- tempered through orthogonalization and Rao--Blackwell optimization -- to achieve systematic variance reduction among base learners, and (2) a second-order recursive ensemble dynamic that mirrors the Fibonacci flow itself, enriching representational depth beyond classical boosting. The resulting methodology is at once rigorous and poetic: a reminder that learning systems flourish when guided by the same intrinsic harmonies that shape the natural world. Through controlled one-dimensional regression experiments using both random Fourier feature ensembles \citep{RahimiRecht2007RFF} and polynomial ensembles, we exhibit regimes in which Fibonacci weighting matches or improves upon uniform averaging and interacts in a principled way with orthogonal Rao--Blackwellization. These findings suggest that Fibonacci ensembles form a natural and interpretable design point within the broader theory of ensemble learning.

Figures (4)

• Figure 1: Sinusoidal regression with random fourier features ensembles: noisy data, true function $f_{\sin}(x)=\sin(2\pi x)$, and the Uniform, Fibonacci, and Orthogonal RB ensemble fits.
• Figure 2: Sinc regression with random fourier features ensembles: noisy data, true function $f_{\mathrm{sinc}}(x)=\sin(x)/x$, and the Uniform, Fibonacci, and Orthogonal RB ensemble fits.
• Figure 3: Sinusoidal regression with polynomial ensembles: noisy data, true function $f_{\sin}(x)=\sin(2\pi x)$, and the Uniform, Fibonacci, and Orthogonal RB ensemble fits.
• Figure 4: Sinc regression with polynomial ensembles: noisy data, true function $f_{\mathrm{sinc}}(x)=\sin(x)/x$, and the Uniform, Fibonacci, and Orthogonal RB ensemble fits.

Theorems & Definitions (30)

• Definition 2.1: Fibonacci Weighting
• Definition 2.2: Orthogonalization Operator
• Theorem 2.1: Fibonacci Variance Dominance
• proof
• Definition 2.3: Fibonacci Conic Hull
• Lemma 2.1: Golden Asymptotic Equidistribution
• proof
• Theorem 2.2: Fibonacci Conic Expansion Theorem
• proof
• Proposition 3.1: Stability Region