[Paper Review] Modified log-Sobolev inequalities for strong-Rayleigh measures

Jonathan Hermon, Justin Salez|arXiv (Cornell University)|Feb 7, 2019

Markov Chains and Monte Carlo Methods4 citations

TL;DR

This paper establishes universal modified log-Sobolev inequalities for reversible Markov chains on the boolean lattice \{0,1\}^n under the stochastic covering property (SCP), a negative dependence condition strictly weaker than strong Rayleigh. The key contribution is a mixing time bound of at most $kn\log\log\frac{1}{\pi(x)}$ for $k$-homogeneous measures, with applications to concentration and sampling via MCMC.

ABSTRACT

We establish universal modified log-Sobolev inequalities for reversible Markov chains on the boolean lattice $\{0,1\}^n$, under the only assumption that the invariant law $π$ satisfies a form of negative dependence known as the stochastic covering property. This condition is strictly weaker than the strong Rayleigh property, and is satisfied in particular by all determinantal measures, as well as any product measure over the set of bases of a balanced matroid. In the special case where $π$ is $k-$homogeneous, our results imply the celebrated concentration inequality for Lipschitz functions due to Pemantle & Peres (2014). As another application, we deduce that the natural Monte-Carlo Markov Chain used to sample from $π$ has mixing time at most $kn\log\log\frac{1}{π(x)}$ when initialized in state $x$. To the best of our knowledge, this is the first work relating negative dependence and modified log-Sobolev inequalities.

Motivation & Objective

• To develop universal functional inequalities for Markov chains on the boolean lattice under negative dependence.
• To bridge the gap between negative dependence and modified log-Sobolev inequalities, a previously unexplored connection.
• To derive explicit mixing time bounds for MCMC samplers under the $k$-homogeneous and stochastic covering property conditions.
• To recover and generalize the Pemantle-Peres concentration inequality for Lipschitz functions via the modified log-Sobolev framework.
• To provide a constructive method for building generators with uniform spectral and functional bounds using coordinate averaging.

Proposed method

• Introduce the stochastic covering property (SCP) as a negative dependence condition that is strictly weaker than strong Rayleigh but includes determinantal and balanced matroid measures.
• Construct a flip-swap generator $Q^{(\ell)}$ for each coordinate $\ell$ using a partitioning scheme based on the value of $x_\ell$, ensuring $\lambda(Q^{(\ell)}), \alpha(Q^{(\ell)}) \geq 1$ and $\Delta(Q^{(\ell)}) \leq n$.
• Average the generators over all coordinates to form $Q^\star = \frac{1}{n}\sum_{\ell=1}^n Q^{(\ell)}$, preserving spectral and functional bounds.
• Use the Herbst argument to derive sub-Gaussian concentration from the modified log-Sobolev inequality, linking $\alpha(Q)$ to tail bounds via the quadratic variation $v(f)$.
• Apply the functional inequality to derive a mixing time bound $t_{\textsc{mix}}(x;\varepsilon) \leq \frac{1}{\alpha(Q)}\left(\log\log\frac{1}{\pi(x)} + \log\frac{1}{2\varepsilon^2}\right)$.
• Establish that for $k$-homogeneous $\pi$, the generator $Q^\star$ satisfies $-Q^\star(x,x) \leq 2k$, enabling the final mixing time bound $kn\log\log\frac{1}{\pi(x)}$.

Experimental results

Research questions

• RQ1 Can modified log-Sobolev inequalities be established for reversible Markov chains on the boolean lattice under a negative dependence condition weaker than strong Rayleigh?
• RQ2 Does the stochastic covering property (SCP) suffice to derive universal functional inequalities and mixing time bounds for MCMC samplers?
• RQ3 Can the Pemantle-Peres concentration inequality for Lipschitz functions be recovered and generalized via modified log-Sobolev inequalities?
• RQ4 What is the optimal mixing time bound for MCMC samplers targeting $k$-homogeneous measures under the SCP?
• RQ5 How can coordinate-wise generators be averaged to yield a global generator with uniform spectral and functional bounds?

Key findings

• The paper establishes a modified log-Sobolev inequality for any reversible Markov chain on $\{0,1\}^n$ whose invariant measure $\pi$ satisfies the stochastic covering property.
• For $k$-homogeneous measures, the mixing time of the constructed MCMC sampler is at most $kn\log\log\frac{1}{\pi(x)}$ when initialized at state $x$, improving upon prior bounds.
• The Herbst argument is applied to derive a sub-Gaussian concentration inequality with a variance proxy depending on the maximal one-sided quadratic variation $v(f)$.
• The stochastic covering property is shown to be strictly weaker than the strong Rayleigh property, yet sufficient to imply the desired functional inequalities.
• The construction of coordinate-specific generators $Q^{(\ell)}$ and their averaging to $Q^\star$ ensures uniform lower bounds on the modified log-Sobolev constant $\alpha(Q^\star) \geq 1$.
• The framework recovers the Pemantle-Peres concentration inequality as a special case when $\pi$ is $k$-homogeneous, demonstrating the generality and strength of the approach.