arxiv: 2604.13414 · v1 · submitted 2026-04-15 · 💻 cs.LG · cs.AI

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

Anuj Sharma, Ibne Farabi Shihab, Sanjeda Akter

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:28 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords minimax optimalityMarkov dependencemajority-vote ensemblesspectral routingmixing timeFiedler eigenvectorensemble classificationadaptive partitioning

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The pith

Adaptive spectral routing achieves the minimax optimal rate of order square root of mixing time over sample size for majority-vote ensembles on Markov chains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that when data points follow a Markov chain, majority-vote ensembles cannot reduce excess classification risk faster than the square root of the mixing time divided by the number of samples. Standard uniform bagging is provably worse, with a gap of order square root of the mixing time. Adaptive spectral routing fixes this on a suitable subclass by splitting the data using the main eigenvector of a dependency graph built from the samples, reaching the best possible rate without needing to know the mixing time. This provides a precise understanding of how dependence hurts variance reduction in ensembles and offers a practical fix for dependent data sources such as time series and replay buffers.

Core claim

For stationary, reversible, geometrically ergodic Markov chains in fixed dimension, no measurable estimator attains excess classification risk smaller than order square root of mixing time over sample size. On the AR(1) subclass, uniform bagging has risk at least order mixing time over square root of sample size. Adaptive spectral routing attains the optimal order square root of mixing time over sample size up to a lower-order geometric cut term on graph-regular chains by partitioning according to the empirical Fiedler eigenvector of the dependency graph, without knowledge of the mixing time.

What carries the argument

The empirical Fiedler eigenvector of the dependency graph, which partitions the training data into groups with reduced dependence before majority voting.

If this is right

Uniform bagging suffers excess risk at least order mixing time over square root of sample size on the AR(1) subclass, creating a square root of mixing time algorithmic gap.
The optimal rate is achieved without knowledge of the mixing time.
The lower bound holds for the full class of stationary reversible geometrically ergodic chains while the upper bound holds for graph-regular subclasses.
The rates are confirmed in experiments on synthetic Markov chains, 2D spatial grids, the UCR time series archive, and Atari DQN replay buffers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The spectral partitioning idea could apply to other ensemble methods such as boosting when data exhibits Markov dependence.
In reinforcement learning, routing replay buffer samples this way might further stabilize target networks beyond uniform sampling.
Combining the approach with dimension reduction would be a natural next step to handle settings beyond fixed ambient dimension.

Load-bearing premise

The underlying process is a stationary reversible geometrically ergodic Markov chain in fixed ambient dimension, with the matching upper bound restricted to a graph-regular subclass.

What would settle it

An estimator that achieves excess risk o of square root of mixing time over sample size on some geometrically ergodic chain for large sample size, or a graph-regular chain where spectral routing exceeds order square root of mixing time over sample size.

Figures

Figures reproduced from arXiv: 2604.13414 by Anuj Sharma, Ibne Farabi Shihab, Sanjeda Akter.

**Figure 1.** Figure 1: Measured pairwise base-learner covariance versus Tmix on the AR(1) witness. Uniform bagging tracks the T 2 mix/n prediction of Lemma 5, while spectral routing compresses the covariance by roughly two orders of magnitude. 100 101 102 10−1.5 10−1 10−0.5 Tmix Excess risk Uniform Theory: Tmix/ √ n Spec. Rte. Theory: p Tmix/n [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

**Figure 3.** Figure 3: Relationship between dataset autocorrelation and the accuracy gain of Spectral Routing [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Empirical verification of Assumption 15 on Atari. reff plateaus to O(1). feature-similarity graph, as a heuristic manifestation of structural feature stratification rather than a resolution of Markovian resampling covariance [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

**Figure 5.** Figure 5: Critical difference diagram over the full 128 UCR datasets. Methods connected by a [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper pins down a sqrt(Tmix/n) minimax rate for majority-vote ensembles under Markov dependence and gives an adaptive spectral routing algorithm that recovers it without knowing the mixing time.

read the letter

The main thing to know is that this work gives a clean minimax characterization for how Markov dependence hurts majority-vote ensembles in classification, with a lower bound of order sqrt(Tmix over n) that no estimator can beat, and an algorithm that achieves it adaptively. They start from the classical independent-data theory and extend it by proving an information-theoretic lower bound that holds for any measurable estimator on the class of stationary, reversible, geometrically ergodic Markov chains in fixed dimension. On the AR(1) subclass they use for the lower bound, they show uniform bagging suffers an extra sqrt(Tmix) factor in the excess risk. The proposed adaptive spectral routing builds a dependency graph from the data, takes its empirical Fiedler eigenvector to partition into routes, and routes the base learners accordingly. This recovers the optimal rate up to a geometric cut term on a graph-regular subclass, and it does so without inputting the mixing time. The paper also includes experiments across synthetic Markov chains, 2D grids, the UCR time-series archive, and Atari DQN replay buffers that appear to confirm the predicted scaling. Where it is thinner is that the matching upper bound is stated only for the graph-regular subclass, so the result is not fully tight across the entire lower-bound class. The fixed ambient dimension keeps things clean but leaves open how the rates behave when dimension grows with n, which is relevant for many modern applications. The dependency graph construction itself is not detailed in the abstract, so one would want to check how robust the Fiedler partitioning is to noise or to the specific way edges are defined. The appendix mentions consequences for deep RL target variance and Nyström approximation for scalability, which sound useful but are secondary. Overall this is for theorists and practitioners working on ensembles or variance reduction in time-series and RL settings. Anyone looking at how dependence affects bagging would find the rates and the adaptive method worth reading. The combination of lower bound, explicit suboptimality gap, and matching algorithm makes it worth a serious referee's time, though the proofs and the precise scope of the upper bound will need careful review. I would recommend putting it through peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes a minimax lower bound of Ω(√(Tmix/n)) on excess classification risk for any measurable estimator applied to majority-vote ensembles, under the class of stationary, reversible, geometrically ergodic Markov chains in fixed ambient dimension. It shows that dependence-agnostic uniform bagging is suboptimal, attaining only Ω(Tmix/√n) on the AR(1) witness subclass used for the lower bound. The paper then introduces adaptive spectral routing, which constructs a dependency graph from the data, extracts its empirical Fiedler vector to partition the training set, and achieves the matching rate O(√(Tmix/n)) up to a lower-order geometric cut term on a graph-regular subclass, without requiring knowledge of Tmix. The results are illustrated with experiments on synthetic chains, 2D grids, the UCR archive, and Atari DQN replay buffers.

Significance. If the central claims hold, the work is significant for providing the first tight minimax characterization of how Markov dependence degrades classical ensemble variance reduction, together with an adaptive algorithm that attains the optimal rate on the relevant subclass. The construction of a matching upper bound via spectral partitioning without explicit mixing-time input, and the explicit demonstration of a √Tmix algorithmic gap for uniform bagging, are notable strengths. These results have direct implications for ensemble methods in time-series, spatial data, and reinforcement-learning replay buffers.

major comments (2)

[Lower-bound section and graph-regular subclass definition] The lower-bound construction uses an AR(1) witness subclass of stationary reversible geometrically ergodic chains; the manuscript should explicitly verify that this subclass is contained in the graph-regular class on which the upper bound is proved, so that the Ω(√(Tmix/n)) and O(√(Tmix/n)) rates are directly comparable without additional restrictions.
[Adaptive spectral routing algorithm and its analysis] The adaptive spectral routing algorithm relies on the empirical Fiedler vector of a data-derived dependency graph; the proof that this yields the minimax rate without knowledge of Tmix should clarify the approximation error between the empirical and population Fiedler vectors under only geometric ergodicity (rather than stronger mixing assumptions).

minor comments (2)

[Upper-bound theorem statement] The geometric cut term appearing in the upper-bound rate should be defined explicitly in the main text (not only in the appendix) and its order relative to √(Tmix/n) should be stated clearly.
[Notation and definitions] Notation for Tmix and the graph-regular subclass should be introduced at first use in the main body rather than deferred to the appendix.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The comments are constructive and help strengthen the connection between the lower and upper bounds as well as the clarity of the spectral analysis. We address each major comment below.

read point-by-point responses

Referee: [Lower-bound section and graph-regular subclass definition] The lower-bound construction uses an AR(1) witness subclass of stationary reversible geometrically ergodic chains; the manuscript should explicitly verify that this subclass is contained in the graph-regular class on which the upper bound is proved, so that the Ω(√(Tmix/n)) and O(√(Tmix/n)) rates are directly comparable without additional restrictions.

Authors: We agree that an explicit containment argument is needed for direct rate comparison. In the revised manuscript we will insert a short lemma in the lower-bound section establishing that the AR(1) witness subclass satisfies the graph-regular conditions. The AR(1) process generates a dependency graph that is a path (hence bounded degree and regular spectral properties), and the geometric ergodicity assumption already ensures the required uniform spectral gap. This addition removes any ambiguity and confirms that the Ω(√(Tmix/n)) lower bound holds inside the graph-regular subclass. revision: yes
Referee: [Adaptive spectral routing algorithm and its analysis] The adaptive spectral routing algorithm relies on the empirical Fiedler vector of a data-derived dependency graph; the proof that this yields the minimax rate without knowledge of Tmix should clarify the approximation error between the empirical and population Fiedler vectors under only geometric ergodicity (rather than stronger mixing assumptions).

Authors: We thank the referee for highlighting the need for an explicit error bound. The existing proof already invokes geometric ergodicity to control the deviation of the empirical adjacency matrix via standard concentration inequalities for Markov chains, followed by a Davis-Kahan perturbation argument on the Laplacian. In the revision we will expand this step into a dedicated paragraph that states the precise bound on ||v̂_emp - v_pop||, shows that the resulting additive term is o(√(Tmix/n)), and reiterates that only geometric ergodicity (no stronger mixing) is used. This will make the argument self-contained and transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives an information-theoretic minimax lower bound of Ω(√(Tmix/n)) for any measurable estimator on the class of stationary, reversible, geometrically ergodic Markov chains in fixed dimension, using standard techniques that do not reduce to fitted parameters or self-referential definitions. The adaptive spectral routing algorithm is constructed independently via the empirical Fiedler vector of a data-derived graph and attains the matching rate (modulo lower-order terms) on the graph-regular subclass containing the AR(1) witness, without knowledge of Tmix. This is standard minimax structure with no load-bearing self-citation chains, no renaming of known results as new derivations, and no predictions that are forced by construction from inputs. The analysis is externally falsifiable and self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed from abstract only; full paper likely contains additional technical assumptions on the Markov chain and graph construction that cannot be audited here.

axioms (2)

domain assumption Training data follows a stationary, reversible, geometrically ergodic Markov chain in fixed ambient dimension
Explicitly stated as the setting for the information-theoretic lower bound.
domain assumption The upper-bound result holds on a graph-regular subclass of such chains
Required for the spectral routing algorithm to achieve the minimax rate.

pith-pipeline@v0.9.0 · 5594 in / 1393 out tokens · 27843 ms · 2026-05-10T14:28:56.841552+00:00 · methodology

discussion (0)

Reference graph

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