Approximating distributions in stochastic learning.

Area of Science:

• Computational Neuroscience
• Machine Learning
• Statistical Physics

Background:

• Many on-line machine learning algorithms and biological spike-timing-dependent plasticity (STDP) rules involve stochastic dynamics modeled by Markov processes.
• Describing these systems requires analyzing probability densities, often using Chapman-Kolmogorov or master equations, which are analytically intractable.
• The nonlinear Fokker-Planck equation (FPE) is a common approximation but is limited and potentially flawed for jump processes.

Purpose of the Study:

• To develop a robust perturbation expansion method for approximating probability densities and their moments in complex stochastic systems.
• To provide an accessible alternative to the FPE, particularly for systems exhibiting jump processes.
• To validate the new method's accuracy against Monte Carlo simulations in biologically relevant and machine learning contexts.

Main Methods:

• Developed a perturbation expansion based on the system size expansion from statistical physics.
• Applied the method to calculate equilibrium distributions for two STDP learning rules and a nonlinear machine learning problem.
• Compared the perturbation series results with Monte Carlo simulations.

Main Results:

• The perturbation expansion provides accurate approximations for probability densities and moments.
• The method demonstrates good agreement with Monte Carlo simulations, even in regimes where the FPE breaks down.
• Successfully applied to biologically observed STDP learning rules and a nonlinear machine learning problem.

Conclusions:

• The developed perturbation expansion offers a reliable and accessible approach for analyzing complex stochastic systems.
• This method overcomes limitations of the FPE, especially for systems with jump processes.
• The approach has broad applicability across neuroscience, machine learning, and statistical physics.