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

TL;DR

This paper characterizes the limits of ensemble methods under Markov dependence, proposing an adaptive spectral routing algorithm that achieves minimax optimal risk bounds without prior knowledge of dependence parameters.

Contribution

It provides a minimax theoretical framework for Markov-dependent data and introduces an adaptive spectral routing algorithm that attains optimal rates.

Findings

Uniform bagging is suboptimal under Markov dependence.

The proposed spectral routing achieves the minimax rate $ ilde{O}( frac{ ext{Tmix}}{ oot n})$.

Experiments validate the theoretical predictions on various datasets.

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.