Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Decentralized Hidden Markov Modeling with Equal Exit Probabilities

Dongyan Sui, Haitian Zheng, Siyang Leng, Stefan Vlaski

TL;DR

This work tackles distributed tracking of a dynamically evolving environment where the true state follows a Markov chain with equal exit probabilities. It introduces the Diffusion $\\alpha$-HMM, a nonlinear log-belief-ratio dynamic with a single tunable parameter $\\alpha$, and shows its connection to Adaptive Social Learning via a linearized form. The authors establish convergence guarantees for a deterministic reference system, proving the existence and uniqueness of a fixed point under steady-state conditions (Theorem 1) and showing contraction of the nonlinear update, which leads to an explicit upper bound on the asymptotic error probability (Theorem 2). Numerical experiments comparing Diffusion $\\alpha$-HMM, its linearized variant, and ASL demonstrate the benefits of nonlinearity for adaptability across dynamic environments, while also highlighting regimes where step-size tuning can mitigate noise effects.

Abstract

Social learning strategies enable agents to infer the underlying true state of nature in a distributed manner by receiving private environmental signals and exchanging beliefs with their neighbors. Previous studies have extensively focused on static environments, where the underlying true state remains unchanged over time. In this paper, we consider a dynamic setting where the true state evolves according to a Markov chain with equal exit probabilities. Based on this assumption, we present a social learning strategy for dynamic environments, termed Diffusion $α$-HMM. By leveraging a simplified parameterization, we derive a nonlinear dynamical system that governs the evolution of the log-belief ratio over time. This formulation further reveals the relationship between the linearized form of Diffusion $α$-HMM and Adaptive Social Learning, a well-established social learning strategy for dynamic environments. Furthermore, we analyze the convergence and fixed-point properties of a reference system, providing theoretical guarantees on the learning performance of the proposed algorithm in dynamic settings. Numerical experiments compare various distributed social learning strategies across different dynamic environments, demonstrating the impact of nonlinearity and parameterization on learning performance in a range of dynamic scenarios.

Decentralized Hidden Markov Modeling with Equal Exit Probabilities

TL;DR

This work tackles distributed tracking of a dynamically evolving environment where the true state follows a Markov chain with equal exit probabilities. It introduces the Diffusion -HMM, a nonlinear log-belief-ratio dynamic with a single tunable parameter , and shows its connection to Adaptive Social Learning via a linearized form. The authors establish convergence guarantees for a deterministic reference system, proving the existence and uniqueness of a fixed point under steady-state conditions (Theorem 1) and showing contraction of the nonlinear update, which leads to an explicit upper bound on the asymptotic error probability (Theorem 2). Numerical experiments comparing Diffusion -HMM, its linearized variant, and ASL demonstrate the benefits of nonlinearity for adaptability across dynamic environments, while also highlighting regimes where step-size tuning can mitigate noise effects.

Abstract

Social learning strategies enable agents to infer the underlying true state of nature in a distributed manner by receiving private environmental signals and exchanging beliefs with their neighbors. Previous studies have extensively focused on static environments, where the underlying true state remains unchanged over time. In this paper, we consider a dynamic setting where the true state evolves according to a Markov chain with equal exit probabilities. Based on this assumption, we present a social learning strategy for dynamic environments, termed Diffusion -HMM. By leveraging a simplified parameterization, we derive a nonlinear dynamical system that governs the evolution of the log-belief ratio over time. This formulation further reveals the relationship between the linearized form of Diffusion -HMM and Adaptive Social Learning, a well-established social learning strategy for dynamic environments. Furthermore, we analyze the convergence and fixed-point properties of a reference system, providing theoretical guarantees on the learning performance of the proposed algorithm in dynamic settings. Numerical experiments compare various distributed social learning strategies across different dynamic environments, demonstrating the impact of nonlinearity and parameterization on learning performance in a range of dynamic scenarios.

Paper Structure

This paper contains 8 sections, 4 theorems, 117 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction and Related Work
  2. Diffusion $\alpha$-HMM
  3. Dynamics of Log-belief Ratio in Steady State
  4. Convergence Analysis
  5. Numerical Experiments
  6. Proof of Lemma 1
  7. Proof of Theorem 1
  8. Proof of Theorem 2

Key Result

Lemma 1

The fixed points of the dynamical reference system reference_dynamics exist, and satisfy $\hat{x}^\infty_{k}(\theta_m)<-\sum_{\ell\in\mathcal{N}_k}a_{\ell k}d_{\ell}(\theta_m)$.

Figures (3)

  • Figure 1: Comparison of the learning accuracy of agent 5, which has no distinguishing capability, under different network topologies and algorithms in Scenario 1. The accuracy is evaluated as a function of $\alpha$ for different fixed values of $\sigma$. LD $\alpha$-HMM refers to the linearized Diffusion $\alpha$-HMM.
  • Figure 2: Comparison of the learning accuracy of agent 5, which has no distinguishing capability, under different transition intervals $T_0$ and algorithms in Scenario 2. The accuracy is evaluated as a function of $\alpha$ for different fixed values of $\sigma$. LD $\alpha$-HMM refers to the linearized Diffusion $\alpha$-HMM.
  • Figure 3: Comparison of agent 5's belief evolution on the true state across the three algorithms under two different settings.

Theorems & Definitions (9)

  • Remark 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 1: Contractions
  • Lemma 2: Contraction mappinghunter2001applied