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Best Ergodic Averages via Optimal Graph Filters in Reversible Markov Chains

Naci Saldi

TL;DR

The paper tackles slow convergence of ergodic (Birkhoff) averages in reversible Markov chains by reframing the problem through graph signal processing on the state-space graph. It defines a graph variation and a graph Fourier transform, treats the ergodic iteration as a graph filter, and designs three optimal low-pass polynomial filters—Bernstein, Chebyshev, and Legendre—to accelerate convergence toward $\pi(f)\mathbf{1}$. Theoretical results show that Bernstein, Chebyshev, and Legendre filters can be constructed with provable optimality properties, and numerical experiments on random walks and Glauber chains demonstrate that Chebyshev and Legendre filters yield substantial speedups over the traditional ergodic average, with Bernstein giving modest gains. This work highlights a practical framework for speeding up ergodic averages using graph-signal-processing techniques and suggests directions for extending to more general state spaces and applications.

Abstract

In this paper, we address the problem of finding the best ergodic or Birkhoff averages in the ergodic theorem to ensure rapid convergence to a desired value, using graph filters. Our approach begins by representing a function on the state space as a graph signal, where the (directed) graph is formed by the transition probabilities of a reversible Markov chain. We introduce a concept of graph variation, enabling the definition of the graph Fourier transform for graph signals on this directed graph. Viewing the iteration in the ergodic theorem as a graph filter, we recognize its non-optimality and propose three optimization problems aimed at determining optimal graph filters. These optimization problems yield the Bernstein, Chebyshev, and Legendre filters. Numerical testing reveals that while the Bernstein filter performs slightly better than the traditional ergodic average, the Chebyshev and Legendre filters significantly outperform the ergodic average, demonstrating rapid convergence to the desired value.

Best Ergodic Averages via Optimal Graph Filters in Reversible Markov Chains

TL;DR

The paper tackles slow convergence of ergodic (Birkhoff) averages in reversible Markov chains by reframing the problem through graph signal processing on the state-space graph. It defines a graph variation and a graph Fourier transform, treats the ergodic iteration as a graph filter, and designs three optimal low-pass polynomial filters—Bernstein, Chebyshev, and Legendre—to accelerate convergence toward . Theoretical results show that Bernstein, Chebyshev, and Legendre filters can be constructed with provable optimality properties, and numerical experiments on random walks and Glauber chains demonstrate that Chebyshev and Legendre filters yield substantial speedups over the traditional ergodic average, with Bernstein giving modest gains. This work highlights a practical framework for speeding up ergodic averages using graph-signal-processing techniques and suggests directions for extending to more general state spaces and applications.

Abstract

In this paper, we address the problem of finding the best ergodic or Birkhoff averages in the ergodic theorem to ensure rapid convergence to a desired value, using graph filters. Our approach begins by representing a function on the state space as a graph signal, where the (directed) graph is formed by the transition probabilities of a reversible Markov chain. We introduce a concept of graph variation, enabling the definition of the graph Fourier transform for graph signals on this directed graph. Viewing the iteration in the ergodic theorem as a graph filter, we recognize its non-optimality and propose three optimization problems aimed at determining optimal graph filters. These optimization problems yield the Bernstein, Chebyshev, and Legendre filters. Numerical testing reveals that while the Bernstein filter performs slightly better than the traditional ergodic average, the Chebyshev and Legendre filters significantly outperform the ergodic average, demonstrating rapid convergence to the desired value.
Paper Structure (12 sections, 8 theorems, 96 equations, 4 figures)

This paper contains 12 sections, 8 theorems, 96 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Reversible Markov Chains
  3. Eigenvalues of Reversible Markov Chains
  4. Graph Signal Processing
  5. Filter Design with Bernstein Polynomials
  6. Filter Design with Chebyshev Polynomials
  7. Filter Design with Legendre Polynomials
  8. Numerical Experiments
  9. Random Walk on a Graph
  10. Glauber Chain
  11. Conclusion
  12. Acknowledgements

Key Result

Lemma 1

Let $P$ be a reversible Markov chain with respect to $\pi$. Then, we have

Figures (4)

  • Figure 1: A circle with $p$ integers placed evenly around it, where $p$ is an odd number.
  • Figure 2: A maximum absolute error as a function of degree of the filter.
  • Figure 3: A state transition of a Glauber chain on the cycle. Up arrow means $+1$, down arrow means $-1$.
  • Figure 4: A maximum absolute error as a function of degree of the filter.

Theorems & Definitions (26)

  • Definition 1
  • Example 1
  • Example 2
  • Lemma 1
  • Theorem 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Lemma 2
  • proof
  • ...and 16 more