Abstract
Majority-vote ensembles achieve variance reduction by averaging over diverse, approximately independent base learners. When training data exhibits Markov dependence, as in time-series forecasting, reinforcement learning (RL) replay buffers, and spatial grids, this classical guarantee degrades in ways that existing theory does not fully quantify. We provide a minimax characterization of this phenomenon for discrete classification in a fixed-dimensional Markov setting, together with an adaptive algorithm that matches the rate on a graph-regular subclass. We first establish an information-theoretic lower bound for stationary, reversible, geometrically ergodic chains in fixed ambient dimension, showing that no measurable estimator can achieve excess classification risk better than \Omega(\sqrt{\Tmix/n}). We then prove that, on the AR(1) witness subclass underlying the lower-bound construction, dependence-agnostic uniform bagging is provably suboptimal with excess risk bounded below by \Omega(\Tmix/\sqrt{n}), exhibiting a \sqrt{\Tmix} algorithmic gap. Finally, we propose \emph{adaptive spectral routing}, which partitions the training data via the empirical Fiedler eigenvector of a dependency graph and achieves the minimax rate \mathcal{O}(\sqrt{\Tmix/n}) up to a lower-order geometric cut term on a graph-regular subclass, without knowledge of \Tmix. Experiments on synthetic Markov chains, 2D spatial grids, the 128-dataset UCR archive, and Atari DQN ensembles validate the theoretical predictions. Consequences for deep RL target variance, scalability via Nystr\"om approximation, and bounded non-stationarity are developed as supporting material in the appendix.
