# Markov Chain Lifting and Distributed ADMM

## TL;DR

This paper reveals that a lifting operation can accelerate convergence in Markov chains and shows that a similar lifting analogy explains faster convergence of distributed ADMM for quadratic problems, offering new insights.

## Contribution

It introduces a novel analogy between Markov chain lifting and distributed ADMM, explaining convergence speedups in quadratic optimization.

## Key findings

- Lifting reduces Markov chain convergence time.
- Distributed ADMM can be viewed as a lifting of Gradient Descent.
- Lifting potentially always improves convergence speed.

## Abstract

The time to converge to the steady state of a finite Markov chain can be greatly reduced by a lifting operation, which creates a new Markov chain on an expanded state space. For a class of quadratic objectives, we show an analogous behavior where a distributed ADMM algorithm can be seen as a lifting of Gradient Descent algorithm. This provides a deep insight for its faster convergence rate under optimal parameter tuning. We conjecture that this gain is always present, as opposed to the lifting of a Markov chain which sometimes only provides a marginal speedup.