# Distributed Fractional Bayesian Learning for Adaptive Optimization

arXiv: https://arxiv.org/abs/2404.11354 · HTML: https://mlsanity.com/paper/2404.11354/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 59 references that are in the map.
- First posted on arXiv 17 Apr 2024.

## Builds on (most cited first, 21 of 21)

- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Fast Convergence Rates for Distributed Non-Bayesian Learning](https://mlsanity.com/paper/1508.05161/) (2015)
- [Social Learning and Distributed Hypothesis Testing](https://mlsanity.com/paper/1410.4307/) (2014)
- [LUCIDGames: Online Unscented Inverse Dynamic Games for Adaptive Trajectory Prediction and Planning](https://mlsanity.com/paper/2011.08152/) (2020)
- [BlueFog: Make Decentralized Algorithms Practical for Optimization and Deep Learning](https://mlsanity.com/paper/2111.04287/) (2021)
- [Exploiting Problem Structure in Optimization under Uncertainty via Online Convex Optimization](https://mlsanity.com/paper/1709.02490/) (2017)
- [Adaptive Sequential Stochastic Optimization](https://mlsanity.com/paper/1610.01970/) (2016)
- [Distributed Optimization for Smart Cyber-Physical Networks](https://mlsanity.com/paper/1906.10760/) (2019)
- [Distributed Personalized Gradient Tracking with Convex Parametric Models](https://mlsanity.com/paper/2008.04363/) (2020)
- [Distributed Coupled Multi-Agent Stochastic Optimization](https://mlsanity.com/paper/1712.08817/) (2017)
- [On the solution of stochastic optimization and variational problems in imperfect information regimes](https://mlsanity.com/paper/1402.1457/) (2014)
- [On the resolution of misspecified convex optimization and monotone variational inequality problems](https://mlsanity.com/paper/1408.5532/) (2014)
- [On the analysis of inexact augmented Lagrangian schemes for misspecified conic convex programs](https://mlsanity.com/paper/1608.01879/) (2016)
- [Distributed Stochastic Optimization under Imperfect Information](https://mlsanity.com/paper/1509.03925/) (2015)
- [A Study on the Power Parameter in Power Prior Bayesian Analysis](https://mlsanity.com/paper/2204.06165/) (2022)
- [Stochastic Adaptive Linear Quadratic Differential Games](https://mlsanity.com/paper/2204.08869/) (2022)
- [Intention Communication and Hypothesis Likelihood in Game-Theoretic Motion Planning](https://mlsanity.com/paper/2209.12968/) (2022)
- [A Tutorial on Distributed (Non-Bayesian) Learning: Problem, Algorithms and Results](https://mlsanity.com/paper/1609.07537/) (2016)
- [Multi-agent Bayesian Learning with Adaptive Strategies: Convergence and Stability](https://mlsanity.com/paper/2010.09128/) (2020)
- [Distributed Non-Bayesian Learning for Games with Incomplete Information](https://mlsanity.com/paper/2303.07212/) (2023)
- [A cutting-surface consensus approach for distributed robust optimization of multi-agent systems](https://mlsanity.com/paper/2309.03519/) (2023)

## Abstract (arXiv)

This paper considers a distributed adaptive optimization problem, where all agents only have access to their local cost functions with a common unknown parameter, whereas they mean to collaboratively estimate the true parameter and find the optimal solution over a connected network. A general mathematical framework for such a problem has not been studied yet. We aim to provide valuable insights for addressing parameter uncertainty in distributed optimization problems and simultaneously find the optimal solution. Thus, we propose a novel distributed scheme, which utilizes distributed fractional Bayesian learning through weighted averaging on the log-beliefs to update the beliefs of unknown parameter, and distributed gradient descent for renewing the estimation of the optimal solution. Then under suitable assumptions, we prove that all agents' beliefs and decision variables converge almost surely to the true parameter and the optimal solution under the true parameter, respectively. We further establish a sublinear convergence rate for the belief sequence. Finally, numerical experiments are implemented to corroborate the theoretical analysis.