Ensemble Learning under Markov Dependence

Minimax Optimality and Spectral Routing for Majority-Vote Ensembles under Markov Dependence

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 Ω(√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 Ω(Tmix/√n), exhibiting a √Tmix algorithmic gap. Finally, we propose adaptive spectral routing, which partitions the training data via the empirical Fiedler eigenvector of a dependency graph and achieves the minimax rate O(Tmix/n) up to a lower-order geometric cut term on a graph-regular sub-class, 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 Nyström approximation, and bounded non-stationarity are developed as supporting material in the appendix.