کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
404337 | 677413 | 2012 | 10 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: Approximating distributions in stochastic learning Approximating distributions in stochastic learning](/preview/png/404337.png)
On-line machine learning algorithms, many biological spike-timing-dependent plasticity (STDP) learning rules, and stochastic neural dynamics evolve by Markov processes. A complete description of such systems gives the probability densities for the variables. The evolution and equilibrium state of these densities are given by a Chapman–Kolmogorov equation in discrete time, or a master equation in continuous time. These formulations are analytically intractable for most cases of interest, and to make progress a nonlinear Fokker–Planck equation (FPE) is often used in their place. The FPE is limited, and some argue that its application to describe jump processes (such as in these problems) is fundamentally flawed.We develop a well-grounded perturbation expansion that provides approximations for both the density and its moments. The approach is based on the system size expansion in statistical physics (which does not give approximations for the density), but our simple development makes the methods accessible and invites application to diverse problems. We apply the method to calculate the equilibrium distributions for two biologically-observed STDP learning rules and for a simple nonlinear machine-learning problem. In all three examples, we show that our perturbation series provides good agreement with Monte-Carlo simulations in regimes where the FPE breaks down.
► Biological STDP and machine online learning both governed by Markov processes.
► Common to adopt Fokker–Planck equation (FPE), not rigorously justified.
► We give well-grounded perturbation expansion for density and for moments.
► Biological and machine learning examples show improvement over FPE.
► Mathematica software available to research community.
Journal: Neural Networks - Volume 32, August 2012, Pages 219–228