In such situations, the successor representation (SR) is a popular framework which supports rapid policy evaluation by decoupling a policy's

In this paper, we provide a comprehensive theoretical analysis of Stochastic Gradient Descent (SGD) and its momentum variants (Polyak Heavy-Ball and Nesterov) for tracking time-varying optima under strong convexity and smoothness. Our finite-time bounds reveal a sharp decomposition of tracking error into transient, noise-induced, and drift-induced components. This decomposition exposes a fundamental trade-off: while momentum is often used as a gradient-smoothing heuristic, under distribution shift it incurs an explicit drift-amplification penalty that diverges as the momentum parameter $β$ approaches 1, yielding systematic tracking lag. We complement these upper bounds with minimax lower bounds under gradient-variation constraints, proving this momentum-induced tracking penalty is not an analytical artifact but an information-theoretic barrier: in drift-dominated regimes, momentum is unavoidably worse because stale-gradient averaging forces systematic lag. Our results provide theoretical grounding for the empirical instability of momentum in dynamic settings and precisely delineate regime boundaries where vanilla SGD provably outperforms its accelerated counterparts.

This paper develops the algorithmic and dynamical foundations of recursive ensemble learning driven by Fibonacci-type update flows. In contrast with classical boosting Freund and Schapire (1997); Friedman (2001), where the ensemble evolves through first-order additive updates, we study second-order recursive architectures in which each predictor depends on its two immediate predecessors. These Fibonacci flows induce a learning dynamic with memory, allowing ensembles to integrate past structure while adapting to new residual information. We introduce a general family of recursive weight-update algorithms encompassing Fibonacci, tribonacci, and higher-order recursions, together with continuous-time limits that yield systems of differential equations governing ensemble evolution. We establish global convergence conditions, spectral stability criteria, and non-asymptotic generalization bounds under Rademacher Bartlett and Mendelson (2002) and algorithmic stability analyses. The resulting theory unifies recursive ensembles, structured weighting, and dynamical systems viewpoints in statistical learning. Experiments with kernel ridge regression Rasmussen and Williams (2006), spline smoothers Wahba (1990), and random Fourier feature models Rahimi and Recht (2007) demonstrate that recursive flows consistently improve approximation and generalization beyond static weighting. These results complete the trilogy begun in Papers I and II: from Fibonacci weighting, through geometric weighting theory, to fully dynamical recursive ensemble learning systems.

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.

Our contribution aims to fill this gap.