Locally Stationary Distributions: A Framework for Analyzing Slow-Mixing Markov Chains (2405.20849v3)

Abstract: Many natural Markov chains fail to mix to their stationary distribution in polynomially many steps. Often, this slow mixing is inevitable since it is computationally intractable to sample from their stationary measure. Nevertheless, Markov chains can be shown to always converge quickly to measures that are locally stationary, i.e., measures that don't change over a small number of steps. These locally stationary measures are analogous to local minima in continuous optimization, while stationary measures correspond to global minima. While locally stationary measures can be statistically far from stationary measures, do they enjoy provable theoretical guarantees that have algorithmic implications? We study this question in this work and demonstrate three algorithmic applications of locally stationary measures: 1. We show that Glauber dynamics on the hardcore model can be used to find independent sets of size $\Omega\left(\frac{\log d}{d} \cdot n\right)$ in triangle-free graphs of degree at most $d$. 2. Let $W$ be a symmetric real matrix with bounded spectral diameter and $v$ be a unit vector. Given the matrix $M = \lambda vv^\top + W$ with a planted rank-one spike along vector $v$, for sufficiently large constant $\lambda$, Glauber dynamics on the Ising model defined by $M$ samples vectors $x \in {\pm 1}^n$ that have constant correlation with the vector $v$. 3. Let $M = A_{\mathbf{G}} - \frac{d}{n}\mathbf{1}\mathbf{1}^\top$ be a centered version of the adjacency matrix where the graph $\mathbf{G}$ is drawn from a sparse 2-community stochastic block model. We show that for sufficiently large constant signal-to-noise ratio, Glauber dynamics on the Ising model defined by $M$ samples vectors $x \in {\pm 1}^n$ that have constant correlation with the hidden community vector $\mathbf{\sigma}$.

• The paper shows that locally stationary distributions enable tractable analysis of slow-mixing Markov chains despite NP-hard convergence challenges.
• It applies the framework using Glauber dynamics for optimization, signal recovery, and community detection with reliable performance guarantees.
• The study provides a novel theoretical foundation that harnesses local stability, offering practical alternatives to full stationary convergence in complex systems.

Analyzing Slow-Mixing Markov Chains Using Locally Stationary Distributions

The paper presented introduces a novel framework for analyzing slow-mixing Markov chains through the concept of "locally stationary distributions." This approach is particularly insightful given the existing challenges associated with Markov chains that do not efficiently converge to their stationary distributions in polynomial time. The research provides both theoretical underpinnings and practical applications of this framework, addressing an important gap in the understanding of Markov chain dynamics.

Summary and Key Contributions

The authors critically assess the situation where many natural Markov chains, employed across various computational tasks, often exhibit slow mixing characteristics. In particular, it is highlighted that some Markov chains cannot rapidly converge to their stationary distributions due to inherent computational complexities such as $\NP$-hardness. The paper pivots on the notion that, despite these slow-mixing properties, Markov chains can exhibit quick convergence to "locally stationary distributions." These are distributions that remain unchanged significantly over short evolutionary windows of the chain.

The key theoretical contribution of this work is showing that these locally stationary distributions can offer significant functional insights and guarantees, comparable at times to results that would be obtained from the true stationary distribution. The authors build strong theoretical foundations for understanding this behavior and translate these into algorithmic implications across three specific applications:

1. Using Glauber dynamics within the hardcore model to identify independent sets of expected size $\Omega\left(\frac{\log d}{d} \cdot n\right)$ in triangle-free graphs. This provides remarkable utility in optimization tasks where configurations must avoid certain conflicts, such as channel assignments in networks.
2. Demonstrating that Glauber dynamics on the Ising model for a symmetric matrix $M$—constructed with a planted rank-one spike—can extract vectors $x$ that maintain a constant correlation with the original vector $v$. This has profound implications for effective signal recovery in high-dimensional statistical inference tasks.
3. In community detection within a sparse 2-community stochastic block model, Glauber dynamics can sample vectors that maintain constant correlation with the hidden community seeds, achieving weak recovery above a critical threshold known as the Kesten-Stigum threshold.

Theoretical Implications

The introduction of locally stationary distributions provides a conceptual breakthrough that draws parallels with local optima in non-convex optimization problems. Similar to how local optima are utilized in optimization despite the hardness of finding global optima, locally stationary distributions offer a feasible alternative in situations where sampling from the true stationary distribution is computationally intractable.

The theoretical results backed by mathematical rigor imply that these locally stationary states can provide viable solutions to various algorithmic problems without necessitating the complete and often infeasible convergence to global stationary distributions. This is akin to harnessing local moments of stability across a Markov chain's evolution as a pragmatic computational strategy.

Future Directions

The fruitful implications of this framework suggest several future directions. Further research could delve into the generalization of these results to a broader class of Markov chains and explore how these insights could be applied to complex, real-world networks and distributed systems. Additionally, the deep connection between local stability and potential global recovery implies that future algorithms could be designed to deliberately exploit such stability in practical applications, such as in machine learning models and information-theoretic tasks.

Moreover, extending the concepts to other models of statistical physics or incorporating them into hybrid algorithms that combine traditional sampling methods with insights from locally stationary distributions could push the boundaries of current computational capabilities.

In conclusion, this work significantly enhances the analytical toolkit available for handling slow-mixing Markov chains, offering both a robust theoretical framework and versatile practical implications that stand to benefit various fields from algorithmic graph theory to statistical inference and machine learning.