[Paper Review] Tail estimates for sums of variables sampled from a random walk

Roy Wagner|ArXiv.org|Aug 30, 2006

Stochastic processes and statistical mechanics5 references4 citations

TL;DR

This paper presents new, elementary tail bounds for sums of functions sampled along a reversible random walk on an undirected graph, using spectral properties like the second-largest eigenvalue and variance. It derives sharper Bernstein- and Bennett-type inequalities than prior work, with explicit dependence on the graph's spectral gap and function variance, enabling precise sample complexity estimates for Monte Carlo-style approximation on expanders.

ABSTRACT

We prove tail estimates for variables $\sum_i f(X_i)$, where $(X_i)_i$ is the trajectory of a random walk on an undirected graph (or, equivalently, a reversible Markov chain). The estimates are in terms of the maximum of the function $f$, its variance, and the spectrum of the graph. Our proofs are more elementary than other proofs in the literature, and our results are sharper. We obtain Bernstein and Bennett-type inequalities, as well as an inequality for subgaussian variables.

Motivation & Objective

• To provide non-asymptotic tail estimates for sums of functions sampled via a reversible Markov chain or random walk on a graph.
• To improve upon existing Bernstein- and Bennett-type inequalities for dependent samples from random walks.
• To quantify the number of samples required to approximate expectations on graphs, especially expanders, with explicit dependence on spectral gap and function variance.
• To offer a more elementary proof framework than prior methods, avoiding advanced tools like Kato's perturbation theory.

Proposed method

• Derives tail bounds using spectral properties of the transition matrix P, particularly the second-largest eigenvalue α and the second-largest absolute eigenvalue β.
• Introduces a novel variational approach to bound the norm of the operator e^{(1/2)rf} P e^{(1/2)rf} in the ℓ²(1/s)-norm.
• Uses coordinate-wise operations and bilinear form decomposition to analyze the operator norm, separating positive and negative eigenvalue contributions.
• Applies Cauchy-Schwarz and eigenvalue decomposition to bound terms involving pσ, pτ, and pσ,τ, especially in the presence of negative eigenvalues.
• Optimizes over a parameter r to minimize the resulting exponential bound, leading to sharp concentration inequalities.
• Derives two main inequalities: one in terms of β² (for general reversible chains) and one in terms of α (for better convergence in non-bipartite cases), with explicit dependence on function variance V and initial distribution q.

Experimental results

Research questions

• RQ1 How can tail bounds for sums of functions along a random walk be improved beyond existing Bernstein-type inequalities in the finite-sample regime?
• RQ2 What role does the spectral gap (via α or β) play in determining the concentration of empirical averages from dependent samples?
• RQ3 Can sharper tail estimates be derived using elementary methods that avoid advanced spectral theory?
• RQ4 How do the variance of the function f and the initial distribution q affect the sample complexity of expectation approximation on expanders?
• RQ5 What is the precise trade-off between function range, variance, and spectral gap in controlling large deviations for random walk sampling?

Key findings

• The paper establishes a new Bennett-type inequality for random walk sums, with a logarithmic term in the exponent that captures deviation behavior beyond the variance.
• It proves a sharper Bernstein-type inequality than previous works, with explicit dependence on the spectral gap (α or β) and function variance V.
• The bound for the probability P(1/n Sn > γ) is expressed as ‖q/√s‖₂ times an exponential term involving r, V, and a correction term Δ(α,r) or Δ(β²,r), with optimization over r.
• For α → 0 (nearly independent sampling), the bound reduces to the standard independent-variable Bennett/ Bernstein form, validating consistency.
• The method yields a new inequality for subgaussian variables sampled via random walks, extending the framework to heavy-tailed and light-tailed settings.
• The proof is more elementary than prior approaches, avoiding Kato’s perturbation theory, and relies on direct operator norm estimation and eigenvalue decomposition.