Adaptive power method for estimating large deviations in Markov chains

Area of Science:

• Statistical Physics
• Nonequilibrium Systems
• Stochastic Processes

Background:

• Large deviation functions characterize fluctuations in physical systems, particularly nonequilibrium processes.
• Stochastic algorithms, like the power method, are adapted for analyzing Markov processes and diffusions.
• Previous work introduced an adaptive power method for risk-sensitive control and continuous-time diffusions.

Purpose of the Study:

• To conduct an in-depth performance analysis of an adaptive power method for learning large deviation functions.
• To investigate the algorithm's convergence near dynamical phase transitions.
• To evaluate the impact of learning rate and transfer learning on convergence speed.

Main Methods:

• Application of a stochastic algorithm based on the power method.
• Adaptive learning of large deviation functions for Markov processes.
• Analysis of convergence rates and the effect of transfer learning.
• Testing the algorithm on a random walk's mean degree on an Erdős-Rényi random graph.

Main Results:

• The adaptive power method demonstrates efficiency in analyzing dynamical phase transitions.
• Convergence speed is studied as a function of the learning rate.
• Transfer learning was incorporated and its effects analyzed.
• The method proved effective for distinguishing between high-degree and low-degree trajectories in the test case.

Conclusions:

• The adaptive power method is a performant and complex-efficient tool for computing large deviation functions.
• The algorithm excels in analyzing systems near dynamical phase transitions.
• It offers advantages over existing algorithms for studying fluctuations in Markov processes.