Optimal prediction of Markov chains with and without spectral gap

TL;DR

This paper investigates the optimal prediction risk for Markov chains with dependent data, revealing how spectral gap influences the rate and providing bounds that connect Markov and iid models.

Contribution

It characterizes the optimal prediction risk for Markov chains with spectral gap considerations and extends results to higher-order chains.

Findings

Optimal prediction risk for k-state Markov chains is (rac{k^2}{n}\u2206 rac{n}{k^2}) for certain k.

Prediction risk matches iid models when spectral gap is not too small.

Extensions to higher-order Markov chains are provided.

Abstract

We study the following learning problem with dependent data: Observing a trajectory of length $n$ from a stationary Markov chain with $k$ states, the goal is to predict the next state. For $3 \leq k \leq O(\sqrt{n})$, using techniques from universal compression, the optimal prediction risk in Kullback-Leibler divergence is shown to be $\Theta(\frac{k^2}{n}\log \frac{n}{k^2})$, in contrast to the optimal rate of $\Theta(\frac{\log \log n}{n})$ for $k=2$ previously shown in Falahatgar et al. (2016). These rates, slower than the parametric rate of $O(\frac{k^2}{n})$, can be attributed to the memory in the data, as the spectral gap of the Markov chain can be arbitrarily small. To quantify the memory effect, we study irreducible reversible chains with a prescribed spectral gap. In addition to characterizing the optimal prediction risk for two states, we show that, as long as the spectral gap is not excessively small, the prediction risk in the Markov model is $O(\frac{k^2}{n})$, which coincides with that of an iid model with the same number of parameters. Extensions to higher-order Markov chains are also obtained.